d. bauer- theory of intense laser-matter interaction

Lecture notes on the

Theory of intense laser-matter interaction

D. Bauer

Max-Planck-Institut für Kernphysik, Heidelberg, [email protected]

June 22, 2006

Contents

1 Introduction 5

2 Classical motion in electromagnetic fields 92.1 The nonrelativistic ponderomotive force . . . . . . . . . . . . . . . . . . . . 9

2.1.1 The ponderomotive force of a standing wave . . . . . . . . . . . . . 112.2 Relativistic dynamics in an electromagnetic wave . . . . . . . . . . . . . . . 14

2.2.1 The oscillation center frame . . . . . . . . . . . . . . . . . . . . . . . . 172.2.2 The adiabatically ramped pulse in the lab frame . . . . . . . . . . . 19

2.3 The relativistic ponderomotive force . . . . . . . . . . . . . . . . . . . . . . . 212.3.1 Example: sin2-pulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Atoms in external fields 273.1 Atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 32

3.2.1 Dyson series, S-matrix, Green’s functions, propagators, and re-solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Important models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3.1 Discrete level coupled to the continuum . . . . . . . . . . . . . . . . 423.3.2 Resonantly coupled discrete levels: Rabi-oscillations . . . . . . . . 47

3.4 Atoms in strong, static electric fields . . . . . . . . . . . . . . . . . . . . . . . 503.4.1 Tunneling ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.5 Atoms in strong laser fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.5.1 Floquet formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.5.2 Non-Hermitian Floquet theory . . . . . . . . . . . . . . . . . . . . . 673.5.3 High-frequency Floquet theory and stabilization . . . . . . . . . . 71

3.6 Strong field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.6.1 Circular polarization and long pulses . . . . . . . . . . . . . . . . . . 813.6.2 Channel closing in above-threshold ionization . . . . . . . . . . . . 823.6.3 Linear polarization and long pulses . . . . . . . . . . . . . . . . . . . 843.6.4 Few-cycle above-threshold ionization . . . . . . . . . . . . . . . . . . 863.6.5 “Simple man’s theory” . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.6.6 High harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . 95

3

4 CONTENTS

Chapter 1

Introduction

In this lecture we shall try to answer the questions

— “What happens if we put electrons, atoms, molecules, clusters, solids, plasmas etc.in an intense laser pulse?”

— “What is the new physics there?”

— “What are the new theoretical methods to deal with the problem?”

— “What is it good for?”

Theoretical research in this area has been strongly motivated by the rapid progress inlaser technology over the last twenty years, and there are good reasons to believe thatthis progress continues. The main goals are to achieve

1. higher laser pulse peak intensities,

2. shorter pulses,

3. shorter wavelengths.

The demand for higher and higher laser pulse intensities is driven by, e.g., inertialconfinement fusion (i.e., nuclear fusion using lasers), “table top” particle acceleration,or particle physics (e.g., particle creation in vacuum using lasers). Shorter pulses areneeded to study fast atomic processes directly in the time domain. Just as femtosecond(1 fs = 10−15 = 0.000000000000001 s) laser technology revolutionized chemistry, thehope is that attosecond (1as = 10−18 s) laser pulses give a new twist on good old atomicphysics. First, successful “proof of principle”-like experiments have been already car-ried out. The shortness of these kind of laser pulses is impressively illustrated by thecomparison

1 fs

1 s' 8min

age of the universe. (1.1)

5

6 CHAPTER 1. INTRODUCTION

Table 1.1: Comparison between a strong laser and sunlight.

field strength [V/m] intensity [W/cm2] excursion [Å] ponderom.potential

[eV]

(at 800 nm)

strong laser 5.14 · 1011 3.51 · 1016 163 2169.2

sun 1000 0.13 3.3 · 10−7 8 · 10−15

Finally, shorter wavelength lasers are desirable to complement (or to offer a cheaperalternative to) synchrotron radiation sources with a vast amount of applications in ma-terial and biological sciences. Moreover, shorter wavelengths λ and shorter pulse du-rations are related since a lower limit for the pulse duration is the laser period 2π/ωwhereω = 2πc/λ is the laser frequency.

What do we mean by strong or intense when we talk about laser pulses? Whether alaser pulse has to be considered strong or not depends on the force it exerts as comparedto other, competing forces. These other forces are typically the binding forces thatprevents the target from falling apart (even without laser). A good example is the forcethat is “seen” by an electron on its first Bohr orbit in the hydrogen atom. This force is

F =−eEH (1.2)

where |EH| = 5.14 · 1011 V/m is the electric field due to the attraction of electron andproton. This electric field can be directly compared with the electric field of a laser pulse

E (r , t ) = Eei(ωt−k·r ), (1.3)

and the latter is at latest to be considered strong if it equals the binding force. The laserintensity is defined as the cycle-averaged modulus of the Poynting vector,

I = c2ε0|E (t )×B(t )|= 1

2cε0E E ∗. (1.4)

Here we use SI units and the relation between the electric field and the magnetic field Bfor an electromagnetic wave in vacuum. The unit of the intensity I is that of an energyflux, namely W/cm2, which is energy passing through an area per time. Plugging inEH yields I = 3.51 · 1016 Wcm−2. This sounds a lot, especially if compared to sunlighton earth (cf. Table 1.1). However, for current laser technology it is rather peanuts,for 1017 Wcm−2 are routinely achieved in many laboratories around the world while1022 Wcm−2 seems to be the maximum achieved so far. The other entries in Table 1.1will be referred to later on.

The shortest pulses generated so far consist of only a few cycles in the “full width halfmaximum” (FWHM) of the pulse at λ = 800 nm. The laser period for this Titanium-Sapphire laser wavelength is T ' 2.7 fs so that the pulse duration is about 5 fs for these

7

shortest laser pulses. During such a short time light travels only a distance of ' 1.5µmin vacuum so that these pulses are not laser beams but rather laser light bullets of afew wavelengths width. The pulse duration determines the time scale which can beresolved directly in the time domain. Since atomic processes occur on the attosecondtime scale (except phenomena involving mainly Rydberg states), 5 fs pulses are still toolong do resolve them. However, in combination with shorter wavelength pulses of at-tosecond duration which can be generated by the 800 nm pulse itself via high harmonicgeneration, femtosecond laser pulses have been already used to study experimentallyattosecond atomic processes such as the Auger decay.

So far, few-cycle laser pulses were only generated at rather modest laser intensitiesbelow 1018 Wcm−2. The most intense pulses with an intensity of 1020 Wcm−2or higherare of picosecond (1ps= 10−12 s) duration or longer. It is instructive to calculate thepower and the energy of such a, e.g., 10ps pulse if it is focused down to, say, (10µm)2:

1020 Wcm−2 · (10µm)2 = 0.1PW (1.5)

where PW stands for petawatt (1015 W), and

0.1PW · 10ps= 1kJ. (1.6)

Currently, megajoule lasers are being constructed in several laboratories around theworld. The light pressures such pulses exert are enormous. From Eq. (1.7) below weestimate pressures of the order of 10 Gigabar. Since dimensionally, pressure is an energydensity, another way of estimating the pressure is 1kJ/[(10µm)2 ·c ·10 ps], which yieldsthe same answer. Such pressures even exceed those in the interior of stars and may offerthe opportunity to study the equation of state of various materials under extreme con-ditions. At this point we do not want to hide that such research is of interest to military,and, in fact, was previously carried out (with limited success) employing undergroundnuclear explosions. The newly constructed “national ignition facility” (NIF) in the USaims at those aspects of intense laser-matter interaction.

Coming back from “boom-boom”-physics to theory, experience shows that stan-dard, quantum mechanical, perturbative approaches start to fail already around1013 Wcm−2 when applied to the ionization of atomic hydrogen in a laser field, for in-stance. The new analytical techniques that have been developed (as well as those stilllacking) will be topics of this lecture.

Many things become more complicated in strong laser fields, however, some othersare getting simpler. There is, for instance, no need to quantize the electromagnetic field.Moreover, many of the new effects observed, such as the plateaus in photoelectron orharmonics spectra and nonsequential double ionization, although utterly unaccessiblein terms of standard perturbation theory, turned out to have rather simple, almost clas-sical explanations. These effects and their theoretical explanation will be also part ofthis lecture.

Finally, there is the relativistic domain which separates the strong laser fields fromthe ultra-strong ones. In the case of a free electron in a laser pulse there is no binding

8 CHAPTER 1. INTRODUCTION

force to compete with. However, there are strong field effects as well. Already at moder-ate laser intensities the so-called ponderomotive force is appreciable, depending on thepulse shape. Ponderomotive effects will be discussed in quite some detail and reappearfrequently during the course of this lecture; in brief: the ponderomotive force in generalpushes charged particles out of regions of high intensity (i.e., the laser focus) and exertspressure on the target. In fact, the light pressure

pl = (1+R)I

c(1.7)

(where R is the reflectivity and c is the velocity of light in vacuum) can be linked tothe ponderomotive force. The ponderomotive force is related to the breakdown of theso-called dipole approximation

E (r , t ) = Ee−i(ωt−k·r ) ' Ee−iωt = E (t ) (1.8)

where the spatial dependence of the field is neglected because the wavelength is so muchlarger than the excursion r (t ) of, say, an electron in the field. The classical motion ofthe electron is governed by the Lorentz force (SI units)

p =−e[E (r , t )+ v ×B(r , t )] (1.9)

(where p, v are the electron momentum and velocity, respectively). Making the dipoleapproximation implies the omittance of the magnetic field so that there is no v × B-force. Hence, the electron is assumed to just oscillate in laser polarization direction (orin the plane of polarization directions if elliptically polarized light is used). However,the velocity of this oscillatory motion increases with the laser intensity and, since thev × B-force is a v/c -effect, from a certain laser intensity on there is no excuse anymorefor neglecting it. This threshold intensity depends on the laser wavelength, and for thecommonly used 800 nm it turns out to be of the order of 1018 Wcm−2. The v ×B-forceis a v/c effect and as such not yet relativistic. Truly relativistic effects are of order (v/c)2or higher and will be discussed in the last part of this lecture.

Chapter 2

Classical motion in electromagneticfields

In this Chapter we shall derive the non-relativistic ponderomotive force and the rela-tivistic equations of motion for a charged particle in a laser pulse of arbitrary intensity.For the case of a travelling wave we also derive the relativistic ponderomotive force.

2.1 The nonrelativistic ponderomotive forceThe Lorentz-force on a particle of charge q is given by (SI units)

F = q[E (r , t )+ v ×B(r , t )]. (2.1)

This force is also relativistically correct. In our case we have in mind E and B makingup the electromagnetic field of a laser pulse, that is,

E (r , t ) = E (r , t )eiωt (2.2)

and, because of∇×E =−∂t B,

B(r , t ) =i

ω∇×E (r , t ), (2.3)

but the derivation also holds for longitudinal waves as they occur in plasmas. Thederivation of the ponderomotive force relies on the possibility to separate the relevanttime scales: the “fast” motion on the time scale of the laser period 2π/ω and a “secular”one due to the ponderomotive force. Hence we assume that E (r , t ), apart from con-taining e−ik·r , describes the laser field envelope, having only a “slow” time dependenceso that ∂t B = iωB to high accuracy.

In lowest order the particle just oscillates around its current position r0 due to theelectric field (the possible slow time-dependence in E is suppressed for notational con-venience):

m r1 = qE (r0)eiωt . (2.4)

9

10 CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS

The corresponding velocity and position are

r1 =−iq

mωE (r0)e

iωt , r1 =−q

mω2E (r0)e

iωt . (2.5)

In the next higher order one has

m r2 = q[(r1 ·∇)E (r0, t )+ r1×B(r0, t )] (2.6)

where the electric field has been expanded around r0. So far we used complex fields andimplicitly understood that the real part has to be taken. Now the calculation becomesnonlinear and this trick is not applicable anymore. Hence we write

m r2 = q− q

4mω2(Ee++ E ∗e−) ·∇(Ee++ E ∗e−) (2.7)

+q

4mω2(−iEe++ iE ∗e−)×∇× (iEe+− iE ∗e−)

= − q2

4mω2[E ·∇E ∗+ E ×∇× E ∗+ c.c.+Ω2ω] (2.8)

with e± = e±iωt and Ω2ω collecting all terms ∼ e±i2ωt which disappear upon averagingover a laser period,

m r2 =−q2

4mω2[E ·∇E ∗+ E ×∇× E ∗+ c.c.]. (2.9)

Using the identity

C ×∇×D +D ×∇×C +C ·∇D +D ·∇C =∇C ·D (2.10)

we finally obtain the nonrelativistic ponderomotive force

Fp = m r2 =−q2

4mω2∇|E (r , t )|2. (2.11)

In E (r , t ) the position r now refers to the so-called oscillation center. We remind that thetime dependence of the envelope must be “slow” as compared to the laser period becauseotherwise the separation of time-scales used in the derivation makes no sense. Theponderomotive force (2.11) can obviously be derived from the ponderomotive potential

Φp(r , t ) =q2

4mω2|E (r , t )|2 (2.12)

which is proportional to the laser intensity and independent of the sign of the parti-cle’s charge: it is always repulsive. Hence all charged particles are expelled from regions

2.1. THE NONRELATIVISTIC PONDEROMOTIVE FORCE 11

of high laser intensity (e.g., the laser focus). However, owing to the mass m in thedenominator the immediate effect on electrons is much larger than on ions. The pon-deromotive force is inverse proportional to the square of the laser frequency, meaningthat its significance increases with increasing laser wavelength.

In order for the derivation being valid, the particle must not oscillate “too much”around r0. More precisely, the conditions read

|r1| c , k · r1 1 (2.13)

for an electromagnetic wave and a longitudinal wave, respectively. The condition forthe electromagnetic wave is exactly the same indicating the breakdown of the dipoleapproximation (1.8). As a consequence, (2.12) is not applicable to relativistically intenselaser pulses. We shall derive in the subsequent Section the relativistic expression for theponderomotive force of a laser pulse.

The ponderomotive potential (at a certain position r ) equals the average quiver en-ergy of the charged particle in the laser field (at that position). In fact, taking r1 from(2.5) yields

Up =1

2m r1

2 =q2

4mω2|E |2. (2.14)

Is there a connection between this mere number Up and the potential Φp? Consider aspatially finite field structure which leads to Φp. Now, think of a particle injected intothis structure with some kinetic energy W0. While strolling through the field structurethe particle will encounter regions of varying Φp. In regions of high Φp it will oscillatewith larger amplitude (high Up) than in regions of low Φp (low Up). After averaging outthe fast time scales, the whole system is conservative so that W +Up =W0. This meansthat the local Up indeed serves as a potential and is just the Φp derived above.

2.1.1 The ponderomotive force of a standing waveLet us consider the standing wave

E (r , t ) = Eeiωt (e−ik·r + eik·r ) = 2Eeiωt cos(k · r ). (2.15)

Comparing this with (2.2) we identify

E (r , t ) = 2E cos(k · r ) (2.16)

and can immediately use expressions (2.11) and (2.12) to give

Φp(r , t ) =q2|E |2mω2

cos2(k · r ), Fp(r , t ) =q2|E |2mω2

k sin(2k · r ). (2.17)

The ponderomotive force will push the particle towards the nodes of the standing wavewhere the field vanishes. Depending on its initial velocity, the particles may be trapped


Figure 2.1: Trajectories of an electron in a standing laser wave (2.15) (polarized along ey , prop-agation directions along ex ) that starts at t = 0 from y = 0 and k x =π/40. The amplitudes wereeA/mc =−e E/mωc =−0.1 (upper left),−0.2 (upper right), and−0.5 (lower). With increasinglaser intensity, the electron is able to escape from the “valley” in which it is released. It can thenbe trapped temporarily in other valleys (upper right and lower plot), and the dynamics can beshown to become chaotic.

inside the “valleys” of the standing wave. If we assume that the propagation directionis ex and expand the force around the position of a node, we obtain a ponderomotiveforce of the form

F “valley′′

p '−Ω2x, Ω2 =2q2|E |2

mc2. (2.18)

The trapped particle will thus undergo harmonic secular motion with a frequency Ωthat is proportional to the electric field (which reminds of the Rabi-frequency in reso-nant laser-atom interaction, to be discussed later on). However, since the frequency in-creases with the field strength, the whole idea of separation of time scales breaks downat some point. In fact, the numerical solution of the (relativistic) equations of motionshows that for sufficiently high field strengths the particle may leave the valley in whichit was trapped. The motion can be shown to be chaotic then. The particle may be tem-porarily trapped in other valleys, leaving them again, and strolling around erratically(cf. Fig. 2.1).

It is clear from the ponderomotive force (2.17) that a standing wave pattern may beused as a grating with a separation of “slits” that is half a laser wavelength (Kapitza-Dirac

2.1. THE NONRELATIVISTIC PONDEROMOTIVE FORCE 13

effect). Particles injected perpendicular to the propagation direction of the two laserbeams will be deflected. If the de Broglie wavelength is chosen properly, a detector onthe opposite side will measure an interference pattern.

Some remarks

In the physics of laser-matter interaction, the importance of the ponderomotive poten-tial is hard to overestimate. In many effects it sets the relevant energy scale, as we shallsee later on. In laser-atom interaction, for instance, Φp(r , t ) determines not only thecut-offs of photoelectron and harmonics spectra but also equals the AC Stark-shift ofthe continuum. In laser-plasma physics, instabilities can be understood in terms of theponderomotive force.

As a historical side remark it should be mentioned that the ponderomotive force hasbeen re-invented several times in the literature although it is derived already in Landau& Lifshitz’s Mechanics volume.

Ponderomotive forces can be also derived for more complex systems than classical,point-like particles. An instructive example (and a good exercise) is to calculate theponderomotive force on two charged particles bound together by a linear force (i.e.,a harmonic oscillator). Such particles with internal degrees of freedom may displaycounter-intuitive behavior, for instance, acceleration into regions of high |E |2 ratherthan repulsion from them. Such kind of ponderomotive forces are used to trap neu-tral atoms in standing laser fields or other electromagnetic fields (“optical traps”) or for“optical tweezers”.

What we did in this Section in order to obtain a potential for the oscillation centeris done in the physics of magnetized plasmas for the so-called guiding center. In thesame way we got rid of the fast but uninteresting oscillations on the time scale of a laserperiod (and twice the laser period), the fast cyclotron motion around the field lines of astrong magnetic field is eliminated, and what remains is the equation of motion of the“guiding center”.

The method of separation of time scales was originally invented for the study ofcelestial mechanics. This is also where the term secular motion comes from. The effect ofJupiter on the earth’s motion around the sun, for instance, is negligible on the time scaleof a few years but may have to be taken into account on longer time scales. One practicaladvantage of the separation of time scales is for numerical simulations. Imagine youwant to simulate one second of the laser plasma dynamics using a fluid code. All effectsyou are interested in happen on the time scale of, say, microseconds or longer. However,there is this oscillatory motion in the laser field on a ten order of magnitude shorter timescale. Resolving this with the fluid code would lead to run-times of years. However,all you need are the secular effects generated by the laser pulse. The ponderomotivepotential comes to your rescue: there only the laser pulse envelope enters while the timescales of the laser period (and faster) are eliminated.


Problem 1.1 Calculate the ponderomotive force of an elliptically polarized laser pulse.

2.2 Relativistic dynamics of a charged particle in anelectromagnetic wave

Let us consider the motion of a particle of charge q and mass m in an electromagneticfield of the form

A(r , t ) = Aσ(η)P (η), η=ωt − k · r = kµxµ, A · k = 0. (2.19)

Here, A(r , t ) is the vector potential of amplitude A, σ(η) carries the fast time-dependence, and P (η) is the slowly varying envelope. The phase η is a relativisticinvariant, as it is evident from η = kµxµ where we use common relativistic nota-tion (µ = 0,1,2,3), xµ = (c t , r ), kµ = (ω/c ,k), the sum convention, and the metricg = diag(1,−1,−1,−1). The fields are

E =−∂tA, B =∇×A. (2.20)

There are various ways to solve the equations of motion for a charged particle in the field(2.19). A particularly elegant method is used in Landau & Lifshitz’s Classical Theory ofFields, based on the relativistic Hamilton-Jacobi equation.

The relativistic Hamiltonian governing the motion of a charged particle in an elec-tromagnetic field given by the scalar potential φ and the vector potential A reads

H = c2Æ

m2c2+(P − qA)2+ qφ. (2.21)

Here, P is the canonical momentum (while p = P − qA is the kinetic momentum). Asin the derivation of the Klein-Gordon equation we square (2.21) and obtain

1

c2(H − qφ)2 = m2c2+(P − qA)2. (2.22)

The goal of the Hamilton-Jacobi method is to find a generating function S, called action,for a canonical transformation to new, constant variables. These constant variables are

2.2. RELATIVISTIC DYNAMICS IN AN ELECTROMAGNETIC WAVE 15

then used to fulfill the initial conditions. The action S, depending on the “old” positions(+ time) and the new canonical momenta (+ energy) has to be chosen such that

E =H =−∂t S, P =∇S. (2.23)

Plugging this into (2.22) yields

(∂c t S + qφ/c)2 = m2c2+(∇S − qA)2. (2.24)

This expression can be written in a covariant manner. Introducing the four vectors

∂ µS = (∂c t S,−∇S) = (−E/c ,−P) =:−Pµ (2.25)

andAµ = (φ/c ,A), (2.26)

eq. (2.24) can be written as

gµν(∂µS + qAµ)(∂ νS + qAν) = m2c2. (2.27)

We are interested in the dynamics of a charged particle in an electromagnetic waveand therefore have

Aµ = (0,A), Aµkµ = 0. (2.28)

With the AnsatzS = Sfree+ Sfield(η) =βµxµ+ Sfield(η) (2.29)

where the βµ play the role of the new, constant momenta (+ energy), we obtain

∂µS =βµ+ ∂µSfield =βµ+ ∂µη∂ Sfield

∂ η=βµ+ kµS ′field (2.30)

where S ′field = ∂ηSfield. The Ansatz with Sfield depending only on the invariant phase η butnot on space and time separately is crucial. The fact that it works makes the problemsoluble at all.

Plugging (2.30) into (2.27) leads to

(βµ+ kµS ′field+ qAµ)(βµ+ kµS ′field+ qAµ) = m2c2. (2.31)

Making use of kµkµ = 0 and Aµkµ = 0, this equation can be solved for Sfield:

Sfield(η) =1

2βµkµ

∫ η

η0

dη′

m2c2− [βµ+ qAµ(η′)][βµ+ qAµ(η′)]

(2.32)


and the total action thus is

S = βµxµ+1

2βµkµ

∫ η

η0

dη′

m2c2− [βµ+ qAµ(η′)][βµ+ qAµ(η′)]

(2.33)

= βµxµ+1

2βµkµ

∫ η

η0

dη′

m2c2−βµβµ− 2qβµAµ(η′)− q2Aµ(η

′)Aµ(η′)

.

The new, constant four-momentum fulfills

βµβµ = m2c2 (2.34)

and is given through the initial conditions. The canonical four-momentum is given by[making use also of (2.34)]

Pµ =−∂ µS =−βµ+ 2qβ ·A(η)+ q2A(η) ·A(η)2β · k kµ (2.35)

with a ·b = aµbµ. The derivatives of the action (2.33) with respect to theβs are constantand give us the trajectory:

∂ S

∂ βµ= xiniµ = xµ−

kµ2(β · k)2

∫ η

η0

dη′

m2c2−β ·β− 2qβ ·A(η′)− q2A(η′) ·A(η′)

− 1

β · k

∫ η

η0

dη′βµ+ qAµ(η

′)

(2.36)

⇒ xµ = xiniµ+1

β · k

∫ η

η0

dη′βµ+ qAµ(η

′)

(2.37)

−kµ

2(β · k)2∫ η

η0

dη′2qβ ·A(η′)+ q2A(η′) ·A(η′) .

In the last line we used (2.34) again.We shall now specialize on an electromagnetic wave

A(η) = Aey sinη, k = kex , (2.38)

i.e.,Aµ = (0,0, Asinη, 0), kµ = (ω/c , k , 0, 0) = (k , k , 0, 0). (2.39)

In this case we have

β ·A=−β2Asinη, A ·A=−A2 sin2η, β · k = k(β0−β1), (2.40)


(the reader should be careful not to mix up upper indices with powers!) and (2.35)becomes explicitly

P 0 =Ec= −β0− 2qβ2Asinη+ q2A2 sin2η

2(β0−β1), (2.41)

P 1 = px = −β1− 2qβ2Asinη+ q2A2 sin2η

2(β0−β1), (2.42)

P 2 = py + qAsinη = −β2, (2.43)

P 3 = pz = −β3. (2.44)

Clearly, the canonical momenta in y- and z-direction are conserved, and the relationbetween canonical and kinetic momentum is included in (2.43).

2.2.1 The oscillation center frameWe are still free to choose the initial conditions βµ. This is equivalent to choose acertain reference frame in which we want to study the dynamics. The only restrictionis β ·β = m2c2. Let us choose the βs in such a way that the oscillation center of theparticle is at rest:

px = py = pz!= 0. (2.45)

From eqs. (2.43) and (2.44) follows

β2 =β3 = 0 (2.46)

so that withβ ·β= m2c2 ⇒ β0 =−

Æm2c2+(β1)2 (2.47)

(minus sign because E in (2.41) must be positive in the free particle case where A= 0).From px = 0 and (2.42) we have

2β1β1+

Æm2c2+(β1)2

= 2qβ2Asinη+ q2A2 sin2η. (2.48)

The cycle average of sinη vanishes while sin2η= 1/2. Equation (2.48) can then be solvedfor β1,

|β1|= 1

4

q2A2

Æm2c2+ q2A2/2

. (2.49)

The sign will be chosen later. The trajectory in the oscillation center frame can becalculated from (2.37) and reads

c t = c tini+β0

k(β0−β1)(η−η0)+

q2A2

2k(β0−β1)2

∫ η

η0

dη′ sin2η′, (2.50)


x1 = xini1+

β1

k(β0−β1)(η−η0)+

q2A2

2k(β0−β1)2

∫ η

η0

dη′ sin2η′, (2.51)

x2 = xini2+

qA

k(β0−β1)

∫ η

η0

dη′ sinη′, (2.52)

x3 = xini3. (2.53)

The difference of (2.50) and (2.51) yields

c t − x1 = c tini− xini1+

η−η0

k(2.54)

so thatη=ωt − k x, η0 =ωtini− k xini, (2.55)

as it should. Since in the oscillation center x1 should only oscillate but not drift, theterm ∼ (η−η0) has to be canceled by the first term of the integral

∫dη sin2η=

1

2η− 1

4sin2η. (2.56)

This is the case if we choose the positive sign in (2.49),

β1 =1

4

q2A2

Æm2c2+ q2A2/2

. (2.57)

Explicitly, we have

β0 =− m2c2+ q2A2/4Æ

m2c2+ q2A2/2, β0−β1 =−

qm2c2+ q2A2/2, (2.58)

and the trajectory is given by [setting rini appropriately and xµ = (c t , x, y, z)]

x = − q2A2

8k(m2c2+ q2A2/2)sin2η, (2.59)

y =qA

kÆ

m2c2+ q2A2/2cosη, (2.60)

z = 0. (2.61)

This describes a figure-eight motion in the plane defined by the polarization vector (ey

in our case) and the propagation direction k/k (ex in our case), as shown in Fig. 2.2. Ifwe define the dimensionless vector potential amplitude as

a =qA

mc(2.62)


Figure 2.2: Figure-eight dynamics of a charged particle in an electromagnetic wave, as seen inthe oscillation center frame. The trajectory in the xy-plane, where x is the propagation directionof the laser pulse, and y is the polarization direction, is shown for a = 0.5 (solid), a = 1 (dotted),a = 10 (dashed), and a = 100 (dashed-dotted). As a increases, the amplitudes k x and ky approachthe calculated values 1/4 and

p2, respectively.

we have

k x =− a2

8(1+ a2/2)︸︷︷︸k x

sin2η, ky =a

Æ1+ a2/2︸︷︷︸

ky

cosη, k z = 0. (2.63)

The size of the figure-eight does not increase infinitely as the laser intensity goes toinfinity:

lima→∞

k x =1

4, lim

a→∞ky =

p2, lim

a→∞

x

y=

1

4p

2' 0.177. (2.64)

Note that we only have the orbit parameterized with the invariant phase η (which isproportional to the proper time). We do not know the explicit expressions for x and yas functions of t . Parameterized with η, the trajectory looks extremely simple. In fact,only η and 2η shows up. If η were justωt we could talk about the fundamental and thesecond harmonic and nothing else. However, since η = ωt − k x and x itself dependson t (or η), all frequencies, that is, all multiples of the laser frequency enter. Thishas consequences for the radiation emitted by such a particle. We neglect the emittedradiation in this Chapter but will come back to it later on.

2.2.2 The adiabatically ramped pulse in the lab frameWe shall now investigate the same situation in the lab frame. We assume that the laserpulse has been ramped up adiabatically fromP = 0 at η→−∞ toP = 1, and that the


Figure 2.3: Particle motion in the lab frame for a = 1. While oscillating in polarization direc-tion, the particle is drifting in propagation direction. The laser pulse is adiabatically ramped upand of constant amplitude afterwards. The particle was initially at rest. Note that there is nobackward motion in x-direction (no loops but spikes).

particle starts from rest. Equations (2.41)–(2.44) hold in the lab frame as well. However,note that four-vectors such as kµ change if we switch from one frame to the other (rela-tivistic Doppler effect). Since it should be clear from the context in which frame we areworking, we suppress explicit indices indicating the frame.

The initial condition p = 0 for η → −∞ implies that β1 = β2 = β3 = 0 andβ0 =−mc . For the trajectory one finds

x =a2

4k

η− 1

2sin2η

, (2.65)

y =a

kcosη, (2.66)

z = 0, (2.67)

where we used the previously introduced dimensionless vector potential amplitude a[cf. (2.62)]. One sees that in polarization direction the particle just oscillates as beforewhile in laser propagation direction it is drifting. An example of such an orbit is shownin Fig. 2.3.

There seems to be a contradiction as far as the excursion of the particle in polariza-tion direction is concerned: in the lab frame this excursion is just proportional to a andthus is, in principle, unlimited. In the oscillation center frame the excursion was limitedtop

2k, despite the fact that a is the same in both frames. The resolution, of course,lies in k. With respect to the lab, the laser pulse is red-shifted in the oscillation centerframe.

Finally, we shall calculate the velocity of the oscillation center frame with respect to

2.3. THE RELATIVISTIC PONDEROMOTIVE FORCE 21

the lab frame. In the latter, eqs. (2.41)–(2.44) can be written as

P 0

mc=E

mc2= 1+

a2

2sin2η,

P 1

mc=

px

mc=

a2

2sin2η, P 2 = P 3 = pz = 0. (2.68)

It is obvious that for our choice of the vector potential, the Lorentz transformation hasto be performed parallel to ex so that

Λµν=

γoc −vocγoc/c 0 0−vocγoc/c γoc 0 0

0 0 1 00 0 0 1

(2.69)

with voc the oscillation center velocity we are looking for and γoc = (1− voc2/c2)−1/2.

Applying this transformation to (2.68), i.e., P ′µ =Λµν

P ν gives us

P ′1

γocmc=−voc

c

1+

a2

2sin2η

+

a2

2sin2η. (2.70)

The condition P ′1 = p ′x = 0 yields

voc =a2

4+ a2c . (2.71)

This is the velocity of the oscillation center of the particle in the lab frame if it was atrest before the pulse arrived.

2.3 The relativistic ponderomotive forceWe shall now derive the relativistic orbit of the oscillation center for a vector poten-tial of the form (2.19). It is desirable to consider the oscillation center as a relativisticpseudo particle. However, it is a priori not given for granted that the oscillation centercoordinates and momenta automatically “behave” in a relativistic, proper way. The cor-rect averaging over a laser period is crucial here. The situation is somewhat similar tothe relativistic center of mass, which is ill-defined if one naively extends the nonrela-tivistic expression and writes R =

∑i riγi mi

/∑

i γi mi

, where γi = (1−v2

i /c2)−1/2,without specifying in which system this expression should be evaluated. This is easilyillustrated by the following example: Imagine two particles of equal rest mass movingwith velocities ±vex in the lab frame, and v very close to c . The observer in the labframe will come to the conclusion that the center of mass (as defined above) is half waybetween both particles and stationary. Let us now transform to the reference frame inwhich one of the two particles is at rest. In this frame the other particle will have aso much higher “effective mass” γm m that the center of mass will be effectively at


the position of this second particle. The two centers of mass measured by the two ob-servers will not transform properly into each other by a Lorentz transformation. Theambiguity can be circumvented by defining the center of mass in the system in which itis at rest. Once determined it can then be transformed to any other system by a Lorentztransformation.

In the case of the relativistic oscillation center of a charged particle in a laser fieldsuch as (2.19) we are in the fortunate situation to have an invariant phase η = k · kover which we can average in an invariant manner. Already in a standing wave there isnot such a single invariant η anymore. In the lab frame, for instance, the two phasesη1 = ωt − k · r and η2 = ωt + k · r show up. In a reference frame moving along kone of the two waves will be red-shifted while the other will be blue-shifted, i.e., in thisframe there will be no standing wave at all. As in the center of mass-problem, in such asituation the oscillation center — if existing — has to be defined in the system where itis at rest, and averaging has to be done over the proper time.

In this lecture we will restrict ourselves to the relativistic ponderomotive force inthe travelling laser pulse where averaging over the invariant phase η can be done in anyframe.

Averaging of (2.35) and (2.37) yields [with βµ = (β0,β)]

roc = r (η) = rini+βη−η0

β · k − kq2

2(β · k)2∫ η

η0

dη′A ·A, (2.72)

poc = P(η) =−β+ kq2A ·A2β · k , (2.73)

Eoc

c=Ec=−β0+ k

q2A ·A2β · k . (2.74)

Here, the averaged phase η is given by

η=ωt − k · roc. (2.75)

The rest mass of the new pseudoparticle called oscillation center is given through

pocµ pocµ =M 2c2, poc

µ = (Eoc/c , poc). (2.76)

For consistency, this mass has also to fulfill

poc = γocM voc, voc = roc, γoc =

1− voc2

c2

−1/2

. (2.77)

For the velocity one finds

voc = (ω− k · voc)droc

dη= (ω− k · voc)

β− kΓ

β · k (2.78)


where the abbreviation

Γ=q2A ·A2β · k (2.79)

has been introduced. In order to solve (2.78) for voc we decompose the oscillation centervelocity and β in components parallel and perpendicular to k,

voc = voc‖k

k+ voc⊥, β=β‖

k

k+β⊥. (2.80)

This yields

voc‖ =ωβ‖− kΓ

β0k − k2Γ, voc⊥ =ω

β⊥β0k − k2Γ

, (2.81)

and

γoc =−β0− kΓ

Æm2c2− 2kΓ(β0−β‖)

. (2.82)

In the same way we decompose poc,

poc = poc‖k

k+ poc⊥ (2.83)

with (2.73) leading to

poc‖ =−β‖+ kΓ, poc⊥ =−β⊥. (2.84)

We then obtainpoc

µ pocµ =β ·β− q2A ·A !=M 2c2 (2.85)

so that, using β ·β= m2c2, the mass of the oscillation center is given by

M =1

c

qm2c2− q2A ·A. (2.86)

It is left as an exercise to the interested reader to check that this mass is indeed consistentwith poc = γocM voc.

Another check is the following: if we transform to the system moving with voc, theenergy E ′oc in this system should be simply M c2. The transformation reads

E ′oc

c= γoc

Eoc

c− voc · poc

c

. (2.87)

Using (2.74), (2.82), and the decompositions for voc and poc above, one readily checksthat indeed

E ′oc =M c2. (2.88)


The relativistic oscillation center pseudo particle possesses a rest mass which is not aconstant but depends on the phase η and thereby on space and time, that is, a mass field.

What remains to be calculated is the ponderomotive force

Fp = poc = (ω− kvoc‖)d

dηpoc. (2.89)

Using (2.73) and (2.79) one obtains

Fp =k

k

ωq2

2(β0− kΓ)

d

dηA ·A. (2.90)

The Minkowski force

F Mp =

d

dτpoc = γocFp (2.91)

(where τ is the proper time) assumes a particularly simple form. Using (2.82) (note thatthe denominator of γoc is proportional to M ) and

d

dη=− 1

k∇ (2.92)

one obtainsF M

p =−∇E ′oc =−c2∇M . (2.93)

The phase-dependent rest mass is the origin of the oscillation center motion. In thenonrelativistic case we saw already that the local quiver motion gives rise to the pon-deromotive potential. Relativistically, this quiver motion is included in the rest mass ofthe oscillation center.

2.3.1 Example: sin2-pulseLet us consider a pulse envelope of the form

P (η) = sin2(εη), 0< ε 1 (2.94)

which may describe a pulsed laser beam in which one pulse lasts from η = nπ/ε toη= (n+ 1)π/ε with n = 0,1,2, . . .. At time t = 0 the particle is assumed to be at rest atr = 0 so that

β= 0, β0 =−mc . (2.95)

Equations (2.72) and (2.73) become in this case

roc =q2A2

4m2c2k

3

8η− 1

4εsin(2εη)+

1

32εsin(4εη)

k

k, (2.96)

poc =q2A2

4mcsin4(εη)

k

k. (2.97)


Figure 2.4: Electron trajectory for a =−0.5 and ε= 0.01. Polarization and propagation direc-tions are ey and ex , respectively. The displacement after the first pulse agrees with formula (2.99)(which yields k · roc1 = 7.36 in this case).

The relativistic ponderomotive force (2.90) is

F sin2

p =

ωmc a2

a2 sin4(εη)+ 4

d

dηsin4(εη) (2.98)

where a = qA/mc has been used again. In the nonrelativistic limit |a| 1 the nonrel-ativistic ponderomotive force (2.11) is recovered (using (2.92) and A2 = E2/ω2). From(2.96) we can immediately infer the displacement of the particle after the first pulse:

roc1 =a2

4k

3

8η1

k

k, η1 =

π

ε. (2.99)

Figure 2.4 shows the trajectory of an electron for a =−0.5 and ε= 0.01 during the firstpulse, confirming formula (2.99).

Problem 2.1 Check that poc = γocM voc.

Further reading: An excellent paper on the subject (including references to importantprevious work) is E. S. SARACHIK and G. T. SCHAPPERT, “Classical Theory ofthe Scattering of Intense Laser Radiation by Free Electrons”, Phys. Rev. D 1, 2738(1970).

Chapter 3

Atoms in external fields

In this Chapter we shall first introduce atomic units and remind the reader of basicquantum mechanical techniques and atomic phenomena such as time-dependent per-turbation theory, resonant interaction (Rabi flopping), Stark-effect etc., before we cometo the actual topic of this lecture, i.e., the strong field effects and the new theoreticalapproaches that have been developed in this context.

3.1 Atomic unitsWe will introduce the atomic units in a somewhat formal way. For beginners, the useof atomic units is sometimes confusing because dimensional checks are not possible ina straightforward way once ħh, me, e have been set “equal to unity”. Another practicalproblem for beginners is to convert the result of a calculation performed using atomicunits back to SI units. If one sticks to SI units and the result of a calculation is, say, 42,one knows that the result is dimensionless. If in atomic units the result is 42 it could beas well 42 ħh, 42 e , 42 me, 42 · 4πε0, 42 ħh me/e , . . .. A good example is the expression forthe nonrelativistic eigenenergies of hydrogen-like ions, which in atomic units is simplygiven by

En =−Z2

2n2, n = 0,1,2, . . . (atomic units) (3.1)

where Z is the nuclear charge. The right-hand-side appears to be dimensionless. Hence,without knowing that the left-hand-side is an energy, there would be no way back toSI units. If one performs the entire calculation (i.e., the solution of the Schrödingerequation) in SI units, one obtains

En =−Z2

2n2

mee4

ħh2(4πε0)2, n = 0,1,2, . . . (SI units). (3.2)

Even without knowing the left-hand-side we can recover from the right-hand-side alonethat the result is an energy, provided we know the SI units of ħh, me, e , and ε0. This

27

28 CHAPTER 3. ATOMS IN EXTERNAL FIELDS

seeming asymmetry between the unit systems arise from the fact that “setting ħh, me, e ,and 4πε0 equal to unity” is more than a change of units. It is like setting kg, m, s, andC to unity.

Let us denote mass, length, time, and charge by M, L, T, and C, respectively. Inatomic units we wish to use the action ħh, the electron mass me, the modulus of theelectron charge e , and 4π times the permittivity of vacuum 4πε0 as the basic units. Therelation between both system of units is established by (“[...]” meaning “units of ...”)

[ħh] = Ma11 La12Ta13Ca14 , (3.3)[me] = Ma21 La22Ta23Ca24 , (3.4)[e] = Ma31 La32Ta33Ca34 , (3.5)

[4πε0] = Ma41 La42Ta43Ca44 . (3.6)

The exponents a41, . . . ,a44, for instance, are determined by noticing that in SI unitsthe dimension of ε0 is Coulomb per Volt and meter. Volt is a derived unit. BecauseCoulomb times Volt is an energy one has

V=ML2

CT2, (3.7)

and hence[4πε0] =M−1L−3T2C2. (3.8)

Similarly, the remainder of the dimensional matrix is easily calculated and reads

A=

a11 a12 a13 a14a21 a22 a23 a24a31 a32 a33 a34a41 a42 a43 a44

=

1 2 −1 01 0 0 00 0 0 1−1 −3 2 2

. (3.9)

Now we want to know the values (in SI units) of one atomic mass, length, time, andcharge unit, denoted byM ,L , T , and C , respectively. Those are later needed for thetransformation back to SI units. Mass and charge are trivial:

M = me, C = e . (3.10)

Let us calculate the value of the atomic length unit:

L = ħh b21 mb22e e b23(4πε0)

b24 . (3.11)

Plugging in (3.3)–(3.6)and using the values for the as in (3.9) lead to

L =Mb21+b22−b24 L2b21−3b24T−b21+2b24Cb23+2b24 , (3.12)

3.1. ATOMIC UNITS 29

giving us four equations for the four b s. Since L is a length, 2b21 − 3b24 = 1, andthe exponents of mass, time, and charge must be zero. One finds b21 = 2, b22 = −1,b23 =−2, b24 = 1 so that

L = ħh24πε0

mee2= 0.5292 · 10−10m, (3.13)

which is the Bohr radius a0. The corresponding calculation for T is left as an exercise.The inverse dimensional matrix is then established through

M = ħh b11 mb12e e b13(4πε0)

b14 , (3.14)

L = ħh b21 mb22e e b23(4πε0)

b24 , (3.15)

T = ħh b31 mb32e e b33(4πε0)

b34 , (3.16)

C = ħh b41 mb42e e b43(4πε0)

b44 (3.17)

with

B =

b11 b12 b13 b14b21 b22 b23 b24b31 b32 b33 b34b41 b42 b43 b44

=

0 1 0 02 −1 −2 13 −1 −4 20 0 1 0

. (3.18)

It is readily checked thatA ·B = B ·A= 1. (3.19)

As an example for the transition from SI to atomic units, let us now formally rewritethe time-dependent Schrödinger equation in position space representation

iħh ∂∂ tΨ(r , t ) =HΨ(r , t ) (3.20)

in atomic units. We have that (with the values of primed symbols in atomic units)

ħh = ħh ′ML2

T = ħh ′ħh ⇒ ħh ′ = 1, (3.21)

t = t ′T = t ′ħh3(4πε0)

2

mee4, (3.22)

Ψ = Ψ′L −3/2 =Ψ′ ħh24πε0

mee2

!−3/2

, (3.23)

H = H ′ML 2

T 2=H ′ mee4

ħh2(4πε0)2. (3.24)


Plugging this into (3.20) (and taking the primed wavefunction as a function of theprimed variables) simply yields

iħh ′ ∂∂ t ′Ψ′(r ′, t ′) =H ′Ψ′(r ′, t ′), (3.25)

i.e., the same equation primed. However, since ħh ′ = 1 [cf. (3.21)] this simplifies to

i∂

∂ t ′Ψ′(r ′, t ′) =H ′Ψ′(r ′, t ′). (3.26)

The ħh effectively disappeared or, more precisely, is hidden in the units. This is whatis actually meant by “setting ħh = me = e = 1”. In the same way ħh disappeared in thisexample, me, e , and 4πε0 are effectively removed. If, for instance,

H =p2

2me

− Ze2

4πε0 r+ eE · r , (3.27)

that is, the Hamiltonian governing the electron of a hydrogen-like ion in an electric fieldE , in atomic units

H ′ =p ′2

2− Z

r ′+E ′ · r ′ (3.28)

holds.Finally, we give an example how to return to SI units after a busy calculation em-

ploying atomic units. Let us suppose the result of this calculation is a an electric fieldstrength

E ′ = f (α′,β′, . . .)ei(ω′ t ′−p′·r ′) (3.29)

and f is some function of the entities α′, β′, . . . which can be converted to the SI valuesα, β, . . ., giving rise to some overall factor K = ħhσmη

e eξ (4πε0)ζ . Since an electric field

strength in SI units is Volt per meter, and Volt is related to mass, length, etc. through(3.7) we have

E = E ′MLCT 2

= E ′m2

e e5

ħh4(4πε0)3. (3.30)

Hence

E = E ′m2

e e5

ħh4(4πε0)3=K

m2e e5

ħh4(4πε0)3

f (α,β, . . .)ei(ω′ t ′−p′·r ′). (3.31)

Obviouslyω′t ′− p′ · r ′ is dimensionless in atomic units, as is ωt in SI units, so that

ωt =ω′t ′ (3.32)

but what about p′ · r ′? In SI units p · r is not dimensionless but an action. Well, we have

p = p′MLT = p′

mee2

ħh4πε0

, r = r ′L = r ′ħh24πε0

mee2(3.33)

3.1. ATOMIC UNITS 31

so that

p′ · r ′ = p · rħh , (3.34)

which, in fact, is dimensionless. Hence the expression (3.29) reads in SI units

E =m2+η

e e5+ξ

ħh4−σ (4πε0)3−ζ f (α,β, . . .)ei(ωt−p·r/ħh). (3.35)

We conclude this section by giving the explicit expressions and values for frequentlyoccurring entities in laser-atom interaction:

atomic mass unitM = me = 9.1094 · 10−31 kg, (3.36)

atomic length unitL =ħh24πε0

e2me

= a0 = 0.5292 · 10−10m, (3.37)

atomic time unit T =ħh3(4πε0)

2

mee4= 2.4189 · 10−17s= 0.024 fs, (3.38)

atomic charge unit C = e = 1.6022 · 10−19C, (3.39)atomic action unit = ħh = 1.0546 · 10−34J, (3.40)

atomic permittivity unit = 4πε0 = 4π · 8.8542 · 10−12CV−1m−1, (3.41)

atomic energy unit =mee4

ħh2(4πε0)2= 4.3598 · 10−19J= 27.21eV, (3.42)

atomic velocity unit =e2

ħh4πε0

= 2.1877 · 106ms−1 = αc , (3.43)

atomic el. field strength unit =m2

e e5

ħh4(4πε0)3= 5.1422 · 1011Vm−1, (3.44)

atomic intensity unit =e12m4

e

8παħh9(4πε0)6= 3.5095 · 1020Wm−2 (3.45)

= 3.5095 · 1016Wcm−2.

Here, the fine structure constant

α=e2

ħh4πε0c=

1

137.04(3.46)

has been introduced. The value of the light velocity in vacuum in atomic units equalsthe inverse fine structure constant. Note that with ħh, e , me, and ε0 alone it is notpossible to construct a dimensionless entity (as it is impossible with kg, m, s, and C).One needs a fifth building brick, which is c in the case of the fine structure constant.


A useful formula to convert the field strength E in atomic units (a.u.) to a laserintensity I in the commonly used Wcm−2 is

(I in Wcm−2) = 3.51 · 1016× (E2 in a.u.). (3.47)

Problem 3.1 Calculate the value of the atomic frequency unit in Hz.

Problem 3.2 Calculate the value of the atomic magnetic field unit in Tesla.

3.2 Time-dependent perturbation theoryIn this Section we briefly review time-dependent perturbation theory. We considera quantum mechanical system such as an atom, for instance, interacting with a time-dependent field, e.g., an electromagnetic field. In lowest order perturbation theory weexpect transitions between the unperturbed states to occur. The Hamiltonians of interestare of the form

H (t ) = H0+W (t ), W (t ≤ 0) = 0. (3.48)

Here we assume that the perturbation W (t ) is switched off for times t ≤ 0. We furtherassume that the solution of the unperturbed problem is known, that is,

H0|φn⟩= En|φn⟩ (3.49)

with given eigenenergies En and eigenstates |φn⟩. The unperturbed states (includingthe continuum states, if present) form a complete basis so that the full solution of thetime-dependent Schrödinger equation (TDSE)

i∂t |ψ(t )⟩= H (t )|ψ(t )⟩ (3.50)

can be expanded as

|ψ(t )⟩=∑∫

ncn(t )|φn⟩e−iEn t . (3.51)

Here we use atomic units and factor the unperturbed time-evolution ∼ e−iEn t out ofcn(t ) (interaction picture). If at time t = 0 the system is in a single, unperturbed statewe have

|ψ(0)⟩= |φi⟩ ⇒ cn(0) = δni. (3.52)

At later times, the probability for finding the system in the state |φf⟩ of H0 is

wi→f(t ) = |cf(t )|2. (3.53)

3.2. TIME-DEPENDENT PERTURBATION THEORY 33

Inserting the expansion (3.51) in the TDSE (3.50) and multiplying from the left by ⟨φm|yields

idcm

dt=∑∫

nWmn(t )cn(t )e

i(Em−En)t , Wmn(t ) = ⟨φm|W (t )|φn⟩. (3.54)

Formal integration of (3.54) gives

cm(t ) = cm(0)+1

i

∫ t

0dt ′∑∫

nWmn(t

′)ei(Em−En )t′cn(t

′), (3.55)

which is an integral equation because the coefficient cm for a certain m also appears un-der the integral within the sum over n. Such integral equations may be solved iterativelyby replacing the cns under the integral by the expression (3.55) itself. If the perturbationis small (i.e., the Wmns are small on the relevant energy scale), the series converges andone can stop at some sufficiently high order. In first order one simply has

cm(t ) = cm(0)+1

i

∫ t

0dt ′∑∫

nWmn(t

′)eiωmn t ′cn(0), ωmn = Em −En. (3.56)

In second order we have

cm(t ) = cm(0)+1

i

∫ t

0dt ′∑∫

nWmn(t

′)eiωmn t ′ (3.57)

×

cn(0)+1

i

∫ t ′

0dt ′′

∑∫

uWnu(t

′′)eiωnu t ′′cu(0)

!.

If the final state of interest was not populated at time t = 0 and ci = 1, the first orderexpression (3.56) becomes

cf(t ) =1

i

∫ t

0dt ′Wfi(t

′)eiωfi t′. (3.58)

Now let us assume that the perturbation is of the form

W (t ) = V sinωt or W (t ) = V cosωt . (3.59)

In the sin-case, for instance, one has

Wfi(t ) =Vfi sinωt =Vfi

2i

eiωt − e−iωt

, Vfi = ⟨φf|V |φi⟩ (3.60)

so that

c sinf (t ) =

Vfi

2i

1− ei(ωfi+ω)t

ωfi+ω− 1− ei(ωfi−ω)t

ωfi−ω

!, (3.61)


and

w sini→f(t ) =

|Vfi|24

1− ei(ωfi+ω)t

ωfi+ω− 1− ei(ωfi−ω)t

ωfi−ω

2

. (3.62)

In the same way one finds for the cos-expression in (3.59)

wcosi→f(t ) =

|Vfi|24

1− ei(ωfi+ω)t

ωfi+ω+

1− ei(ωfi−ω)t

ωfi−ω

2

. (3.63)

The case of a constant perturbation can be obtained by setting ω = 0:

wconsti→f (t ) =

|Vfi|2ω2

fi

1− eiωfi t2= |Vfi|2 F (t ,ωfi), F (t ,ωfi) =

sin(ωfit/2)

ωfi/2

2

. (3.64)

The results for the sin and cos-like perturbations can be written as (upper sign for sin,lower for cos)

w sin,cosi→f(t ) =

|Vfi|24

A+∓A−2 (3.65)

where

A± =1− ei(ωfi±ω)t

ωfi±ω=−i ei(ωfi±ω)t/2

sin[(ωfi±ω)t/2](ωfi±ω)/2

. (3.66)

If we assume that ω,ωfi > 0 and t ω−1, only the resonant term A− is importantand the highly oscillatory and small anti-resonant term A+ can be neglected. This is theso-called rotating wave approximation (RWA). We then obtain

wRWAi→f (t ) =

|Vfi|24

F (t ,ωfi−ω). (3.67)

The function F is defined in (3.64). It is illustrated in Fig. 3.1 for ωfi = 1 and differenttimes. One sees that the maximum increases in time as t 2 while the resonance width

∆ω = 4π/t (3.68)

decreases in time. The resonance width defines the energy resolution with which wecan determine a transition frequency∆E using a sinusoidal perturbation,

t∆E ' 1. (3.69)

This is the time-energy uncertainty relation. Note that it is of completely different origincompared to uncertainty relations for noncommuting operators (such as the position-momentum uncertainty relation, for instance).

Let us now discuss the validity of our first order perturbation theory results. Onone hand, the time t must not be too big, because at resonance one has

wresi→f(t ) =

|Vfi|24

t 2. (3.70)


(a) (b) (c)

Figure 3.1: The function F (t ,ωfi −ω) vs ω for ωfi = 1 and (a) t = 10, (b) t = 30, and (c)t = 100. The maximum value t 2 occurs at resonance ω = ωfi, the width of the central peak is∆ω = 4π/t .

Since wresi→f(t ) is a probability (i.e., a number between zero and one) and the perturbation

is supposed to be small

|Vfi|24

t 2 1 ⇒ t 1

|Vfi|(3.71)

must hold. On the other hand, the RWA to be valid requires

t 1

|ωfi|' 1

ω(3.72)

because at early times A+ and A− are of the same magnitude (transient dynamics).Hence,

1

|ωfi| t 1

|Vfi|⇒ |ωfi| |Vfi| (3.73)

which confirms our statement above, that “small perturbation” means that the valuesof the relevant matrix elements Wmn are small compared to the “relevant energy scale”,which here is the transition energyωfi.

If the final state lies in the continuum and is characterized by a set of parameters α(e.g., quantum numbers `, m, wave vector k or energy E etc.) one has to integrate overall the possible final states to obtain the transition probability,

wi→f(t ) =∫

α∈Df

dα |⟨α|ψ(t )⟩|2 . (3.74)


We now choose the energy E as one of these parameters describing the final state, andthe set β comprises the remaining parameters. Introducing the density of states ρ(β,E )we can write

dα= ρ(β,E )dβdE , (3.75)

and (3.74) turns into

wi→f(t ) =∫

(β,E )∈Df

dβdE ρ(β,E ) |⟨β,E|ψ(t )⟩|2 . (3.76)

In the case of a constant perturbation, for instance, we obtain

wi→f(t ) =∫

(β,E )∈Df

dβdE ρ(β,E )⟨β,E|V |φi⟩

2

F (t ,E −Ei). (3.77)

If t is sufficiently large, and ρ(β,E )⟨β,E|V |φi⟩

2

varies much slower over an energyinterval 4π/t than F , the integration over energy in (3.77) can be performed using

limt→∞

F (t ,E −Ei) =πt δ[(E −Ei)/2] = 2πt δ(E −Ei) (3.78)

so that

wi→f(t ) = 2πt∫

β∈Df

dβ⟨β,Ef = Ei|V |φi⟩

2ρ(β,Ef = Ei) if Ei ∈Df (3.79)

and zero otherwise.The transition probability per unit time (i.e., the rate) is given by

Γi→f =dwi→f

dt(3.80)

and is time-independent in the case of a constant perturbation. If we suppress the otherparameters β to be integrated over, we simply have

Γconst.i→f = 2π

⟨Ef = Ei|V |φi⟩2ρ(Ef = Ei). (3.81)

In the case of a sinusoidal perturbation we obtain

Γsin,cosi→f

= 2π⟨Ef = Ei+ω|V |φi⟩

2ρ(Ef = Ei+ω). (3.82)

Equations (3.81) and (3.82) are “Fermi’s Golden Rule” applied to a constant perturbation(which conserves energy) and one photon absorption into the continuum, that is, singlephoton ionization or the photo effect, respectively.


In the case of a laser field, V is proportional to r · E where E is the electric fieldamplitude (not an operator!). Hence

⟨Ef = Ei+ω|V |φi⟩2 ∼ |E |2 ∼ I , (3.83)

i.e., the rate will be proportional to the laser intensity.In nth order n photons may contribute to the ionization process. Because in nth

order the matrix elements Wmn appear n times in the expression for cf, the nth orderionization rate is proportional to I n and thus fulfills

Γ(n)i→f= σn I n (3.84)

where the σn are called generalized cross sections.

Problem 3.3 Calculate the density of states ρ(E ) for a free, spinless particle of mass m.

Problem 3.4 Calculate the SI units of the generalized cross sections σn in (3.84).

3.2.1 Dyson series, S-matrix, Green’s functions, propagators, andresolvents

Let us consider the time-evolution operator U corresponding to the TDSE (3.50),

|ψ(t )⟩= U (t , t ′)|ψ(t ′)⟩. (3.85)

The operator U fulfills the TDSE,

i∂t U (t , t ′) = H (t )U (t , t ′), (3.86)

and formal “solutions” of this equation are given by

U (t , t ′) = U0(t , t ′)− i∫ t

t ′dt ′′ U (t , t ′′)W (t ′′)U0(t

′′, t ′) (3.87)

= U0(t , t ′)− i∫ t

t ′dt ′′ U0(t , t ′′)W (t ′′)U (t ′′, t ′) (3.88)


where U0 is the time-evolution operator associated to the unperturbed Hamiltonian H0.Equations (3.87) and (3.88) are integral equations since U appears also on the right handside under the time integrals [cf. (3.55)]. Iteration yields

U (t , t ′) = U0(t , t ′)+ (−i)∫ t

t ′dt ′′ U0(t , t ′′)W (t ′′)U0(t

′′, t ′) (3.89)

+(−i)2∫ t

t ′dt ′′∫ t ′′

t ′dt ′′′ U0(t , t ′′)W (t ′′)U0(t

′′, t ′′′)W (t ′′′)U0(t′′′, t ′)+ · · · .

Introducing the perturbation WIP in the interaction picture

WIP(t , t ′) = U †(t , t ′)W (t )U (t , t ′) = U (t ′, t )W (t )U (t , t ′), (3.90)

making use ofU0(t

′′, t ′′′) = U0(t′′, t ′)U0(t

′, t ′′′) (3.91)

and multiplying (3.89) by U0(t′, t ) from the left yields

U0(t′, t )U (t , t ′) = 1+(−i)

∫ t

t ′dt ′′WIP(t

′′)+ (−i)2∫ t

t ′dt ′′∫ t ′′

t ′dt ′′′WIP(t

′′)WIP(t′′′)+ · · · .

(3.92)Here we suppressed the second time index of WIP (which is always t ′). We now rewritethis series in the following way:

U0(t′, t )U (t , t ′) = 1+(−i)

∫ t

t ′dt ′′WIP(t

′′) (3.93)

+(−i)2∫ t

t ′dt ′′∫ t

t ′dt ′′′θ(t ′′− t ′′′)WIP(t

′′)WIP(t′′′)+ · · · (3.94)

= 1+(−i)∫ t

t ′dt ′′WIP(t

′′) (3.95)

+(−i)2

2

∫ t

t ′dt ′′∫ t

t ′dt ′′′

¦θ(t ′′− t ′′′)WIP(t

′′)WIP(t′′′) (3.96)

+θ(t ′′′− t ′′)WIP(t′′′)WIP(t

′′)©+ · · ·

=∞∑

n=0

(−i)n

n!

∫ t

t ′dt1 · · ·

∫ t

t ′dtn T

¦WIP(t1) · · ·WIP(tn)

©. (3.97)

In the last line we introduced the time ordering operator T. The expansion of the timeevolution operator in terms of integrals over time-ordered products of the perturbation(in interaction picture representation) is called Dyson series. The last expression (3.97)is often written in the form

U0(t′, t )U (t , t ′) = Te−i

∫ tt ′ dt ′′ WIP(t

′′) . (3.98)


If the WIPs commute at different times this expression reduces to

exp

−i∫ t

t ′dt ′′WIP(t

′′)

,

i.e., the problem is boiled down to the solution of the single integral∫ t

t ′ dt ′′WIP(t′′).

Unfortunately, in general [WIP(t1),WIP(t2)] 6= 0 in most of the interesting cases.Let us now define an initial and a final state as

|i⟩= |ψi(−∞)⟩, |f⟩= |ψf(−∞)⟩, (3.99)

and let us assume that the perturbation W is absent for t →±∞. The probability for atransition from the initial to the final state is

wi→f = |Mif|2 (3.100)

with

Mif = ⟨ψf(∞)|U (∞,−∞)|ψi(−∞)⟩ (3.101)

= ⟨ψf(−∞)|U †0 (∞,−∞))U (∞,−∞)|ψi(−∞)⟩ (3.102)

= ⟨f|U0(−∞,∞)U (∞,−∞)|i⟩ (3.103)

=: ⟨f|S |i⟩= Sfi. (3.104)

Sfi is the so-called S-Matrix. Note that S is the left hand side of (3.98) for t ′→−∞ andt →∞.

Let us come back to the integral equation for U (3.88). Multiplication by θ(t − t ′)and insertion of θ functions under the integral so that the integration limits can beextended to ±∞ yields

U (t , t ′)θ(t− t ′) = U0(t , t ′)θ(t− t ′)−i∫ ∞

−∞dt ′′ U0(t , t ′′)θ(t− t ′′)W (t ′′)U (t ′′, t ′)θ(t ′′− t ′).

(3.105)Defining the retarded Green’s functions

K+(t , t ′) = U (t , t ′)θ(t − t ′) , K0+(t , t ′) = U0(t , t ′)θ(t − t ′), (3.106)

turns (3.105) into

K+(t , t ′) = K0+(t , t ′)− i∫ ∞

−∞dt ′′ K0+(t , t ′′)W (t ′′)K+(t

′′, t ′). (3.107)

It is easy to show [using (3.86), (3.106), and ∂tθ(t ) = δ(t )] that K+ fulfills the inhomo-geneous TDSE

(i∂t − H )K+(t , t ′) = iδ(t − t ′). (3.108)


The advanced Green’s function is defined through

K−(t , t ′) =−U (t , t ′)θ(t ′− t ). (3.109)

It fulfills the same Eq. (3.108) (but for different boundary conditions).Let us now assume that H is time-independent so that

K+(τ) = e−iHτθ(τ), τ = t − t ′. (3.110)

This assumption may appear strange, especially if we have in mind interactions withstrong laser fields (which are time-dependent electromagnetic fields). However, if wequantize the electromagnetic field, the system atom + field becomes conservative, andH would have no explicit time-dependence. It is always—at least in principle—possibleto extend the system under study in such a way that the perturbing driver becomes partof the system, making the total system conservative. In this sense the TDSE is a specialcase of the time-independent Schrödinger equation and not the other way round. TheTDSE arises when we divide bigger, conservative systems into smaller non-conservativesystems. The ultimate system would be the entire universe (including parallel ones, ofcourse) with no room for “external” time-dependent drivers whatsoever.

We introduce a new operator G+ which is proportional to the Fourier-transformedGreen’s function,

K+(τ) = − 1

2πi

∫ ∞

−∞dE e−iEτG+(E ), (3.111)

G+(E ) = −i∫ ∞

−∞dτ eiEτK+(τ). (3.112)

We can then switch from the time-domain to the energy-domain (or frequency-domain)and back by working either with K+(τ) or G+(E ), respectively. Using (3.110) we obtain

G+(E ) = −i∫ ∞

0dτ ei(E−H )τ (3.113)

= −i limη→0+

∫ ∞

0dτ ei(E−H+iη)τ, η≥ 0 (3.114)

⇒ G+(E ) = limη→0+

1

E − H + iη(3.115)

and in the same way

G−(E ) = limη→0+

1

E − H − iη. (3.116)

G± are called propagators.


The Fourier-transform of (3.107) for a time-independent W gives us

G+(E ) = G0+(E )−∫ ∞

−∞dτ eiEτ

∫ ∞

−∞dt ′′ K0+(t , t ′′)W K+(t

′′, t ′) (3.117)

= G0+(E )−∫ ∞

−∞dτ∫ ∞

−∞dt ′′ eiEτ1K0+(τ1)W eiEτ2K+(τ2) (3.118)

= G0+(E )−∫ ∞

−∞dτ1

∫ ∞

−∞dτ2 eiEτ1K0+(τ1)W eiEτ2K+(τ2). (3.119)

The last step follows from

τ1 = t − t ′′, τ2 = t ′′− t ′ ⇒∂ (τ, t ′′)

∂ (τ1,τ2)

= 1. (3.120)

We thus haveG+(E ) = G0+(E )+ G0+(E )W G+(E ) (3.121)

which could have been immediately derived from the identity

1

A=

1

B+

1

B(B −A)

1

A. (3.122)

Equation (3.121) is an algebraic equation as compared to the integral equations for K+and U .

Finally, we introduce the resolvent

G(z) =1

z − H, z ∈C. (3.123)

Obviously, as z→EG±(E ) = lim

η→0+G(E ± iη). (3.124)

Since θ(x)+θ(−x) = 1 we can write

U (τ) = K+(τ)− K−(τ) (3.125)

=1

2πi

∫ ∞

−∞dE e−iEτ[G−(E )− G+(E )] (3.126)

=1

2πi

∫

C++C−

dz e−izτ G(z). (3.127)

The integration contours are depicted in Fig. 3.2. Discrete eigenenergies appear as poleson the real z-axis, continua correspond to cuts.


C+

C −

cutspoles

z

Figure 3.2: Illustration of the integration contours C± in (3.127) in the z-plane. For τ > 0 thecontribution from C− is zero, for τ < 0 the contribution from C+ is zero. Eigenenergies of Hare located along the real axis (discrete ones indicated by crosses, a continuum by a thick line.

For the resolvent (3.121) becomes

G(z) = G0(z)+ G0(z)W G(z) (3.128)

which can be iterated,

G(z) = G0(z)+ G0(z)W G0(z)+ G0(z)W G0(z)W G0(z)+ · · · . (3.129)

3.3 Important modelsIn this Section we will go through two important, exactly soluble models. The firstone shows how discrete states cease to exist if they are coupled to a continuum. Thesecond one discusses two resonantly (or almost resonantly) coupled discrete states andthe population transfer between them (Rabi-oscillations). It is a nice example for asoluble problem where the previously introduced perturbation theory is not applicable.

3.3.1 Discrete level coupled to the continuumStarting point is the Hamiltonian

H = H0+W . (3.130)

We allow for a single discrete state |ϕ⟩ and the continuum states. We now discretize thecontinuum by introducing a box of length L. The continuum states |k⟩ will then bespaced by δ, and the density of states is 1/δ so that Fermi’s Golden Rule (3.81) yields

Γ= 2πw2 1

δ, lim

δ→0

w2

δ= const. (3.131)

Herewk := ⟨k|W |ϕ⟩ (3.132)

3.3. IMPORTANT MODELS 43

and w is the matrix element for which Ek = Eϕ, as required by (3.81). We assume

⟨k|H0|k⟩= Ek = kδ, −∞< k <∞, k ∈Z, (3.133)

Eϕ = 0 ⇒ Eϕ = Ek=0, (3.134)

⟨ϕ|W |ϕ⟩= ⟨k|W |k ′⟩= 0, (3.135)

and thatwk = ⟨k|W |ϕ⟩= ⟨ϕ|W |k⟩= w, (3.136)

that is, all these transition matrix elements are equal and real. We are now looking forthe new energies Eµ and states |ψµ⟩ fulfilling

H |ψµ⟩= Eµ|ψµ⟩. (3.137)

By multiplying from the left with ⟨k| and ⟨ϕ|, and making use of Eϕ = 0 and

1= |ϕ⟩⟨ϕ|+∑

k

|k⟩⟨k|, (3.138)

we obtainEk⟨k|ψµ⟩+w⟨ϕ|ψµ⟩= Eµ⟨k|ψµ⟩, (3.139)

w∑

k

⟨k|ψµ⟩= Eµ⟨ϕ|ψµ⟩ (3.140)

so that

⟨k|ψµ⟩= w⟨ϕ|ψµ⟩Eµ−Ek

, Eµ−Ek 6= 0 (3.141)

and

Eµ =∑

k

w2

Eµ−Ek

, (3.142)

which is the desired (although implicit) equation for the new eigenenergies Eµ. It canbe rewritten

Eµ =∑

k

w2

Eµ−δk=

w2

δ

∑k

1

z − k=

w2

δ

π

tanπz, z =

Eµδ

. (3.143)

Hence,

Eµ =πw2

δ tan(πEµ/δ)⇒ 1

tan(πEµ/δ)=

2EµΓ

, Γ=2πw2

δ. (3.144)


Figure 3.3: Graphical determination of the eigenvalues Eµ of H . The abscissa of each intersec-tion point Mk is an eigenvalue. Let Mk be the point of intersection whose abscissa is betweenkδ and (k+1)δ (the abscissa of M−k being between−(k+1)δ and−kδ). The associated eigen-value is denoted Eκ = Eµ where the Greek index κ corresponds to the Roman index k of theunperturbed states. The two intersections between−δ and δ arise from the unperturbed states|k = 0⟩ and |ϕ⟩. [Figure taken from COHEN-TANNOUDJI et al., Atom-Photon Interactions,(Wiley, New York, 1998)].


Finding the new eigenenergies Eµ amounts to find the intersections of y = ax withy = 1/ tan(b x)where a = 2/Γ and b =π/δ, and x = Eµ. This is shown in Fig. 3.3. Thenew eigenvalues Eµ are interspersed between the old ones. The larger |k| the closer arethe new eigenvalues to the old ones. If Eµ Γ one has Eµ 'Ek .

Using the normalization condition∑

k

|⟨k|ψµ⟩|2+ |⟨ϕ|ψµ⟩|2!= 1 (3.145)

we find for the projections of the new states on the unperturbed states

⟨ϕ|ψµ⟩ =1+

∑

k ′

w

Eµ−Ek ′

!2−1/2

, (3.146)

⟨k|ψµ⟩ =w

Eµ−Ek

1+

∑

k ′

w

Eµ−Ek ′

!2−1/2

. (3.147)

The square bracket can be rewritten using

∑k

(z − k)−2 =π2

sin2πz(3.148)

and yields

1+∑

k ′

w

Eµ−Ek ′

!2

= 1+π2w2

δ2

1+ tan−2

πEµδ

. (3.149)

With the help of (3.144) this can be recast in the form

1+∑

k ′

w

Eµ−Ek ′

!2

=1

w2

w2+

Γ2

2

+E 2µ

(3.150)

so that⟨ϕ|ψµ⟩=

wh

w2+Γ2

2+E 2

µ

i1/2. (3.151)

Consider an energy interval [E ,E + dE ] such that

δ dE Γ (3.152)

so that the probability to find the formerly discrete state |ϕ⟩ within this interval reads

dNϕ =∑

Eµ∈[E ,E+dE ]|⟨ϕ|ψµ⟩|2 '

dEδ|⟨ϕ|ψµ⟩|2. (3.153)


We thus havedNϕ

dE =Γ/2π

w2+Γ2

2+E 2

. (3.154)

In the continuum limit δ→ 0 the probability w2 has vanishing measure and

dNϕ

dE =Γ/2π

Γ2

2+E 2

(3.155)

which is a Lorentzian of width Γ around the energy E = Eϕ = 0. We learn from thisstudy that discrete levels cease to exist when they are coupled to a continuum and thatthe continuum states in the vicinity' Γ around the energy of the unperturbed, discretestate |ϕ⟩ retain memory of |ϕ⟩.

We finally show that the decay in this simple model system in exponential. Let usassume that at time t = 0 the system is prepared to be in the state |ϕ⟩,

|ψ(0)⟩= |ϕ⟩=∑µ

wh

w2+Γ2

2+E 2

µ

i1/2|ψµ⟩. (3.156)

As a consequence, the state at a later time will be

|ψ(t )⟩=∑µ

w e−iEµ t

hw2+

Γ2

2+E 2

µ

i1/2|ψµ⟩. (3.157)

Using (3.151) yields

⟨ϕ|ψ(t )⟩ =∑µ

w2 e−iEµ t

w2+Γ2

2+E 2

µ

(3.158)

= δΓ

2π

∑µ

e−iEµ t

w2+Γ2

2+E 2

µ

(3.159)

δ→0=Γ

2π

∫ ∞

−∞dE e−iE t

Γ2

2+E 2

(3.160)

= e−Γt/2, t ≥ 0. (3.161)

For the last step we used the method of residues. The population of the formerly dis-crete level thus decays exponentially according

|⟨ϕ|ψ(t )⟩|2 = e−Γt , t ≥ 0. (3.162)


We stop the discussion of the model at this point. We only mention that it canbe extended toward more than a single discrete state. If, for instance, the populationof the formerly discrete state can either reach the continuum directly or via transitionto another (formerly) discrete state, an interference effect arises which leads to spectralshapes (Fano profiles) depending on the ratio of the relevant coupling strengths w.

The fact that we obtain a simple exponential decay law (3.162) is due to the verysimple nature of this model system. In fact, it can be shown that real systems cannotdecay exponentially on a time scale shorter than the half-time or much longer than thehalf-time. The deviation from exponential decay at short time scales offers the possibil-ity to alter the dynamics of the quantum system by repeated measurements (Zeno andanti-Zeno effect).

Further reading: Time-dependent perturbation theory is covered in every reasonablequantum mechanics text book. Dyson series, S-matrix, Green’s functions, prop-agators etc. are usually covered in books on quantum field theory or many-bodytheory. Models and techniques relevant to (weak) laser-atom interaction are dis-cussed in CLAUDE COHEN-TANNOUDJI, JACQUES DUPONT-ROC, GILBERTGRYNBERG, Atom-Photon Interactions, (Wiley, New-York, 1998) and in FARHADH. M. FAISAL, Theory of Multiphoton Processes, (Plenum, New York, 1987).

3.3.2 Resonantly coupled discrete levels: Rabi-oscillationsLet us consider the Hamiltonian of a two-level system coupled to a laser field in dipoleapproximation,

H (t ) = H0+W (t ) =ωa|a⟩⟨a|+ωb |b ⟩⟨b | − qE z cosωt . (3.163)

The charge of the particle in atomic units is q (i.e., q =−1 for an electron), andωa,ωbare the energies of the states |a⟩ and |b ⟩, respectively. Plugging the Ansatz

|ψ(t )⟩= ca(t )e−iωa t |a⟩+ cb (t )e

−iωb t |b ⟩ (3.164)

into the TDSEi∂t |ψ(t )⟩= H (t )|ψ(t )⟩ (3.165)

and multiplication from the left by ⟨a| and ⟨b |, respectively, leads to

ica(t ) = −1

2E qcb (t ) ei∆t ⟨a|z |b ⟩, (3.166)

icb (t ) = −1

2E qca(t ) e−i∆t ⟨b |z |a⟩. (3.167)


Here, we introduced the detuning

∆=ωa −ωb −ω =ωab −ω, (3.168)

made use of ⟨a|z |a⟩ = ⟨b |z |b ⟩ = 0 (assuming that |a⟩ and |b ⟩ have well-defined par-ity), and neglected the anti-resonant terms ∼ e±i(ωab+ω)t (RWA, see Sec. 3.2) assumingwithout loss of generality thatωab > 0 andω> 0. Setting

qE⟨a|z |b ⟩= E |q⟨a|z |b ⟩|︸︷︷︸ΩR

e−iϕ =ΩR e−iϕ, ΩR= E |q⟨a|z |b ⟩| (3.169)

(3.166) and (3.167) become

ca(t ) = i1

2ΩR ei∆t−iϕ cb (t ), (3.170)

cb (t ) = i1

2ΩR e−i∆t+iϕ ca(t ). (3.171)

The solutions are

ca(t ) =

a1 eiΩt/2+ a2 e−iΩt/2

ei∆t/2, (3.172)

cb (t ) =

b1 eiΩt/2+ b2 e−iΩt/2

e−i∆t/2 (3.173)

where

Ω=qΩ2

R+∆2 . (3.174)

The constants a1, a2, b1, b2 are determined through the initial conditions,

a1 =1

2Ω

(Ω−∆)ca(0)+ΩR e−iϕcb (0)

, (3.175)

a2 =1

2Ω

(Ω+∆)ca(0)−ΩR e−iϕcb (0)

, (3.176)

b1 =1

2Ω

(Ω+∆)cb (0)+ΩR eiϕca(0)

, (3.177)

b2 =1

2Ω

(Ω−∆)cb (0)−ΩR eiϕca(0)

. (3.178)

We thus have

ca(t ) =

ca(0)

cosΩt

2

− i∆

ΩsinΩt

2

(3.179)

+iΩR

Ωe−iϕcb (0) sin

Ωt

2

ei∆t/2,

cb (t ) =

cb (0)

cosΩt

2

+

i∆

ΩsinΩt

2

(3.180)

+iΩR

Ωeiϕca(0) sin

Ωt

2

e−i∆t/2,


which is remarkably complicated considering the fact that we deal only with two states.In the case of exact resonance∆= 0 one has Ω=ΩR. Assuming further that at time

t = 0 the system is in state |b ⟩ so that cb (0) = 1 and ca(0) = 0, Eqs. (3.179) and (3.180)simplify to

ca(t ) = ie−iϕ sinΩRt

2

, (3.181)

cb (t ) = cosΩRt

2

(3.182)

so that the probability to find the system, e.g., in the state |b ⟩ is

wb (t ) = |cb (t )|2 = cos2ΩRt

2

=

1

2[1+ cosΩRt] , (3.183)

i.e., it oscillates with the frequency ΩR. These oscillations are called Rabi oscillations (orRabi floppings).

In first order time-dependent perturbation theory we saw that the depopulationof the initial state at resonance goes ∼ t 2 which corresponds to the first term of theexpansion of cosΩRt . Obviously, perturbation theory is not adequate to describe thepopulation transfer between the two states.

Rabi floppings are employed experimentally to prepare atoms in a certain desiredstate using lasers that are tuned to the transition of interest. If the pulse duration Tp

is chosen such that ΩRTp = π (so-called π-pulses) the, e.g., ground state population istransferred completely to the desired state. As long as the detuning is small compared totransition energies to other states, the two-level approximation is adequate. In the caseof nonvanishing detuning ∆ 6= 0 the population transfer is not complete (i.e., |ca(t )|2never reaches unity).

The Rabi frequency ΩR is proportional to the dipole matrix element [cf. (3.169)].Hence, the higher the field strength, the faster is the population transfer between thestates. However, the so-called AC Stark effect modifies the energies of the two states sothat a laser frequency that equals the unperturbed level spacingωab will be detuned.

As an example, we shall finally calculate the Rabi-frequency ΩR for the resonantlydriven 1s→2p transition in atomic hydrogen for a driving laser of field amplitude E =9 · 108 V/m. We have for the dipole matrix element

⟨a|z |b ⟩=∫

dr r 2∫

dΩφ∗2p0

rY ∗10(Ω)︸︷︷︸2p0

r cosϑ︸︷︷︸z

φ1s

rY00

︸︷︷︸1s

(3.184)

where Y`m(Ω) are the spherical harmonics and Ω is the solid angle (in this context).Using

φ1s = 2r e−r , φ2p0=

r 2

2p

6e−r/2, Y00 =

1p

4π, (3.185)


and

z = r cosϑ = r

È4π

3Y10(Ω), (3.186)

we obtain

⟨a|z |b ⟩= 1p

3

∫dr

r 4

p6

e−3r/2 =256p

2 243' 0.7449. (3.187)

In SI units one has

ΩR=256p

2 243

a0e E

ħh ' 5.4 · 1013 s−1 (3.188)

(with a0 the Bohr radius). For comparison: one atomic frequency unit corresponds to4.1 ·1016 s−1, and the laser frequency at 800 nm is 2.3×1015 s−1. Hence, Rabi-flopping isstill “slow” for the considered field strength both on the inner-atomic time scale and onthe time scale of the laser period.

An atom driven resonantly emits at the frequenciesω andω±ΩR (Mollow-spectrum,fluorescence). Corrections stemming from the anti-resonant terms can be taken intoaccount in a perturbative manner (Bloch-Siegert shifts). It is common to introduce in thecontext of Rabi-floppings so called dressed states. These states form a new basis in whichthe Hamiltonian becomes diagonal at resonance. We postpone the introduction of thedressed states until the discussion of the Floquet approach.

Further reading: Rabi-oscillations are discussed in all Quantum Optics text books.We followed MARLAN O. SCULLY and M. SUHAIL ZUBAIRY, Quantum Optics,(Cambridge University Press, Cambridge, 1997). A historic paper on the sub-ject (employing an interesting application of continued fractions) is S.H. AUT-LER AND C.H. TOWNES, Stark Effect in Rapidly Varying Fields, Phys. Rev. 100,703 (1955). A recent review is ULRICH D. JENTSCHURA and CHRISTOPH H.KEITEL, Radiative corrections in laser-dressed atoms: formalism and applications,Annals of Physics 310, 1 (2004).

3.4 Atoms in strong, static electric fieldsAlthough we assume the external electric field to be static in this Section, the followinganalysis is useful for atoms in laser fields too, as will become clear below. The Hamilto-nian governing an electron moving in a Coulomb potential −Z/r and a static electricfield E = Eez reads

H =p2

2+Veff, Veff =−

Z

r+ E z . (3.189)

3.4. ATOMS IN STRONG, STATIC ELECTRIC FIELDS 51

Veff

V

z

Ez

−Z/r"over barrier"

"tunneling"

Figure 3.4: Effective potential Veff in field direction. The unperturbed Coulomb potential andthe field potential are also shown separately. Depending on the initial (and possibly Stark-shifted)state, the electron may either escape via tunneling or classically via “over-barrier” ionization.

The effective potential Veff describes a tilted Coulomb potential (see Fig. 3.4). A per-turbative treatment of the problem can be found in almost all quantum mechanics oratomic physics text books (Stark effect). In first order (linear Stark effect) the degener-acy with respect to the angular quantum number ` is removed while the degeneracy inthe magnetic quantum number m is maintained. The non-degenerate ground state isonly affected in second order (quadratic Stark effect). It is down-shifted since the poten-tial widens in the presence of the field. In the case of the hydrogen atom (Z = 1) thisdown-shift is given by∆E =−9E2/4.

Let us first point out that, strictly speaking, there exist no discrete, bound statesanymore even for the tiniest electric field. This is because even a very small field givesrise to a potential barrier (see Fig. 3.4) through which the initially bound electron maytunnel. The electric field couples all bound states to the continuum and thus, as we havelearnt in Section 3.3.1, all discrete states become resonances with a finite line width.Mathematically speaking, the spectrum of the Hamiltonian (3.189) is unbound frombelow. However, since the barrier for small fields is far out, the probability for tunnelingis extremely low (note that the tunneling probability goes exponentially down with thedistance to be tunneled) and the states are “quasi-discrete”.

A strong increase in the ionization probability is expected when the electron caneven escape classically, that is, when it does not have to tunnel. In a zeroth order ap-proximation (which, in fact, is wrong for hydrogen-like ions, as will be discussed below)this so-called critical field Ecrit may be estimated as follows: assuming that the initial en-ergy of the electron does not change (i.e., Stark effect negligible), classical over-barrier


ionization sets in when the barrier maximum coincides with the energy level of theelectron. The position of the barrier is (for E > 0) located at

zbarr =−È

Z

E(3.190)

and the energy at the barrier maximum is

Vbarr =−2p

ZE . (3.191)

Hence, if we restrict ourselves to the ground state of hydrogen-like ions, we require that

E =−Z2

2!=−2

ÆZEcrit ⇒ Ecrit =

Z3

16. (3.192)

Because of the strong Z -dependence of the critical field even with the most intense lasersavailable today it is not possible to fully strip heavy elements (see Problem 3.5 below).For hydrogen-like ions Eq. (3.192) even underestimates the critical field by more than afactor of two, as will be shown soon.

The Schrödinger equation with the Hamiltonian (3.189) separates in parabolic coor-dinates (ξ ,η,ϕ),

ξ = r + z, η= r − z, r =1

2(ξ +η), z =

1

2(ξ −η), 0≤ ξ ,η. (3.193)

Here, r is the radial coordinate, and ϕ is the usual azimuthal angle (as in spherical, polaror cylindrical coordinates). Cuts of contours of constant ξ and η in the x z -plane areshown in Fig. 3.5. The Hamiltonian in parabolic coordinates reads

H =− 2

ξ +η

∂ξ (ξ ∂ξ )+ ∂η(η∂η)

− 1

2ξ η∂ 2ϕ− 2Z

ξ +η+ E

ξ −η2

. (3.194)

Plugging the AnsatzΨ= f1(ξ ) f2(η)e

imϕ (3.195)

into the Schrödinger equation EΨ = HΨ and multiplying by (ξ + η)/2 leads to anequation that can be decoupled into

d

dξ

ξ

d f1

dξ

+E

2ξ − m2

4ξ− E

4ξ 2

f1+Z1 f1 = 0, (3.196)

d

dη

η

d f2

dη

+E

2η− m2

4η+

E

4η2

f2+Z2 f2 = 0 (3.197)

where Z1, Z2 are separation constants fulfilling

Z1+Z2 = Z . (3.198)


0.1

1

2

3

4

5

6

7

8

9

0.1

1

2

3

4

5

6

7

8

9

Figure 3.5: Illustration of parabolic coordinates. Cuts of contours ξ = const. (dashed, valuesgiven next to the lines) and η= const. (solid) in the x z-plane (azimuthal symmetry with respectto the z-axis!).

Division by 2ξ and 2η, respectively, yields the two Schrödinger equations−1

2

d2

dξ 2+

1

ξ

d

dξ− m2

4ξ 2

− Z1

2ξ+

E

8ξ

f1 =

E4

f1, (3.199)

−1

2

d2

dη2+

1

η

d

dη− m2

4η2

− Z2

2η− E

8η

f2 =

E4

f2 (3.200)

which have the same shape like two Schrödinger equations in cylindrical coordinatesand the potentials

Vξ =−Z1

2ξ+

E

8ξ , Vη =−

Z2

2η− E

8η. (3.201)

Both potentials have a Coulombic part and a linear contribution, like Veff in (3.189).However, because ξ ,η ≥ 0, the potential Vξ has only bound states (we assume E > 0).The potential Vη instead displays a barrier. Hence, in parabolic coordinates ionizationhappens with respect to the η coordinate while the electron is expected to remain ratherconfined in ξ . The consequences of this in Cartesian coordinates can be understoodwith the help of Fig. 3.5: confinement to a region ξ < ξmax implies preferred electronemission toward negative z with a lateral spread that can be estimated by the confiningcontour ξmax. The potentials Vξ and Vη are called “uphill” and “downhill potential”,respectively. They are illustrated in Fig. 3.6. Given an energy E one finds a sequenceof Z1 for which the solution of the Schrödinger equation in ξ , Eq. (3.199), leads tonormalizable bound states. This sequence can be labelled by the number of nodes inf1 for ξ > 0, n1 = 0,1,2, . . . (note that Z1 < 0 is also possible). The second equation


V

ξ,η

Vξ

Vη

"downhill"(ionization)

(bound motion)"uphill"

Figure 3.6: Illustration of the potentials Vξ and Vη [Eqs. (3.201)]. The “uphill potential” Vξ

(for E > 0) supports only bound states while the “downhill potential” Vη displays a barrierthrough which the electron may tunnel.

(3.200) has to be solved for Z2 = Z−Z1 and the same energy E . This is possible becausethe corresponding Hamiltonian Hη [i.e., the square bracket in (3.200)] is neither boundfrom below nor from above.

In the field-free case E = 0 the two Schrödinger equations (3.199) and (3.200) areidentical, and the relation between the “usual” principal quantum number n and theparabolic quantum numbers n1, n2 are as follows:

ni +|m|+ 1

2= n

Zi

Z, n1+ n2+ |m|+ 1= n, i = 1,2. (3.202)

Instead of working directly with the parabolic coordinates ξ and η, one can performan additional, simple coordinate transformation

u =p

2ξ , v =p

2η (3.203)

which, after multiplication of the new Schrödinger equation by (u2+ v2)/4, and plug-ging in the Ansatz Ψ=Φu(u)Φv(v)e

imϕ yields

−1

2

1

u∂u(u∂u)−

m2

u2

+

1

2Ω2u2− 1

4g u4

Φu = Z1Φu , (3.204)

−1

2

1

v∂v(v∂v)−

m2

v2

+

1

2Ω2v2+

1

4g v4

Φv = Z2Φv , (3.205)


where, again, Z = Z1+Z2 is used, and Ω and g are defined as

1

2Ω2 =−E

4, E ≤ 0, g =−E

4. (3.206)

The Schrödinger equations (3.204) and (3.205) have the shape of two-dimensional oscil-lators (with radial coordinates u and v, respectively) of frequency Ω and with a quarticperturbation that is proportional to the electric field. In the field-free case, the Coulombproblem is mapped to two two-dimensional oscillators where, however, the energy as-sumes the role of the oscillator frequency, and the nuclear charge (splitted into Z1 andZ2) assumes the role of the energy. The transformation to the coordinates (u, v,ϕ) cor-responds to the Kustaanheimo-Stiefel transformation.

Let us now evaluate an improved critical field for the case of hydrogen-like ions. Asmentioned above, formula (3.192) underestimates the critical field by more than a factorof two.

In the unperturbed case (g = 0) and for m = 0 (groundstate) the solutions to (3.204)and (3.205) are Gaussians. We therefore use

Φu(u) =r

au

πe−au u2/2 (3.207)

(and analogous for v) as trial functions with parameters au and av . The “energy” then is

Z1(g ) = 2π∫ ∞

0du uΦ∗u HuΦu =

1

au

Ω2

2−

a2u

2

!+ au −

g

2a2u

. (3.208)

Minimizing this energy yields up to first order in g

au =Ω

1− g

Ω3

, av =Ω

1+

g

Ω3

. (3.209)

The oscillator “energies” are

Z1(g ) = Ω

1− g

2Ω3

, Z2(g ) = Ω

1+

g

2Ω3

. (3.210)

Note that this is consistent with the fact that the linear Stark effect vanishes for theground state since

Z1(g )+Z2(g ) = 2Ω (3.211)

is independent of the field g . Since Z1+Z2 = Z we have Z = 2Ω which is [see (3.206)]equivalent to E =−Z2/2, as it should for the ground states of hydrogen-like ions.

In physical coordinates the variationally determined wave function for, e.g., hydro-gen (Z = 1) reads

ΨH(r ) =1− 4E2

pπ

e−r e−2E ·r , (3.212)


i.e., the unperturbed wave function is multiplied by a “deformation factor”.If E > 0 we have g < 0 and vice versa. Let us assume g > 0 so that the u-oscillator

displays a barrier while the v-oscillator does not. The barrier is located at ubarr =Ω/pg

and the energy at the barrier maximum isΩ2u2barr/2− g u4

barr/4=Ω4/(4g ). Claiming that

at the critical field strength the energy of the u-state coincides with the barrier-energygives us

Z1(g ) = Ω−g

2Ω2=Ω4

4g⇒ gcrit =Ω

3(1− 2−1/2)' 0.3Ω3 (3.213)

which translates [using (3.206)] to

E H−likecrit = (

p2− 1)E 3/2. (3.214)

In the case of atomic hydrogen one obtains E H−likecrit = 0.147 instead of the 0.0625 pre-

dicted by (3.192). The wrong prediction of (3.192) is due to the erroneous assumptionthat one can consider the electron motion in z-direction and in lateral direction as beingindependent. Instead, the problem separates in parabolic coordinates! However, sincethe “exceptional” symmetry of hydrogen-like ions is broken in many-electron atoms,the over-barrier formula given (and to be derived) in Problem 3.6 below [Eq. (3.241)] isuseful and quite accurate for many practical applications.

3.4.1 Tunneling ionizationGoing one step beyond a classical over-barrier analysis amounts to take tunneling intoaccount. This can be done in a semi-classical way. Let us consider the tunneling of theelectron in atomic hydrogen through the barrier of the “downhill potential” in Fig. 3.6.We assume the electron is initially in the 1s ground state. The Schrödinger equation(3.200) reads for m = 0, Z2 = 1/2, E =−1/2

−1

2

d2

dη2+

1

η

d

dη

− 1

4η− E

8η

f2(η) = −1

8f2(η). (3.215)

Substitutingχ (η) =pη f2(η) (3.216)

yields the Schrödinger equation

∂ 2χ

∂ η2+−1

4+

1

2η+

1

4η2+

1

4Eηχ = 0. (3.217)

Comparison with

−1

2

∂ 2χ

∂ η2+V (η)χ = εχ (3.218)


1/ E

η 0

Figure 3.7: Plot of the potential V (η) [Eq. (3.219)] for E = 0.0075. The tunnel “exit” η1 forsufficiently low fields E is in good approximation given by η1 ' 1/E . The matching point η0 islocated inside the barrier where 1 η0 1/E holds.

shows that we effectively deal with one-dimensional motion of an electron in the poten-tial

V (η) =−1

2

1

2η+

1

4η2+

1

4Eη

(3.219)

with total energy

ε=−1

8. (3.220)

The potential V (η) is of the form depicted in Fig. 3.7.We now match at a position η0 inside the barrier,

1 η0 1/E (3.221)

the “left” quasi-classical wave function

χleft(η) =−iCp|p|

exp

∫ η

η0

p(η′)dη′

!(3.222)

with the “right” quasi-classical outgoing wave function

χright(η) =Cp

pexp

i∫ η

η0

p(η′)dη′− iπ/4

!(3.223)


where

p(η) =Æ

2[ε−V (η)] =

s−1

4+

1

2η+

1

4η2+

1

4Eη. (3.224)

The semi-classical approximation breaks down at the classical turning point η1 ' 1/Esince p(η1) = 0. In general, semi-classical wave functions are accurate as long as the deBroglie wave length

λdB =2πħh

p(3.225)

is small compared to the length scale characterizing changes in the potential (i.e., thepotential should be sufficiently “flat”). For vanishing momentum p the de Brogliewave length is infinite so that the semi-classical approximation necessarily breaks down.However, for the calculation of the probability flux out of the potential the disagree-ment between the semi-classical wave function and the exact wave function in a narrowregion around the classical turning point η1 plays no role.

For the determination of the normalization constant C we set the left wave functionat position η0 equal to the unperturbed wave function so that

pη0

1pπ

e−(ξ+η0)/2 =− iCp|p0|

(3.226)

with p0 = p(η0). The “uphill” coordinate ξ appears as a parameter here which willbe integrated out later on; in other words: the wave function is assumed to retain itsground state shape with respect to ξ (i.e., the Stark effect is neglected). We obtain forthe right wave function

χright(η,ξ ) = i

sη0|p0|π p(η)

e−(ξ+η0)/2 exp

i∫ η

η0

p(η′)dη′− iπ/4

!(3.227)

so that

|χright(η,ξ )|2 = η0|p0|π p(η)

exp

−ξ −η0+ 2ℜ

i∫ η

η0

p(η′)dη′!

(3.228)

where ℜ denotes the real part. Because of (3.221) we can expand p(η) in ε= 1/η,

p(η) =

1

2

pEη− 1+

1

ηp

Eη− 1+ · · ·

!outside barrier, η > η1

1

2

ip

1− Eη+1

iηp

1− Eη+ · · ·

!inside barrier, η < η1

. (3.229)

Since Eη0 1 it is sufficient to take |p0| = 1/2. In order to keep the leading termsdependent on E in the prefactor as well as in the exponent, we set in the denominator


of the prefactor in (3.228) p(η) = (p

Eη− 1)/2. In the exponent we integrate inside thebarrier and have to keep both terms in the expansion (3.229). We thus obtain

|χright(η,ξ )|2 = η0

πp

Eη− 1e−(ξ+η0) exp

−∫ η1

η0

pEη− 1− 1

ηp

Eη− 1

dη

!.

(3.230)The integral can be solved. Using η1 ' 1/E and η0E 1 we obtain for the probabilitydensity outside the barrier

|χright(η,ξ )|2 = 4e−ξ

πEp

Eη− 1e−2/(3E). (3.231)

The total probability current through a plane perpendicular to the z-axis is

w =∫ ∞

0| f1(ξ ) f2(η)|2vz2πρdρ (3.232)

where f1(ξ ), f2(η) are the wave functions introduced in (3.195), vz is the velocity inz-direction and ρ is the radial cylindrical coordinate. The ξ -dependent part | f1|2 isincluded in our |χright(η,ξ )|2 so that

w =∫ ∞

0

|χright(η,ξ )|2η

vz2πρdρ. (3.233)

With z = (ξ −η)/2'−η/2 for small ξ and large η, we estimate for vz

1

2v2

z + E z ' 1

2v2

z −1

2Eη ⇒ vz '

pEη− 1 (3.234)

so that

w =∫ ∞

0


pEη− 1 2πρdρ. (3.235)

Finally, with

ρ=Æξ η ⇒ dρ= d

Æξ η=

1

2

ηpξ η

dξ +1

2

ξpξ η

dη' 1

2

Èη

ξdξ (3.236)

(where the last step again follows from η ξ ) we arrive at

w =∫ ∞

0


pEη− 1 2π

Æξ η

1

2

Èη

ξdξ =

∫ ∞

0

4

Ee−2/(3E) e−ξ dξ (3.237)

⇒ w =4

Ee−2/(3E). (3.238)


1/16 0.147

Figure 3.8: The LANDAU-rate (3.238) vs field strength E . The vertical lines indicate the (herewrong) over-barrier field strength (3.192) (1/16, dashed) and the correct (3.214) 0.147 (solid),respectively.

This is the LANDAU-rate for tunneling ionization of atomic hydrogen from the groundstate. Its derivation is given as an exercise in the Quantum Mechanics volume of LAN-DAU & LIFSHITZ! It has been checked numerically that (3.238) is exact in the limit oflow field strengths E while it overestimates ionization as the over-barrier field strengthis approached. Figure 3.8 shows w vs the field strength E .

It is, of course, desirable to extend the above calculation to laser fields, to morecomplex atoms, and to higher field strengths. All directions have been pursued, andthere exists a vast amount of literature on tunneling ionization (see “Further reading”below). Here we have restricted ourselves to the “generic” case of atomic hydrogenwhere the typical tunneling exponent ∼ E−1 already arises, as derived above.

The most commonly used tunneling formula is the so-called ADK-formula (namedafter AMMOSOV, DELONE, AND KRAINOV although it was, even more generally, de-rived much earlier by POPOV):

w =C 2n∗ f (`, m)

Z2

2n∗2

s3E n∗3

πZ3

2Z3

En∗3

2n∗−|m|−1

exp

− 2Z3

3n∗3E

(3.239)

with

Cn∗ = 2e

n∗

n∗

(2πn∗)−1/2, f (`, m) =(2`+ 1)(`+ |m|)!2|m||m|! (`− |m|)!

, n∗ =Z

Æ2Eip

,

(3.240)and e = 2.71828. n∗, `, and m are the (effective) quantum numbers of the initial statewith ionization potential Eip. The behavior of w as a function of E is again dominated

3.5. ATOMS IN STRONG LASER FIELDS 61

by the exponent ∼ E−1. The varying laser field is taken into account by cycle-averagingE(t ). Hence, by E in (3.239) the laser electric field amplitude is meant. The expressionfor circular polarization is obtained by multiplying (3.239) with (πZ3/3En∗3)1/2. Theexpression is supposed to be increasingly accurate as E Z3/16n4, n∗ 1, and ` n∗.Experimentalists are used to compare their measured results for ion yields with thecorresponding ADK-prediction. Often, the agreement is satisfactory if one allows foran adjustment of the laser intensity (multiplication of the latter by factors between, say,0.5 and 2).

Problem 3.5 Estimate the laser intensity needed to fully strip Uranium.

Problem 3.6 Show (using an over-barrier estimate) that the (classically) expected ap-pearance intensity Iapp,Z for a charge state Z of an atom with ionization potentialEip,Z is given by

Iapp,Z =E 4

ip,Z

16Z2. (3.241)

Further reading: The hydrogen atom in parabolic coordinates is treated in manyAtomic Physics and Quantum Mechanics textbooks (i.e., LANDAU & LIF-SHITZ’S Quantum Mechanics volume). The correct over-barrier field strength ofhydrogen-like ions is calculated in D. BAUER, Ejection energy of photoelectrons instrong field ionization, Phys. Rev. A 55, 2180 (1997). A review of tunneling ion-ization is given by V.S. POPOV in Tunnel and multiphoton ionization of atoms andions in a strong laser field (Keldysh theory), Physics Uspekhi 47, 855 (2004).

3.5 Atoms in strong laser fieldsThere are at least three different energy scales (and the related time scales) in the inter-action physics of atoms in strong laser fields: (i) the ionization potential Eip = |E |, (ii)the photon energyω, and (iii) the ponderomotive energy Up [see Eq. (2.14)]. The pulseduration may introduce an additional laser-related time-scale while the energy spectrumof the atom, through typical transitions, could introduce an additional species-relatedtime-scale. If one ignores the two latter parameters, the atomic species enter through|E | only.

In caseω> |E | Up or |E |>ωUp perturbation theory in lowest nonvanishingorder (LOPT) can be applied. In contrast, when with the increasing laser intensitythe regime |E | > Up > ω is reached, non-perturbative effects such as above-threshold


ionization (ATI) or channel-closing take place. This regime is commonly referred to as(nonperturbative) multiphoton ionization (MPI). Finally, increasing the intensity further(or decreasing the photon energy) one arrives at Up > |E |>ω. Translated into the time-domain this implies that both the inner-atomic time-scale and the ionization dynamicsare “fast” with respect to a laser period. If this is the case, a quasi static field ionizationpicture may be applied where, at the instant of ionization t ′, the electron “feels” aneffective potential which is the sum of the Coulomb (or effective core) potential and theinstantaneous potential of the laser, as depicted in Fig. 3.4. If the field reaches the criticalfield estimated above, the electron may escape classically over the barrier [over-barrieror barrier suppression ionization (OBI) and (BSI), respectively]. Below the critical fieldstrength the electron can escape via tunneling through the barrier (tunneling ionization).

MPI and tunneling are defined byUp

|E | < 1 (MPI),Up

|E | > 1 (tunneling). (3.242)

KELDYSH introduced a parameter γ as the ratio of “tunneling time” and laser period,or, expressed in frequencies, γ = ω/ωt with the tunneling frequency ωt estimated byωt = E/

p2|E | where E is the electric field |E |. Hence, one has the Keldysh parameter

γ =ω

ωt

=

√√√√ |E |2Up

, (3.243)

and conditions (3.242) can be also stated as (neglecting a prefactorp

2 which does notmatter here)

γ > 1 (MPI), γ < 1 (tunneling). (3.244)Numerous strong laser-atom experiments operate around γ ' 1 or at γ > 1 and are thusnot in the tunneling domain. Taking, for instance, the case of atomic hydrogen in an800 nm and 1014 W/cm2 laser pulse one finds γ ' 1.1. This is a typical value for ATImeasurements.

What are the differences between the ionization dynamics in the MPI and in thetunneling domain? Since in the tunneling regime the process is fast compared to alaser period, significant ionization occurs during a single half laser cycle, predominantlyaround the electric field maximum because the barrier is lowest then. Furthermore, intunneling ionization the quiver amplitude E/ω2 of the freed electron in the laser fieldis large compared to the atomic dimension while in MPI this is not the case. This hasconsequences for the rescattering dynamics which is responsible for various effects, suchas the ATI plateau, high-harmonic generation, and nonsequential ionization, as will bediscussed below.

3.5.1 Floquet formalismWhen we discussed atoms in strong, static electric fields in Sec. 3.4 we made use of thefact that in the tunneling regime the laser field may be considered “quasi-static” since


the inner-atomic dynamics is much faster than a laser period. What else could we takeadvantage of in the treatment of atoms in laser fields in order to render it tractable? Ifthe laser pulse duration is long, i.e., if the pulse contains many laser cycles, we may ingood approximation consider it infinitely long. As a consequence the Hamiltonian willthen be periodic,

H (t ) = H0+W (t ), W (t +T ) = W (t ), T =2π

ω, (3.245)

and the TDSE will be a partial differential equations with periodic coefficients. Thistype of problem has been studied by FLOQUET more than 120 years ago (see “Furtherreading” below). The Floquet theorem tells us that the TDSE

H (t )|Ψ(t )⟩= 0, H (t ) = H (t )− i∂t (3.246)

has solutions of the form

|Ψ(t )⟩= e−iεt |Φ(t )⟩, |Φ(t +T )⟩= |Φ(t )⟩, (3.247)

i.e., the wave function |Φ(t )⟩ is periodic (while |Ψ(t )⟩ itself is not!). Note that the Blochtheorem used in solid state physics to treat particle motion in periodic potentials is theFloquet theorem applied to spatially periodic systems. Inserting (3.247) into (3.246)leads to the eigenvalue equation

H (t )|Φ(t )⟩= ε|Φ(t )⟩. (3.248)

ε is called quasi-energy. Note that if ε and |Φ(t )⟩ solve (3.248), then also

ε′ = ε+mω, |Φ(t )⟩′ = eimωt |Φ(t )⟩, m ∈Z (3.249)

do so. Let |α⟩ be the solution of the unperturbed problem

H0|α⟩= E 0α|α⟩. (3.250)

Because of the periodicity of |Φ(t )⟩ we can expand

|Ψ(t )⟩= e−iεt |Φ(t )⟩= e−iεt∞∑

n=−∞

∑α

Φ(n)α|α⟩e−inωt (3.251)

where the expansion coefficients Φ(n)α

are time-independent. Inserting (3.251) into(3.246) gives ∑

nα

[H (t )− ε− nω]Φ(n)α|α⟩e−inωt = 0. (3.252)

Multiplying from the left with ⟨β|, eimωt , and integrating T −1∫ T

0 dt yields∑nα

[⟨β|H (m−n)|α⟩− (ε+mω)δnmδαβ]Φ(n)α= 0 (3.253)


with the time-independent Hamiltonian

H (m−n) =1

T

∫ T

0H (t )ei(m−n)ωt dt . (3.254)

Introducing the Floquet state

|αn⟩= |α⟩⊗ |n⟩, ⟨t |n⟩= einωt (3.255)

we can recast (3.253) into∑nα

[⟨βm|HF|αn⟩− ε⟨βm|αn⟩]Φ(n)α= 0 (3.256)

where HF is the Floquet Hamiltonian whose matrix elements read

⟨βm|HF|αn⟩= ⟨β|H (m−n)|α⟩−mωδmnδαβ. (3.257)

Hence, we obtain the eigenvalue equation

∑nα

[⟨βm|HF|αn⟩Φ(n)α= εΦ(m)

β(3.258)

or, in matrix notation,HFΦ= εΦ. (3.259)

Here, HF and Φ are an infinite matrix and an infinite vector, respectively. In practice,the size of the system has to be truncated, of course.

In order to illustrate the Floquet approach let us return to the two-state problem ofSec. 3.3.2 where the Hamiltonian reads

H (t ) =ωa|a⟩⟨a|+ωb |b ⟩⟨b |︸︷︷︸H0

− qE z cosωt︸︷︷︸−W (t )

, ωa >ωb . (3.260)

We thus have

H (n) =1

T

∫ T

0dt H (t )einωt

= H0δn0−qE z

2T

∫ T

0dt einωt

eiωt + e−iωt

= H0δn0−qE z

2(δn,−1+δn1), (3.261)

i.e., a tridiagonal Hamiltonian in the “photon subspace”. For (3.257) we obtain

⟨βm|HF|αn⟩=⟨β|H0|α⟩−mωδαβ

δnm −

1

2qE⟨β|z |α⟩(δm,n−1+δm,n+1) (3.262)


so that (using ⟨β|H0|α⟩=ωαδαβ) (3.258) reads

∞∑n=−∞

|b ⟩∑|α⟩=|a⟩

(ωα−mω)δαβδmn −

1

2qE⟨β|z|α⟩(δm,n−1+δm,n+1)

Φ(n)α= εΦ(m)

β.

(3.263)Defining

A=−1

2qE⟨a|z |b ⟩=−1

2ΩRe−iϕ, E (m)

α=ωα−mω, (3.264)

the corresponding matrix equation has the following structure (e.g., around m = 0):

E (−1)a A

E (−1)b

A∗

A E (0)a AA∗ E (0)

bA∗

A E (1)aA∗ E (1)

b

Φ(−1)aΦ(−1)

bΦ(0)aΦ(0)

bΦ(1)aΦ(1)

b

= ε

Φ(−1)aΦ(−1)

bΦ(0)aΦ(0)

bΦ(1)aΦ(1)

b

. (3.265)

In Sec. 3.3.2 we applied the rotating wave approximation (RWA). In the Floquet frame-work this corresponds to allow only for transitions between Floquet states |αn⟩ and|βm⟩ of equal energies E (n)

α= E (m)

β. Hence,

E (n)a =ωa − nω !=ωb −mω = E (m)b

(3.266)

⇒E (n)a −E(m)b=∆=ωa −ωb︸︷︷︸

ωab

− (n−m)︸︷︷︸!=1

ω (3.267)

where ∆ is the detuning (3.168), and we are only interested in one-photon transitionsbetween |a⟩ and |b ⟩ (so that n − m = 1). It would have been smarter to choose theopposite sign for n in the Fourier expansion (3.251) since then the energy would be,more intuitively, the atomic energy plus the number of photons (as if we quantized theelectromagnetic field). However, in most of the literature on Floquet theory the signconvention is like in (3.251). Equation (3.265) now reduces to the 2× 2 equation

E (n−1)

bA∗

A E (n)a

Φ(n−1)

bΦ(n)a

= εΦ(n−1)

bΦ(n)a

. (3.268)

The eigenvalues are

ε(n)1,2 =1

2(ωa +ωb )+ω

1

2− n± 1

2

qΩ2

R+∆2. (3.269)


ωa

ωb

ωa ωb( + )/2

n=

n=

n=

n= −1

0

1

2

Ω

Ω

Ω

Ω

Figure 3.9: Illustration of Eq. (3.269) for∆= 0. The coupling of the two unperturbed levels band a to the laser field gives rise to an infinite manifold of pairs of field-dressed levels (labelledby n). Because of (3.249) all the different ns are equivalent. The field dressed levels are separatedby the energy (3.174) (which equals ΩR for the vanishing detuning considered here).

The first term is the energy half way between |b ⟩ and |a⟩, the second term is the energyof the “photon field”, and the third term gives rise to two levels, separated by the fre-quency (3.174). Figure 3.9 illustrates the situation for vanishing detuning ∆ = 0. Theenergy levels (3.269) are called field dressed energy levels.

Let us finally calculate the field dressed states for our two-state problem in RWA andvanishing detuning. Because of (3.249) we are free to choose, say, n = 1. We also can,without loss of generality, set ωb = 0. Vanishing detuning ∆= 0 then implies ωa =ω.Moreover we assume A to be real. Then

ε1,2 =±1

2ΩR (3.270)

and the eigenvalue problem reduces to

0 AA 0

Φ(0)

bΦ(1)a

= ε1,2

Φ(0)

bΦ(1)a

. (3.271)

The matrix

M=1p

2

−1 11 1

=M−1 (3.272)

diagonalizes (3.271) and yields

1

2

ΩR 00 −ΩR

Ψ−Ψ+

= ε1,2

Ψ−Ψ+

(3.273)

where

Ψ− =1p

2(Φ(1)a −Φ

(0)b), Ψ+ =

1p

2(Φ(1)a +Φ

(0)b) (3.274)

are the field-dressed states or Floquet states.


3.5.2 Non-Hermitian Floquet theory

As long as we are dealing only with discrete states the quasi energy ε is real. If, onthe other hand, we allow for transitions into the continuum, e.g., via (multiphoton)ionization, the quasi-energies become complex,

ε= ε0+∆ε− iΓ

2. (3.275)

Here, ε0 is the unperturbed energy,∆ε is the AC Stark shift, and Γ is the ionization rate[cf. Eq. (3.162)]. One may wonder why a Hermitian Floquet Hamiltonian should yieldcomplex eigenvalues. The reason for complex quasi-energies lies in the boundary con-ditions. Decaying dressed bound states or dressed resonances must fulfill the so-calledSiegert boundary conditions. Instead of explicitly taking these boundary conditions intoaccount one may apply the complex dilation (also called complex scaling) method.

Let us write (3.251) in position space representation as

Ψ(r , t ) = e−iεtΦ(r , t ) = e−iεt∞∑

n=−∞e−inωt

∑N LM

Φ(n)N LM

1

rS (κ)N L(r )YLM (Ω) (3.276)

where S (κ)N L are Sturmian functions, YLM (Ω) are spherical harmonics, N LM are principal,angular momentum, and magnetic quantum number, respectively, and κ is a scalingparameter which may be complex. Sturmian functions proved useful in calculationsinvolving Coulomb potentials because they have the following properties (independentof κ):

∫ ∞

0SN L

1

rSN ′L dr = 0 for N ′ 6=N (3.277)

∫ ∞

0SN LSN L dr = 1, (3.278)

and their overlap matrix is tridiagonal. Note that none of the Sturmian functions underthe integrals in (3.277), (3.278) have to be taken complex conjugated.

Sturmian functions can be expressed in terms of Laguerre polynomials (note thatthe definition of the Sturmians is not unique in the literature),

S (|κ|)N L (r ) =

s|κ|(N − 1)!

(N + L)(N + 2L)!(2|κ|r )L+1e−|κ|r L2L+1

N−1 (2|κ|r ) (3.279)

where we specialized on a real and positive κ= |κ| for the moment.Complex scaling amounts to the transformation of the position space coordinate

r → reiθ (3.280)


where θ is a real angle. Imagine we express the matrix elements of the non-HermitianFloquet Hamiltonian in (3.258) in position space representation. The complex scaledFloquet Hamiltonian is obtained as

HF(θ)(r ) = HF(e

iθr ). (3.281)

Transforming the Floquet Hamiltonian in this manner turns out to be equivalent tocalculate the matrix elements of HF using the complex Sturmians

S (κ)N L(r ) =

s−iκ(N − 1)!

(N + L)(N + 2L)!(−2iκr )L+1eiκr L2L+1

N−1 (−2iκr ) (3.282)

if the rotation angle θ and the complex scaling parameter κ are related through

θ=π

2− argκ. (3.283)

The Siegert boundary condition will be automatically fulfilled if κ lies within the firstquadrant of the complex κ-plane and is chosen “properly” (see example in Fig. 3.11).

So what does complex scaling do to the energy spectrum? Note that κ is, dimension-ally, a momentum. From (3.283) can be inferred that the angle of κ with the imaginaryaxis of the complex κ-plane is θ. Because energy is∼ κ2 we see from (3.283) that energyeigenstates (previously all on the real energy-axis, of course) will be rotated by 2θ intothe lower half of the complex energy plane. In the absence of the perturbation and if κis chosen properly, only the continuum states are rotated (about the continuum thresh-old) while the bound states remain on the real axis. However, if the perturbation (thelaser field) is present, also the formerly discrete, real levels acquire an imaginary contri-bution to their quasi-energy ε, namely the −iΓ/2 in Eq. (3.275). This is illustrated inFig. 3.10.

We mentioned above that κ has to be chosen “properly”. What do we mean bythat? The modulus |κ| should be chosen such that the states of interest are well rep-resented with as few as possible Sturmian functions in the numerical basis set. SinceS (|κ|)N L (r ) ∼ e−|κ|r , |κ| is obviously related to the width of the states in position space (ifwe are interested in the shift of the H(1s) state, for instance, |κ| close to one would bea reasonable choice). argκ is directly related to θ [cf. Eq. (3.283)]. It turns out that thequasi-energies become quasi-stationary up to high accuracy (i.e., up to 6 or more dec-imal places) within a certain θ-interval. An illustrative example is shown in Fig. 3.11.

Figure 3.12 shows how resonances (where the photon energy fits exactly with a tran-sition energy) are related to avoided crossings of the dressed energies. Let us consider anexperiment where we drive atomic hydrogen with a laser frequency around the 1s↔2p-resonance atω =−1/8− (−1/2) = 0.375. Let us first keep the driving field strengthvery low so that the Rabi-frequency is small. From Fig. 3.9 we then expect at reso-nance the two dressed states being very close to one of the unperturbed states. In fact,


x x x x xx2θ Re

Im ε

ε

discrete states

...

continuumcontinuumthreshold

xRe

Im ε

εx x xxx

(b)(a)

continuumnew

threshold

Figure 3.10: Illustration of the effect of complex scaling on the eigenenergies. Without per-turbation (a) the discrete states remain on the real energy-axis while the continuum is rotatedby 2θ about the continuum threshold into the lower half of the complex energy plane. Withperturbation (b) the discrete states acquire a negative imaginary contribution −iΓ/2 where Γ isthe ionization rate. They may also be shifted horizontally (∆ε, AC Stark effect).

Figure 3.11: Quasivariational determination of an optimal value of θ in the DC Stark case(E = F = 0.1, ω = 0) in the complex energy-plane. In the case of N = 10 basis functions (peratomic symmetry) a stationary point near θ = 0.4 is found. [From A. MAQUET, SHIH-I CHU,and WILLIAM P. REINHARDT, Stark ionization in dc and ac fields: An L2 complex-coordinateapproach, Phys. Rev. A 27, 2946 (1983).]


Figure 3.12: Real part of the quasi-energy ε for laser-driven atomic hydrogen as a function ofthe laser frequency ω and for three different driver amplitudes E = F , as given in the plot.Avoided crossings occur at the 1s ↔ 2p-resonance frequency ω = 0.375. The separations ofthe quasi-energies at resonance (indicated by the colored arrows) equal the corresponding Rabi-frequencies. [From A. MAQUET, SHIH-I CHU, and WILLIAM P. REINHARDT, Stark ionizationin dc and ac fields: An L2 complex-coordinate approach, Phys. Rev. A 27, 2946 (1983), coloredarrows added by the author of these Lecture Notes.]

Fig. 3.12 explains how this comes about. For a small driving field far off any resonancethe dressed energies are close to the unperturbed ones. As the resonance at 0.375 is ap-proached the dressed energy (which once was the 2p-energy −1/8) “collides” with the1s-energy. However, the energies do not cross. Instead, as ω becomes > 0.375 the for-mer 2p state assumes the role of the 1s state while the former 1s state continues like the2p state for ω < 0.375. The separation of the two quasi-energy levels at resonance isgiven by Ω. As expected, the separation at resonance increases with increasing driverstrength since Ω∼ |E |= F . However, the resonance frequency will also shift due to theAC Stark effect [i.e., the∆ε in (3.275)].

Figure 3.13 shows the ionization rate of H(1s) for λ= 1064 nm laser light as a func-tion of the laser intensity, calculated using the non-Hermitian Floquet method. At thiswavelength at least 12 photons must be absorbed by the electron in order to escape.The lowest order perturbation theory (LOPT) results for S excess photon-ionization(i.e., (12+ S)-photon ionization) are included in Fig. 3.13. They by far overestimatethe ionization rate. The exact Floquet-result displays an interesting structure whichcan be explained in terms of so-called Freeman resonances. As the AC Stark up-shift ofthe continuum (which is given by the ponderomotive potential Up) increases, the min-


Figure 3.13: Ionization rate vs laser intensity for H(1s) irradiated by linearly polarized lightof wavelength λ = 1064 nm. Dashed curves are partial rates for (12+ S)-photon ionization ob-tained within LOPT. The arrows indicate the intensities at which the real part of the 1s Floqueteigenvalue crosses the 13- and 14-photon ionization thresholds. [From R.M. POTVLIEGE andROBIN SHAKESHAFT, Multiphoton processes in an intense laser field: Harmonic generation andtotal ionization rates for atomic hydrogen, Phys. Rev. A 40, 3061 (1989).]

imum number of photons required for ionization increases from 12 to 13 (first arrow)and to 14 (second arrow). After, with increasing intensity, these thresholds are passed,structures appear, indicating a strong enhancement of the ionization rate at certain laserintensities. It turns out that this is due to Rydberg states that are brought into (12+ S)-photon resonance with the ground state via the AC Stark effect (Freeman resonances).

3.5.3 High-frequency Floquet theory and stabilization

The coupling W (t ) to the laser field in (3.245) reads in dipole approximation and veloc-ity gauge

W (t ) = p ·A(t )+ 1

2A2(t ) (3.284)

with A(t ) the vector potential. The transformation of the wave function

|Ψ(t )⟩= e−i2

∫ t−∞A2(t ′)dt ′e−iα(t )·p|ΨKH(t )⟩ (3.285)

removes the A2-term and transforms to the system of an electron oscillating with anexcursion

α(t ) =∫ t

−∞A(t ′)dt ′ (3.286)


in the laser field (Kramers-Henneberger transformation). In this system the nuclear po-tential appears to oscillate. The TDSE then reads

i∂t |ΨKH(t )⟩=

p2

2+V [r +α(t )]

|ΨKH(t )⟩. (3.287)

The Kramers-Henneberger Hamiltonian

HKH(t ) =p2

2+V [r +α(t )] (3.288)

is, for an infinitely long laser pulse, periodic as well so that we may apply the Floquettheorem. Introducing the time-averaged Kramers-Henneberger potential

VKH(α, r ) =1

T

∫ T

0V [r +α(t )]dt , α=max |α(t )|, (3.289)

which is the zero-frequency contribution in a Fourier-expansion of the potential,Eq. (3.253) can be written as

−(ε+mω)Φ(m)β+∑α

⟨β| p2

2+VKH(α, r )|α⟩Φ(m)

α+∑αn

n 6=m

⟨β|V (m−n)|α⟩Φ(n)α= 0. (3.290)

If the laser frequency is large compared to the relevant inner-atomic transitions, we mayneglect the third term so that we are left with an equation diagonal in the photon index,which corresponds to the solution of the time-independent Schrödinger equation

ε|ΨKH⟩=

p2

2+VKH(α, r )

|ΨKH⟩. (3.291)

If it is possible to transfer the entire electron population to the bound states of VKH(α, r )there will be no ionization whatsoever. For intense fields where α 1 Bohr radius thepotential VKH(α, r ) looks very different from the unperturbed nuclear potential sinceit has a double-well structure with the minima close to the classical turning points ±α.If electronic probability density is trapped in this potential the ionization rate decreasesdespite increasing laser field strength. This has indeed been observed in numerical sim-ulations and is called adiabatic stabilization. The stabilization effect survives also for a“real” laser pulse with an up- and a down-ramp (dynamical stabilization).

Figure 3.14 illustrates stabilization of a one-dimensional model atom employing asoft-core binding potential V (x) = −(x2 + ε)−1/2 with ε = 1.9 (leading to a bindingenergy of −0.5). The full TDSE was solved. The laser pulse of frequency ω = 2.5 wasramped over 10 cycles up to E = 62.5 and thereafter held constant. The excursion of afree electron in this field is α(t ) = 10sinωt . The cycle-averaged Kramers-Henneberger


Ene

rgy

(a.u

.)T

ime

(cyc

les)

x (a.u.)

(a)

(b)

Figure 3.14: Adiabatic stabilization in a one-dimensional model atom. The cycle-averagedionic potential in a reference frame where a freely oscillating electron is at rest is depicted in(a) (solid line). The three lowest energy levels in this “dressed” potential and the correspondingprobability densities are also plotted. In (b) a shadowgraph of the probability density, obtainedfrom the full solution of the time-dependent Schrödinger equation, is shown. The probabilitydensity remains trapped in the effective potential. Low-frequency Rabi-floppings are responsiblefor the oscillatory pattern. Note that the time scale of these oscillations is small compared to thelaser period.


Figure 3.15: Lifetime of the H atom in the ground state according to the high-frequency Flo-quet theory, vs intensity, at various laser frequenciesω; circular polarization. Numbers adjacentto points on the curves are the corresponding values of α. The descending branches of the curvescorrespond to LOPT, the ascending ones to adiabatic stabilization (the latter can be “trusted”as the laser frequency increases beyond the ionization potential 0.5). [From M. PONT AND M.GAVRILA, Stabilization of atomic hydrogen in superintense, high-frequency laser fields of circularpolarization, Phys. Rev. Lett. 65, 2362 (1990).]

potential, its three lowest levels and the corresponding probability densities are indi-cated in Fig. 3.14a. The potential has the above mentioned double-well shape with theminima close to the classical turning points ±α. In Fig. 3.14b a shadowgraph of theprobability density obtained from the TDSE solution is shown. The probability den-sity remains confined within the two turning points. Only during the up-ramping ofthe pulse some density escapes. Obviously, not only a single dressed state is occupiedsince the probability density distribution oscillates in time. It can be shown that thisoscillation is due to Rabi floppings between several of the dressed states. Note that thetime-scale of this dynamics is slow compared to the laser period.

Figure 3.15 shows the lifetime (i.e., the inverse ionization rate) of atomic H incircularly laser pulses of various frequencies and intensities as predicted by the high-frequency Floquet theory. With increasing laser intensity the lifetime first decreases(i.e., ionization increases). This is the expected behavior from LOPT. Then, however,the lifetime passes through a minimum (the “death valley”) before it increases again (i.e.,ionization is reduced).

So far, adiabatic stabilization with the electron starting from the ground state wasobserved in numerical simulations only. This is because there are no sufficiently strong


lasers available yet at short wavelengths. The photon energy ω has to exceed the ion-ization potential, and the laser intensity must be strong enough in order to lead to thetwo minima in the time-averaged potential so that the probability density is kept faraway from the nucleus (where absorption of laser energy is efficient) most of the time.Since the elongation α is inversely proportional to ω2, the laser intensity necessary foradiabatic stabilization increases with ω. However, with the free electron lasers (FELs)under development, for example at DESY in Hamburg, the regime of adiabatic stabi-lization should become accessible experimentally. Stabilization of Rydberg atoms bythe same mechanism was already demonstrated.

There are (at least) three effects which counteract to adiabatic stabilization. Firstly,there is the so-called “death-valley” effect: an atom experiences during the rising edgeof a strong laser pulse field intensities in which violent ionization may occur and henceno more electrons are left to be stabilized once the optimal stabilization condition isreached. Secondly, inner electrons might be excited or removed by few photon processesinduced by the incident high frequency radiation. Then the question arises whether anouter electron can stabilize although there are electron holes in the lower lying shells.Thirdly, at high frequencies and high intensities the dipole approximation breaks down,and the magnetic v × B-force pushes the electron into the propagation direction, thusenhancing ionization. These three effects will probably make the experimental verifica-tion of stabilization of atoms initially in the ground state a formidable task. However,in case this kind of stabilization will be achieved, a new and interesting kind of matteris formed: pseudo atoms with charge clouds extending over tens or more atomic units.

Problem 3.7 How does Fig. 3.9 look for non-vanishing detuning?

Further reading: A recent review of Floquet methods is given in SHIH-I CHU andDIMITRY A. TELNOV, Beyond the Floquet theorem: generalized Floquet formalismsand quasienergy methods for atomic and molecular multiphoton processes in intenselaser fields, Phys. Rep. 390, 1 (2004). A Floquet-code can be downloaded from theComputer Physics Communications-archive: R.M. POTVLIEGE, STRFLO: a pro-gram for time-dependent calculations of multiphoton processes in one-electron atomicsystems I. Quasienergy spectra and angular distributions, Comput. Phys. Comm.114, 42 (1998). The original paper on what later became the Floquet theoremis G. FLOQUET, Sur les equations differentielles lineares à coefficients periodique,Ann. Ecol. Norm. Sup. 12, 47 (1883). The fact that Floquet states can be inter-preted as the quantum field states of the electromagnetic field was pointed out inJ.H. SHIRLEY, Solution of the Schrödinger Equation with a Hamiltonian Periodicin Time, Phys. Rev. 138, B979 (1965).


A relatively recent review on stabilization (by one of the eminent persons in thefield) is MIHAI GAVRILA, Atomic stabilization in superintense laser fields, J. Phys.B: At. Mol. Opt. Phys. 35, R147 (2002). Dynamic stabilization of a two-electronmodel system has been studied in D. BAUER and F. CECCHERINI, Electron cor-relation vs stabilization: A two-electron model atom in an intense laser pulse, Phys.Rev. A 60, 2301 (1999). Two-color stabilization in circularly polarized laser fieldswas investigated in D. BAUER and F. CECCHERINI, Two-color stabilization ofatomic hydrogen in circularly polarized laser pulses, Phys. Rev. A 66, 053411 (2002).Very recently, two-electron stabilization beyond the dipole approximation hasbeen studied in ANDREAS STAUDT and CHRISTOPH H. KEITEL, Two-electronionization and stabilization beyond the dipole approximation, Phys. Rev. A 73,043412 (2006).

3.6 Strong field approximation

The strong field approximation (SFA) and its extensions are the theoretical workhorsein strong field laser-atom and laser-molecule interaction. It is sometimes referred to asKeldysh-Faisal-Reiss (KFR) theory because of the papers of these authors (see “Furtherreading” below).

As the laser field is far from being a small perturbation, “conventional” perturbationtheory is not applicable. The SFA does neither consider the laser field being smallcompared to the binding forces nor does it assume the contrary at all times during theinteraction. Instead, the SFA’s assumptions consist basically of considering the bindingpotential dominant until ionization whereas the laser field takes over after ionization.

The SFA has been applied to ionization, harmonic generation, and non-sequentialionization. The beauty of the SFA lies, besides in its predictive power, in the possibilityto interpret its equations in intuitively accessible terms, as will be seen below. However,there are “problems” and limits as well, to be discussed in the last part of this Section.

Let us start with an electronic eigenstate of the field-free Hamiltonian. The electronmay at time ti be in the ground state |Ψ0(ti)⟩ with energy E0 < 0, for instance, and thelaser field is not yet switched on. Now let us consider the matrix element

Mp(tf, ti) = ⟨Ψp(tf)|U (tf, ti)|Ψ0(ti)⟩, (3.292)

which governs the probability wi→f = |Mp(tf, ti)|2 to find the electron at time tf in thescattering state |Ψp(tf)⟩ where p is the asymptotic momentum far away from the atom(where the measurement is performed). We assume that at time tf the laser field is offagain. U (t , t ′) = U †(t ′, t ) is the time-evolution operator associated with the TDSE

i∂

∂ t|Ψ(t )⟩= H (t )|Ψ(t )⟩, H (t ) =

1

2[ p +A(t )]2+ V (r ) (3.293)

3.6. STRONG FIELD APPROXIMATION 77

where A(t ) is the vector potential describing the laser field (in dipole approximation).The minimum coupling Hamiltonian H (t ) can be splitted in various ways:

i∂

∂ t|Ψ(t )⟩= [H0+W (t )]|Ψ(t )⟩= [H (V)(t )+ V (r )]|Ψ(t )⟩, (3.294)

with

H0 =p2

2+ V (r ), H (V)(t ) =

p2

2+W (t ), (3.295)

and W (t ) the interaction with the laser field,

W (t ) = p ·A(t )+ 1

2A2(t ) (velocity gauge). (3.296)

The gauge transformation of the potentials (both scalar potential φ and vector po-tential A) and the wave function |Ψ(t )⟩

A′ =A+∇χ (r , t ), φ′ =φ− ∂ χ (r , t )

∂ t, |Ψ′(t )⟩= e−iχ (r ,t )|Ψ(t )⟩ (3.297)

where χ (r , t ) is an arbitrary differential scalar function, leaves the electric and the mag-netic field unchanged:

E =−∂tA−∇φ= E ′, B =∇×A= B ′. (3.298)

This gauge invariance offers the possibility to choose a gauge that suits us best, e.g.,as far as computational simplicity is concerned. However, this statement only holdstrue as long as all approximations we make do not destroy the gauge invariance. Un-fortunately, the standard SFA does break the gauge invariance. Transformation to theso-called length gauge is achieved by choosing

χ (r , t ) =−A(t ) · r . (3.299)

Because of ∇χ = −A the vector potential is “transformed away” while φ′ = −∂tχ =−E · r . The Hamiltonian in length gauge reads

H ′(t ) =p2

2+ V (r )−φ′(r , t ) =

p2

2+ V (r )+E (t ) · r (3.300)

(one could also absorb V (r ) in φ and φ′). Note that the transformation of the wavefunction

|Ψ′(t )⟩= eir ·A(t )|Ψ(t )⟩ (3.301)

can be interpreted as a translation in momentum space. In fact, while in velocity gaugethe quiver momentum is effectively subtracted from the kinetic momentum, leading to


a canonical momentum different from the kinetic momentum, in length gauge kineticand canonical momentum are equal.

From (3.300) we can infer

W ′(t ) = E (t ) · r (length gauge) (3.302)

with E (t ) =−∂tA(t ).H0 describes the unperturbed atom and seemingly does not depend on the gauge

chosen, i.e., H0 = H ′0. However, one should bear in mind that the momentum p in H0

is not the kinetic momentum (which, in atomic units, equals the velocity) while in H ′0

(length gauge) it is.The Volkov-Hamiltonian H (V)(t ) governs the free motion of the electron in the laser

field. It fulfills in velocity gauge

i∂

∂ t|Ψ(V)(t )⟩= H (V)(t )|Ψ(V)(t )⟩= 1

2[p+A(t )]2|Ψ(V)(t )⟩. (3.303)

Thanks to the dipole approximation the Volkov-Hamiltonian is diagonal in momentumspace. The solution of (3.303) is thus readily written down:

|Ψ(V)(t )⟩= e−iSp(t ,ti)|p⟩, Sp(t , ti) =1

2

∫ t

ti

dt ′ [p+A(t ′)]2 (3.304)

where |p⟩ are momentum eigenstates, ⟨r |p⟩ = eip·r/(2π)3/2. Note that the lower in-tegration limit ti affects the overall phase of the Volkov solution only. As mentionedabove, the transition to the length gauge corresponds to a translation in momentumspace. It is thus easy to check that in length gauge one has

|Ψ(V)′(t )⟩= e−iSp(t ,ti)|p+A(t )⟩ (length gauge) (3.305)

with the same action Sp(t , ti) as in (3.304).Let us now continue to derive the SFA transition matrix element. The time evolu-

tion operator U (t , t ′) satisfies the TDSE (3.294),

i∂t U (t , t ′) = [H0+W (t )]U (t , t ′). (3.306)

Its formal solution is given by the integral equations [cf. (3.87) and (3.88)]

U (t , t ′) = U0(t , t ′)− i∫ t

t ′dt ′′ U (t , t ′′)W (t ′′)U0(t

′′, t ′), (3.307)

= U0(t , t ′)− i∫ t

t ′dt ′′ U0(t , t ′′)W (t ′′)U (t ′′, t ′),


where U0(t , t ′) is the evolution operator corresponding to the TDSE with H0 only.Inserting (3.307) in the matrix element (3.292) leads to

Mp(tf, ti) =−i∫ tf

ti

dt ′ ⟨Ψp(tf)|U (tf, t ′)W (t ′)|Ψ0(t′)⟩ (3.308)

where use of ⟨Ψp(tf)|U0(tf, ti)|Ψ0(ti)⟩ = ⟨Ψp(tf)|Ψ0(tf)⟩ = 0 was made because |Ψp(tf)⟩is a scattering state perpendicular to |Ψ0(tf)⟩, and U0(t

′, ti)|Ψ0(ti)⟩ = |Ψ0(t′)⟩. Since the

propagator U (t , t ′) also satisfies the integral equations

U (t , t ′) = U (V)(t , t ′)− i∫ t

t ′dt ′′ U (V)(t , t ′′)V U (t ′′, t ′), (3.309)

= U (V)(t , t ′)− i∫ t

t ′dt ′′ U (t , t ′′)V U (V)(t ′′, t ′),

where U (V)(t , t ′) is the evolution operator corresponding to the TDSE (3.303), one ob-tains, upon inserting (3.309) in (3.308),

Mp(tf, ti) =−i

∫ tf

ti

dt ′ ⟨Ψp(tf)|U (V)(tf, t ′)W (t ′)|Ψ0(t′)⟩ (3.310)

−i∫ tf

ti

dt ′′∫ tf

t ′′dt ′ ⟨Ψp(tf)|U (V)(tf, t ′)V U (t ′, t ′′)W (t ′′)|Ψ0(t

′′)⟩

.

Using∫ tf

tidt ′′∫ tf

t ′′dt ′ =

∫ tf

tidt ′∫ tf

tidt ′′Θ(t ′ − t ′′) =

∫ tf

tidt ′∫ t ′

tidt ′′ expression (3.310) may

be recast in the form

Mp(tf, ti) = −i∫ tf

ti

dt ′ ⟨Ψp(tf)|U (V)(tf, t ′)

W (t ′)|Ψ0(t′)⟩ (3.311)

−i∫ t ′

ti

dt ′′ V U (t ′, t ′′)W (t ′′)|Ψ0(t′′)⟩

.

Eq. (3.311) is still exact and gauge invariant. Whatever is missed in the first term of(3.311) is included in the second term where the full but unknown time evolution oper-ator U (t ′, t ′′) appears.

Neglecting the second term and replacing the final state |Ψp(tf)⟩ with a plane wave|p⟩ yields the SFA or so-called Keldysh-amplitude

M (SFA)p (tf, ti) =−i

∫ tf

ti

dt ⟨Ψ(V)p (t )|W (t )|Ψ0(t )⟩ (3.312)

where in velocity gauge the Volkov wave |Ψ(V)(t )⟩= U (V)(t , ti)|p⟩ is given by (3.304) andin length gauge by (3.305). The SFA transition amplitude integrates over all ionization


times t where the transition from the bound state |Ψ0(t )⟩ to the Volkov state |Ψ(V)p (t )⟩,mediated by the interaction with the laser field W (t ), may take place. It is thus a bitsurprising, at least at first sight, that the Keldysh amplitude can be recast in a formwhere W (t ) is replaced by V (r ):

M (SFA)p (tf, ti) = −i

∫ tf

ti

dt ⟨Ψ(V)p (t )| H (V)(t )− H0+ V︸︷︷︸W (t )

|Ψ0(t )⟩

= −i∫ tf

ti

dt ⟨Ψ(V)p (t )| − i←∂ t +V − i

→∂ t |Ψ0(t )⟩

= −∫ tf

ti

dtn∂t ⟨Ψ(V)p (t )|Ψ0(t )⟩+ i⟨Ψ(V)p (t )|V |Ψ0(t )⟩

o

= − ⟨Ψ(V)p (t )|Ψ0(t )⟩

tf

ti

− i∫ tf

ti

dt ⟨Ψ(V)p (t )|V |Ψ0(t )⟩. (3.313)

Since the laser is off at the times ti and tf, the first term is just the difference of theFourier-transformed initial state at the two times ti and tf. The latter can be alwayschosen such that the term vanishes. The fact that the boundary term shows up at all isdue to the non-orthogonality of plane waves (introduced within the SFA) and the initialbound state. However, the contribution of the first term vanishes at latest when theasymptotic rate is calculated. Hence, the SFA-amplitude (3.312) may be also written as

M (SFA)p (tf, ti) =−i

∫ tf

ti

dt ⟨Ψ(V)p (t )|V (r )|Ψ0(t )⟩. (3.314)

Following the above interpretation it now seems that ionization is mediated by thebinding potential V (r ), which appears to be absurd. However, upon time-reversal ion-ization turns into (re)combination, for which indeed the nuclear potential is responsi-ble. Moreover, in the above derivation of (3.312) W and V are treated on an equal,symmetrical footing since the essential steps consisted of using the integral equations(3.307) and (3.309).

In velocity gauge where W (t ) = p ·A(t )+A2(t )/2 one can write (3.312) as

M (SFA)p (tf, ti) = −Ψ0(p)e

iSp,E0(t ,ti)

tf

ti

+ iΨ0(p)

p2

2−E0

tf∫

ti

dt eiSp,E0(t ,ti) (3.315)

with the classical action

Sp,E0(t , ti) =

t∫

ti

dt ′

p2

2−E0+W (t ′)

, (3.316)


Ψ0(p, t ) = ⟨p|Ψ0(t )⟩, and Ψ0(p, t ) = exp(−iE0t )Ψ0(p), that is, Ψ0(p) is the Fourier-transformed initial state wave function

Ψ0(p) =1

(2π)3/2

∫d3 r e−ip·rΨ0(r ) (3.317)

and E0 is the initial energy. The first term in (3.315) vanishes when the asymptotic rate

Γp = limT→∞

Mp

2

T, Mp = lim

ti→−∞tf→∞

Mp(tf, ti) (3.318)

is calculated:

M (SFA)p = iΨ0(p)

p2

2−E0

∞∫

−∞

dt eiSp,E0(t ,−∞). (3.319)

3.6.1 Circular polarization and long pulses

In this case one may write the vector potential in dipole approximation as

A(t ) =1

2A[εexp(iωt )+ ε∗ exp(−iωt )] (3.320)

where the polarization vectors ε,ε∗ fulfill ε2 = ε∗2 = 0 and ε · ε∗ = 1, e.g., ε = (ex +iey)/p

2. The factor 1/2 is introduced in order to obtain Up = A2/4, as in the linearlypolarized case. With

[p+A(t )]2 = p2+1

2A2+ 2A|p · ε|cos(ωt −ϕ), (3.321)

where p · ε= |p · ε|exp(−iϕ), the action (3.316) reads

Sp,E0(t ,−∞) =

p2

2−E0+Up

t +

A

ω|p · ε| sin(ωt −ϕ) (3.322)

where we neglected contributions from ti = −∞ since they just affect the irrelevant,overall phase of the transition matrix element. The SFA transition matrix element thenis

M (SFA)p, circ.

= 2πiΨ0(p)∞∑

n=−∞(nω−Up)exp(inϕ) Jn

−A|p · ε|ω

!δ(p2/2−E0+Up− nω),

(3.323)


where use of the Bessel functions Jn(ζ ), obeying

exp[−iζ sin(ωt −ϕ)] =∞∑

n=−∞Jn(ζ )exp[−in(ωt −ϕ)], (3.324)

was made. The time integration in (3.319) leads to the energy-conserving δ-function.Employing (3.318) one obtains the ionization rate

Γ(SFA)p, circ.

= 2π|Ψ0(p)|2∞∑

n=−∞(nω−Up)

2 J 2n

−A|p · ε|ω

!δ(p2/2−E0+Up−nω). (3.325)

Γ(SFA)p, circ.

has the dimension of a density in momentum space per time. To evaluate thesquare of the δ-function we used the “working formula”

δ(Ω) =1

2πlim

T→∞

∫ T /2

−T /2exp(iΩt )dt = lim

T→∞

T

2πfor Ω= 0 (3.326)

(and zero otherwise). For obtaining the total rate Γ the partial rate Γp has to be inte-grated over all final momenta p,

Γ=∫

d3 p Γp =∫

d p dΩ p2Γp =∫

dΩdΓ

dΩ(3.327)

where dΩ = sinϑ dϑ dϕ is the solid angle element. The final rate for ionization withthe electron ejected into the solid angle dΩ is given by

dΓ(SFA)circ.

dΩ= 2π

p8ω5

∞∑n=nmin

(n−Up/ω)2 pnp

2ω|Ψ0(pn)|2 J 2

n

−Apn sinϑp

2ω

!, (3.328)

withpn =

Æ2(nω−Up+E0). (3.329)

The sum in (3.328) runs over all n which yield real pn, starting with the minimumnumber of absorbed photons nmin.

3.6.2 Channel closing in above-threshold ionizationThe increase of nmin with increasing Up is the channel-closing phenomenon. This isillustrated in Fig. 3.16. We expect peaks in the photoelectron spectra at the positions

En =1

2p2

n = nω− (Up+ |E0|), n ≥ nmin (3.330)

where |E0| is the ionization potential and Up+ |E0| is the “effective” ionization potentialdue to the AC Stark shift of the continuum threshold with respect to the ground state


ε0

(a) (b)

short pulse long pulse

short pulse

long pulse

U p

Figure 3.16: Channel closing in the short and long pulse regime. In (a) the laser intensity issmall so that the Up-shift of the continuum threshold is less than a photon energy. In (b) thechannels n = 5 and n = 6 are closed due to the pronounced AC Stark shift of the continuum.The photoelectron spectra look different in the short (blue) and long pulse regime (red) sincethe released electron gains Up in the latter case upon leaving the laser focus.


level. Due to the fact that peaks n > nmin are present—even with higher probabilitythan n = nmin—the name above-threshold ionization has been coined for this strong fieldionization phenomenon. In actual measurements the positions of the peaks depend onthe laser pulse duration. In short-pulse experiments the released electron has no time toleave the focus before the laser pulse is over. In long pulses, instead, the electron has timeto leave the focus and, by doing so, gains the energy Up. As a consequence the Up-termin (3.330) is cancelled and the peak positions in the long-pulse regime are determined by

E (long pulse)n =

1

2p2

n = nω− |E0|, n ≥ nmin (3.331)

with nmin, however, still to be calculated with Up (since before leaving the laser focus,ionization has to occur in the first place). In Fig. 3.16a the Up-shift of the continuumthreshold is small, and the channel n = 5 is responsible for the first peak in the photo-electron spectra. At higher laser intensity, in Fig. 3.16b, the channels n = 5 and n = 6are closed since 6 photons are not sufficient to overcome the effective ionization poten-tial |E0|+ Up. In the long pulse regime peaks corresponding to a certain channel arealways at the same positions in the energy spectra whereas in the short pulse regime thepeaks move with Up. As a consequence, focal averaging reduces the contrast in the shortpulse regime whereas it has less of an effect in the long pulse regime.

In Fig. 3.17 we show the example of an experimental long-pulse spectrum wherelower order peaks are indeed suppressed due to channel closing.

3.6.3 Linear polarization and long pulses

We assume a laser field of the form

A(t ) = Aεcos(ωt ), ε2 = 1. (3.332)

The action (3.316) reads in this case

Sp,E0(t ,−∞) =

p2

2−E0+Up

t − A

ωp · ε sin(ωt )+

A2

8ωsin(2ωt ). (3.333)

Proceeding as in the circular field-case one arrives at

M (SFA)p,lin.= 2πiΨ0(p)

∞∑n=−∞

(nω−Up)Jn

−Ap · εω

,−Up

2ω

!δ(p2/2−E0+Up−nω), (3.334)

Γ(SFA)p,lin.= 2π|Ψ0(p)|2

∞∑n=−∞

(nω−Up)2J 2

n

−Ap · εω

,−Up

2ω

!δ(p2/2−E0+Up− nω),

(3.335)


Up

Figure 3.17: Channel closing in the long-pulse regime. The experimental spectrum was takenfrom R.R. FREEMAN & P.H. BUCKSBAUM, Investigation of above-threshold ionization using sub-picosecond laser pulses, J. Phys. B: At. Mol. Opt. Phys. 24, 325 (1991).


dΓ(SFA)lin.

dΩ= 2π

p8ω5

∞∑n=−∞

(n−Up/ω)2 pnp

2mω|Ψ0(pn)|2J 2

n

−Ap · εω

,−Up

2ω

!. (3.336)

Jn is the generalized Bessel function of integer order defined by

Jn(u, v) =1

2π

∫ π

−πdθ exp[i(u sinθ+ v sin(2θ)− nθ)]. (3.337)

The relation to the ordinary Bessel functions is

Jn(u, v) =∞∑

k=−∞Jn−2k(u)Jk(v). (3.338)

The equation used in the derivation of (3.334) is

∞∑n=−∞

exp(inθ)Jn(u, v) = exp[iu sinθ+ iv sin(2θ)]. (3.339)

Further properties of the generalized Bessel functions may be found in the Appendix Bof the original SFA paper by REISS. In the same article it is also demonstrated how thetypical exponential behavior ∼ exp[−2(2|E0|)3/2/(3|E |)] arises in the case of tunnelingionization through the asymptotic behavior of the generalized Bessel functions Jn.

Problem 3.8 Which channels n are closed in Fig. 3.17?

Further reading: The KFR papers are: L.V. KELDYSH, Zh. Eksp. Teor. Fiz. 47, 1945(1964); [Sov. Phys. JETP 20, 1307 (1965)]; F.H.M. FAISAL, J. Phys. B: Atom.Molec. Phys. 6, L89 (1973); H.R. REISS, Phys. Rev. A 22, 1786 (1980).

3.6.4 Few-cycle above-threshold ionization

In few-cycle pulses the pulse envelope A(t ) varies on a time scale not much slower thanthe period T = 2π/ω corresponding to the carrier frequency ω. Moreover, the carrierenvelope phase φ, governing the shift of the carrier wave with respect to the envelope,affects basically all observables. Hence, instead of (3.332) we write

A(t ) = A(t )ε sin(ωt +φ) (3.340)

with A(t ) a sin2 or a Gaussian envelope, for instance.


In the case of few-cycle pulses it would be very cumbersome to deal with Besselfunctions, as we did above for constant (or slowly varying) A. Instead, we start off withthe still exact matrix element (3.311) and replace once again the final state by a planewave and the full propagator U (t ′, t ) by U (V)(t ′, t ). In that way we obtain the extendedSFA transition matrix element

M (SFA)p (tf, ti) =M (SFA,dir)

p (tf, ti)+M (SFA,resc)p (tf, ti), (3.341)

M (SFA,dir)p (tf, ti) = M (SFA)

p (tf, ti) =−i∫ tf

ti

dt ⟨Ψ(V)p (t )|W (t )|Ψ0(t )⟩, (3.342)

M (SFA,resc)p (tf, ti) = −

∫ tf

ti

dt∫ t

ti

dt ′ ⟨Ψ(V)p (t )|V U (V)(t , t ′)W (t ′)|Ψ0(t′)⟩. (3.343)

In what follows we set ti = 0, and we assume that

A(0) =A(Tp) = 0 (3.344)

where Tp is the laser pulse duration. Relabelling the integration variables and makinguse of the fact that ionization can only happen while the laser is on, (3.342) and (3.343)can be recast in the form

M (SFA,dir)p = −i

∫ Tp

0dtion ⟨Ψ(V)p (tion)|W (tion)|Ψ0(tion)⟩, (3.345)

M (SFA,resc)p = −

∫ Tp

0dtion

∫ ∞

tion

dtresc ⟨Ψ(V)p (tresc)|V U (V)(tresc, tion)W (tion)|Ψ0(tion)⟩. (3.346)

The interpretation of the second term is straightforward: ionization due to interactionwith the laser (W ) occurs at time tion and is restricted to times where the laser is on.After ionization, the electron moves freely in the laser field (governed by U (V)) beforeit rescatters at time tresc with the ionic core (V ). Note that tresc may be > Tp. After the

rescattering event, the electron ends up in the Volkov state |Ψ(V)p ⟩, meaning that it willfinally arrive with a (field free and thus kinetic) momentum p at the detector.

In length gauge the matrix elements read

M (SFA,dir)p = −i

∫ Tp

0dtion⟨p+A(tion)|r ·E (tion)|Ψ0⟩eiSp,E0

(tion), (3.347)

M (SFA,resc)p = −

∫ Tp

0dtion

∫ ∞

tion

dtresc

∫d3k eiSp(tresc)⟨p+A(tresc)|V |k+A(tresc)⟩

×e−iSk (tresc)⟨k+A(tion)|r ·E (tion)|Ψ0⟩eiSk,E0(tion) (3.348)


where

Sp,E0(t ) =

t∫

0

dt ′1

2[p+A(t ′)]2−E0

, Sp(t ) =

1

2

t∫

0

dt ′ [p+A(t ′)]2. (3.349)

Introducing the new variables

t = tresc, τ = tresc− tion (3.350)

the rescattering SFA matrix element (3.348) can be written as

M (SFA,resc)p = −

∫ ∞

0dt∫ t

0dτ∫

d3k eiSp(t )

Vp−k=⟨p|V |k⟩︷︸︸︷⟨p+A(t )|V |k+A(t )⟩ (3.351)

×e−iSk (t )⟨k+A(t −τ)|r ·E (t −τ)|Ψ0⟩

= −∫ ∞

0dt eiSp,E0

(t )∫ t

0dτ∫

d3k Vp−ke−iSk,E0(t ,t−τ) (3.352)

×⟨k+A(t −τ)|r ·E (t −τ)|Ψ0⟩

where

Sk,E0(t , t −τ) =

∫ t

t−τdt ′1

2

k+A(t ′)

2−E0

. (3.353)

The time τ is the time the electron spends in the continuum between ionization andrescattering. The infinite upper limit in the integration over the rescattering time in(3.348) can be restricted to Tp since rescattering after the laser is off will not lead toenergy absorption. As a consequence, the final energy will be within a region stronglydominated by the more probable direct ionization process.

The integration over the intermediate momentum k can be performed using thesaddle-point approximation (stationary phase method) where we seek the stationary mo-mentum ks(t ,τ) contributing most to the k-integration:

∇kSk,E0(t , t −τ) != 0 ⇒ ks(t ,τ) =−α(t )−α(t −τ)

τ(3.354)

with α(t ) the excursion as in (3.286). Hence,

M (SFA,resc)p = −

∫ Tp

0dt eiSp,E0

(t )∫ t

0dτ2π

iτ

3/2

Vp−ks(t ,τ)e−iSs,E0

(t ,t−τ)

×⟨ks(t ,τ)+A(t −τ)|r ·E (t −τ)|Ψ0⟩(3.355)


where the stationary action Ss,E0(t , t −τ) is given by

Ss,E0(t , t −τ) = Sks(t ,τ),E0

(t , t −τ) (3.356)

and the factor [2π/(iτ)]3/2 comes from the saddle-point integration. In actual numericalevaluations the denominator iτ can be either regularized by adding a real, positive ε, orthe lower integration limit 0 for τ can be replaced by ε. As long as ε is sufficiently small,the results are independent of ε.

In the case of a hydrogen-like atom the matrix element needed in (3.355) reads

⟨k|r ·E (t )|Ψ0⟩=−i27/2(2|E0|)5/4k ·E (t )

π(k2+ 2|E0|)3. (3.357)

For rescattering at potentials of the form

V (r ) =−

b +a

r

e−λr (3.358)

the matrix element Vp−k is given by

Vp−k =−2bλ+ aC

2π2C 2, C = (p− k)2+λ2. (3.359)

In few-cycle laser pulses the concept of an ionization rate is not useful since the lat-ter would be time-dependent and sensitive to all details of the pulse (duration, shape,carrier-envelope phase, peak field strength). In experiments one measures the differ-ential ionization probability wp, which is the probability to find an electron of finalenergy Ep = p2/2 emitted in a certain direction, given by the solid angle element dΩp

that is covered by the measuring device. The probability wp is related to the transitionmatrix element Mp through

wp dEp︸︷︷︸p dp

dΩp = |Mp|2 d3 p = |Mp|2 p2d p dΩp (3.360)

so thatwp = p |Mp|2. (3.361)

3.6.5 “Simple man’s theory”The remaining time integral(s) in (3.347) and (3.355) can be either solved numerically orapproximately by using modifications of the saddle-point method with respect to time.We do not want to go into the details but only emphasize here that the SFA transitionmatrix element can be approximated by a sum over the stationary contributions,

M (SFA)p =

∑s

as ,peiSs ,p . (3.362)


As it turns out the saddle-point equations define quantum orbits that are close to theclassical orbits of the so-called simple man’s theory. In the following we will use simpleman’s theory to derive the cut-off laws for the photoelectron spectra.

If an electron is set free at time tion and from thereon does not interact with the ionicpotential anymore, its momentum and position at times t > tion are given by

p(t ) = −∫ t

tion

dt ′E (t ′) =A(t )−A(tion), (3.363)

r (t ) =∫ t

tion

dt ′A(t ′)−A(tion)(t − tion)

= α(t )−α(tion)−A(tion)(t − tion). (3.364)

If the vector potential fulfills (3.344) the momentum at the end of the pulse is deter-mined by the value of the vector potential at the time of ionization, p(Tp) = −A(tion),so that the final energy is

Edir,p(tion) =1

2A2(tion)≤ 2Up (3.365)

because the ponderomotive potential is Up = A2/4. The fact that the direct electrons areclassically restricted to energies up to 2Up is one of the celebrated cut-off laws in strongfield physics.

Let us now allow for one rescattering event, i.e., at the time tresc the electron returnsto the origin (where the ion is located),

ε!> |r (tresc)|= |α(tresc)−α(tion)−A(tion)τ| (3.366)

where ε is a distance small compared to the Bohr radius. Let us assume the extreme caseof 180 back-reflection where the electron changes the sign of its momentum so thatimmediately after the scattering event

p(tresc+) =−[A(tresc)−A(tion)]. (3.367)

At later times we have

p(t > tresc) = −∫ t

tresc

dt ′E (t ′)− [A(tresc)−A(tion)]

= A(t )− 2A(tresc)+A(tion) (3.368)

so that presc(Tp) =−2A(tresc)+A(tion) and

Eresc,p(tresc, tion) =1

2[A(tion)− 2A(tresc)]

2 ≤ 10Up. (3.369)


Up

"direct" electrons

electronsrescattered

final energy contours /

Figure 3.18: Final photoelectron energy Eresc,p vs ionization time tion and rescattering timetresc > tion (black: high value of Eresc,p , white: small value of Eresc,p , contours 2, 10 and 16Uplabelled explicitly). The red branch labelled “rescattered electrons” indicate times tion and trescwhere (3.366) is fulfilled. The highest energy those rescattered electrons can have is 10Up (seered branch touching the 10Up-contour). The inlet shows the final energy (3.365) of the “direct”electrons vs the ionization time. The highest energy there is 2Up.

Because of the condition (3.366) the 10Up cut-off law for the rescattered electrons is notso obvious. However, it can be readily checked numerically by plotting Eresc,p(tresc, tion)vs all possible tresc, tion (where tresc > tion) and then indicating those pairs of tion, tresc thatfulfill (3.366). This is shown in Fig. 3.18.

How good is the strong field approximation?

In order to answer this question we compare the results of an ab initio TDSE solutionwith the corresponding SFA predictions. Apart from the dipole approximation (whichis well applicable for the peak intensities used) the TDSE result is exact. Comparisonswith TDSE results serve as a much more demanding testing ground for approximatetheories such as the SFA than comparison with experiments, mainly because of focalaveraging effects present in experiments, the uncertainties in laser intensity, pulse dura-


(a)θ=πφ=0

Photoelectron energy (a.u.)

φ=0(b)θ=0

Yie

ld (

arb.

u.)

Up

Up

Up

Up

10

10

2 2

Figure 3.19: Photoelectron spectra of the H(1s) electron after irradiation with a 4-cycle laserpulse (ω = 0.056, φ = 0, E = 0.0834). The TDSE and SFA results are drawn solid and dashed,respectively. Panel (a) shows the “left-going” electrons (i.e., opposite to the laser polarizationez ), panel (b) the “right-going” electrons (in ez )-direction. The spectra were adjusted verticallyby multiplication with a single factor in such a way that agreement is best in the cut-off regionfor the right-going electrons. The spectra were not shifted in energy.

tion, and shape, the limited resolution in energy, and the limited dynamic range in theyields.

In Fig. 3.19 the results for a n = 4-cycle pulse of the form E (t ) =Eez sin2[ωt/(2n)]cos(ωt +φ) for 0 < t < n2π/ω, ω = 0.056, E = 0.0834, φ = 0are shown. The agreement between TDSE and SFA results improve with increasingphotoelectron energy. A possible explanation for that might be that slow electronsspend more time in the vicinity of the atomic potential where Coulomb corrections areexpected to be important. As can be seen in Fig. 3.19, the transition regime betweenthe cut-off for the direct electrons at E = 2Up = 1.1 up to energies where rescatteredelectrons start to take over at E ≈ 2.5 is quite smooth in the TDSE spectra whereaspronounced patterns are visible in the SFA results. Moreover, at very low energies thepositions of the local maxima disagree. It has been shown (see “Further reading” below)that the agreement at lower energies improves if the binding potential is made short-range by cutting it at certain distances. This is expected since the crucial assumptionin SFA is that the electron is not affected by the ionic potential anymore once ioniza-tion has occurred. This assumption is well justified for short-range potentials but lessso for long range Coulombic ones as in the H(1s) case. However, the pronounced pat-


(b)

(a)

"direct" electrons

rescattered electrons

same final energy

Figure 3.20: (a) Final energy of direct electrons (black line) vs the ionization time tion. Ifrescattered is allowed, higher energies may occur. The color coding for the rescattered electronsindicates the time spent in the continuum between ionization and rescattering, i.e., tresc − tion(the lighter the color the longer the time). The cut-off energies for the direct and the rescatteredelectrons are 2 and 10Up, respectively. Panel (b) shows the course of the laser field. Ionization isimprobable for small |E(t )|.

tern showing a spiky, downward-pointing structure in the rescattering plateaux betweenenergies ≈ 3 and 6 a.u. is remarkably well reproduced using the SFA.

Interference effects

The spiky structure in Fig. 3.19 is due to quantum interference. For a fixed final mo-mentum p the sum (3.362) is a sum over all quantum orbits that end up with the samemomentum at the detector. It turns out that, most of the time, there are two dominatingcontributions. Depending on their phase-difference those may interfere constructively(local maxima in Fig. 3.19) or destructively (downward-pointing spikes in Fig. 3.19).The corresponding two trajectories of simple man’s theory can be readily calculated.As an example we show the final energy vs the ionization time for a “flat-top” pulse inFig. 3.20. In the lower panel of Fig. 3.20 the course of the electric field is indicated. Letus focus on the time t = 0.5 cycles where the electric field has a maximum and ioniza-tion is therefore most likely. The upper panel shows that a “direct” electron emitted atthat time will have vanishing final energy. In order for a direct, classical electron to havethe maximum energy 2Up it has to be emitted at times where the electric field is zero(which is unlikely in the tunneling and over-barrier regime). However, if an electron is


emitted around the maximum of the electric field and rescatters once, its final energymay be close to 10Up. It is clearly seen that two emission times very close to each otherlead to the same final energy (as indicated in the upper panel). These are the two trajec-tories that interfere. Exactly at the cut-off 10Up those two solutions merge to a singleone. The “travel times” tresc− tion between rescattering and ionization are color-codedfrom black (tresc− tion = 0) to yellow (tresc− tion = 2 cycles).

In very short pulses the situation is more complex than in the regular, flat-top pulsecase. Since the ionization probability is strongly weighted with the modulus of the elec-tric field, only a few “time-windows” may remain “open”, thus affecting the number ofinterfering quantum orbits for a given p. The interference pattern is then very sensitiveto the details of the few-cycle pulse, e.g., to the carrier-envelope phase. Because of the“time-windows” that are opened and closed depending on the parameters of the few-cycle pulse, one may view the setup as a “double slit experiment in time” (see “Furtherreading” below).

Problem 3.9 Write down the SFA amplitude for the direct electrons emitted duringa few-cycle pulse using velocity gauge. What is the difference with respect to(3.347)?

Further reading: A review of few-cycle above-threshold ionization including many ofthe relevant references is D.B. MILOŠEVIC, G.G. PAULUS, D. BAUER, and W.BECKER, Above-threshold ionization by few-cycle laser pulses, J. Phys. B: At. Mol.Opt. Phys. 39 (in press). The gauge problem in SFA has quite some history. Re-cent work on the subject, showing that in the case of atoms the velocity gauge maylead to (even qualitatively) wrong results, is D. BAUER, D.B. MILOŠEVIC, and W.BECKER, Strong-field approximation for intense laser-atom processes: the choice ofgauge, Phys. Rev. A 72, 023415 (2005). A comparison of SFA with TDSE resultsfocussing on the effect of the range of the binding potential has been undertakenin D. BAUER, D.B. MILOŠEVIC, and W. BECKER, On the validity of the strongfield approximation and simple man’s theory, J. Mod. Opt. 53, 135 (2006). Theemergence of the simple man’s theory-orbits in ab initio TDSE results has beendemonstrated in D. BAUER, Emergence of Classical Orbits in Few-Cycle Above-Threshold Ionization of Atomic Hydrogen, Phys. Rev. Lett. 94, 113001 (2005). Theabove-mentioned “double slit in time”-experiment has been published in F. LIND-NER, M.G. SCHÄTZEL, H. WALTHER, A. BALTUŠKA, E. GOULIELMAKIS, F.KRAUSZ, D.B. MILOŠEVIC, D. BAUER, W. BECKER, and G.G. PAULUS, Attosec-ond Double-Slit Experiment, Phys. Rev. Lett. 95, 040401 (2005).


3.6.6 High harmonic generation

When an intense laser pulse of frequency ω1 impinges on any kind of sample usuallyharmonics ofω1 are emitted. A typical signature of the emission spectrum in the case ofstrongly driven atoms, molecules or clusters is that the harmonic yield does not simplyroll-off exponentially with increasing harmonic order. Instead, a plateau is observed.This is a prerequisite for high order harmonic generation (HOHG) being of practicalrelevance as an efficient short wavelength source. As targets for HOHG one may thinkof single atoms, dilute gases of atoms, molecules, clusters, crystals, or the surface of asolid (which is rapidly transformed into a plasma by the laser). In fact, for all thosetargets HOHG has been observed experimentally. Even a strongly driven two-levelsystem displays nonperturbative HOHG. The mechanism generating the harmonicsand its efficiency, of course, vary with the target-type. In the case of atoms the so-called three step model explains the basic mechanism in the spirit of simple man’s theory:an electron is freed by the laser at a certain time t ′, subsequently it oscillates in thelaser field, and eventually recombines with its parent ion upon emitting a photon offrequency

ω = nω1, n ≥ 1. (3.370)

This process is illustrated in Fig. 3.21. If the energy of the returning electron is E , theenergy of the emitted photon is ω = E + |Ef| where Ef is the energy of the level whichis finally occupied by the electron, for example, the groundstate.

Space

Energy

Figure 3.21: Illustration of the three step model for high harmonic generation. An electronis (i) released, (ii) accelerated in the laser field, and (iii) driven back to the ion. There it mayrecombine upon emitting a single photon which corresponds to a multiple of the photon energyof the incident laser light.

From this simple considerations we conclude that the maximum photon energy onecan expect is ωmax = Emax+ |Ef|. Using (3.363) we obtain at the recombination time trec


for the return energy Eret

Eret =1

2p2(trec) =

1

2[A(trec)−A(tion)]

2 (3.371)

where we have to impose [see (3.366)]

|r (trec)|= |α(trec)−α(tion)−A(tion)(trec− tion)|!< ε. (3.372)

Figure 3.22 shows the possible return energies fulfilling (3.371) and (3.372). We inferthat the maximum return energy is around ' 3.2. Closer inspection shows that thenumber is 3.17 so that the cut-off law reads

ωmax = 3.17Up+ |Ef| . (3.373)

LEWENSTEIN showed (see “Further reading” below) that, more precisely, it readsωmax = 3.17Up+ 1.32|Ef|.

Figure 3.22: Return energy as a function of the ionization time. The color coding indicatesthe time between recombination and ionization (the longer this time the lighter the color). Themaximum return energy is ' 3.17Up. The course of the laser field is shown in panel (b).

From simple man’s theory one expects that harmonic generation should be muchless efficient in elliptically polarized laser fields since, classically, the freed electron nevercomes back to its parent ion so that recombination can be considered unlikely. Thisindeed was confirmed experimentally.


Besides the academic interest, harmonic generation in atoms, molecules and clustershas huge practical relevance as an efficient source of intense XUV radiation. This is be-cause (i) the ponderomotive scaling of the cut-off allows to achieve high values of Emaxand thus high harmonic orders n, and (ii), fortunately, the strength of the harmonicsdoes not decay exponentially with the order n but displays, after a decrease over the firstfew harmonics, an overall plateau (at least on the logarithmic scale) up to the cut-off en-ergy 3.17 Up+|Ef|, allowing relatively high intensities at short wavelengths. In fact, highorder harmonics below λ = 4.4 nm, the so-called water-window (between the K-edgesof carbon and oxygen) have been observed experimentally. The intensity of the emit-ted radiation in the plateau region is about 10−6 of the incident laser intensity whichis typically 1015–1018 Wcm−2 in rare gas experiments. Therefore, the intensity of thehigh order harmonics is sufficient for various kinds of applications, such as interferom-etry, for dense plasma diagnostics, holography, high-contrast microscopy of biologicalmaterials, and attosecond spectroscopy or metrology. Attosecond pulses are generatedvia harmonic generation. If the incoming pulse is already short (i.e., consists only ofa few cycles) the harmonic emission is restricted to a narrow time window similar tothe “double slit in time”-experiment mentioned above. As a consequence, the harmonicpulse has a duration that is short compared to the laser period of the incoming pulse(usually a few hundred attoseconds). If the incoming laser pulse is longer, one can con-struct attosecond pulse trains by selecting a few phase-locked harmonics. Attosecondpulses that are generated via harmonic generation have been used to probe the ionizinglaser field itself as well as fast atomic processes such as Auger decay. Figure 3.23 showsan example for experimental high-order harmonic spectra.

For calculating the rate of harmonic emission one may follow the same route as inthe SFA treatment of ionization. Harmonic generation and above threshold ionization(ATI) are complementary to each other: while in harmonic generation the electroncomes back to the ion and eventually recombines, upon emitting radiation, in ATIit rescatters upon which it may gain additional energy. In harmonic generation oneobserves a plateau reaching up to photon energies 3.17 Up + |Ef|. In ATI a plateau isobserved as well—this time with respect to the kinetic energy of the photo electrons—extending up to 10 Up (for one rescattering event).

In classical electrodynamics, the total radiated power by a dipole of charge q is givenby Larmor’s formula

P =2q2

3c3|r |2. (3.374)

Thus, in a semiclassical approach, it appears reasonable to replace the acceleration byits quantum mechanical expectation value and making use of Ehrenfest’s theorem. Oneobtains

P =2q2

3c3

d2

dt 2⟨r⟩2

=2q2

3c3

1

m

*−∂ H

∂ r

+

2

. (3.375)

The last expression on the right-hand side is particularly suited for the numerical evalu-


Figure 3.23: Harmonic production efficiency in Ar and Ne at 2× 1015 and 5× 1014Wcm−2,respectively, using a 7-fs Ti:sapphire laser pulse. [From M. SCHNÜRER et al., Absorption-LimitedGeneration of Coherent Ultrashort Soft-X-Ray Pulses, Phys. Rev. Lett. 83, 722 (1999).]

ation of the harmonic spectra. In order to simplify even further we write the dipole asa Fourier-transform,

d(t ) = q ⟨r⟩= qp

2π

∫ ∞

−∞dω exp(iωt )d(ω). (3.376)

The total emitted energy then is

Erad =∫ ∞

−∞dt P =

2

3c3

∫ ∞

−∞dωω4 |d(ω)|2, (3.377)

and we infer that the yield radiated into a spectral interval [ω,ω+ dω] is

εrad,ω dω ∼ω4 |d(ω)|2 dω. (3.378)

A full quantum mechanical treatment reveals that calculating the harmonic spec-trum emitted by a single atom from the square of the dipole expectation value is actuallyincorrect. Since the expectation value of the number of photons in a mode ω,k (withcreation and annihilation operators a†, a, respectively) at time t is

⟨a†(t )a(t )⟩ = ⟨a†(ti)a(ti)⟩+ 2Cℜ∫ t

ti

dt ′ ⟨r (t ′)a(ti)⟩exp(−iωt ′)


+C 2∫ t

ti

dt ′∫ t

ti

dt ′′ ⟨r (t ′′)r (t ′)⟩exp[iω(t ′− t ′′)], (3.379)

where C is a coupling constant, one sees that if the mode under consideration is notexcited at the initial time t = ti, as it is the case here, only the third term survives. Thisterm accounts for spontaneous emission and scattering. Hence, the harmonic spectrumof a single atom should be calculated from the two-time dipole-dipole correlation func-tion ⟨r (t ′′)r (t ′)⟩ instead of the Fourier-transformed one-time dipole. However, if oneconsiders a sample of N atoms

⟨a†(t )a(t )⟩=C 2N∑

k=1

N∑j=1

∫ t

ti

dt ′∫ t

ti

dt ′′ ⟨rk(t′′)r j (t

′)⟩exp[iω(t ′− t ′′)] (3.380)

results and, by assuming that all these atoms are uncorrelated, that is ⟨rk(t′′)r j (t

′)⟩ '⟨rk(t

′′)⟩⟨r j (t′)⟩, one arrives at

⟨a†(t )a(t )⟩ ≈C 2

N∑

k=1

∫ t

ti

dt ′ ⟨rk(t′)⟩exp(iωt ′)

2

(3.381)

if N 1 is assumed so that the self-interaction terms∼ ⟨rk(t′′)⟩⟨rk(t

′)⟩ contribute negli-gibly. Moreover, if all atoms “see” the same field one obtains simply the absolute squareof N times the single dipole expectation value. Therefore, calculating the harmonicspectra from the Fourier-transformed dipole, although not correct in the single atomresponse case, is a reasonable method when comparison with high-order harmonic gen-eration experiments in dilute gas targets is made. Hence, for the study of macroscopicpropagation effects the dipole expectation value may be inserted as a source in Maxwell’sequations.

SFA for harmonic generation: the Lewenstein-model

The dipole expectation value of a single atom with one active electron (q =−1) is

d(t ) = −⟨Ψ(t )|r |Ψ(t )⟩ (3.382)

= −⟨Ψ0(ti)|U (ti, t ) r U (t , ti)|Ψ0(ti)⟩,where we assumed that at an initial time ti the electron starts in the state |Ψ0(ti)⟩=: |Ψ0⟩.Using (3.307) we obtain

d(t ) = −⟨Ψ0(t )|r |Ψ0(t )⟩ (3.383)

−i∫ t

ti

dt ′ ⟨Ψ0(t′)|W (t ′)U (t ′, t )r |Ψ0(t )⟩

+i∫ t

ti

dt ′ ⟨Ψ0(t )|r U (t , t ′)W (t ′)|Ψ0(t′)⟩

−∫ t

ti

dt ′∫ t

ti

dt ′′ ⟨Ψ0(t′)|W (t ′)U (t ′, t )r U (t , t ′′)W (t ′′)|Ψ0(t

′′)⟩.


The first term vanishes for a spherically symmetric binding potential. The second andthird term are complex conjugates of each other and describe ionization (W ), propaga-tion (U ), and recombination (r ) (i.e., the emission of harmonic radiation) in differenttime-ordering. The last term involves one additional interaction with the laser field. Wewill neglect it here without, however, omitting to point out that it has been discussed inthe literature (see “Further reading” below). Replacing—as in the SFA for ionization—the full time evolution operator U by U (V),

d (L)(t ) = −i∫ t

ti

dt ′ ⟨Ψ0(t′)|W (t ′)U (V)(t ′, t )r |Ψ0(t )⟩+ c.c., (3.384)

we shall recover the Lewenstein-result. In length gauge [W (t ) = E (t ) · r] we obtain(suppressing the +c.c.)

d (L)(t ) =−i∫ t

ti

dt ′∫

d3 p ⟨Ψ0(t′)|E (t ′) · r |p+A(t ′)⟩⟨p+A(t )|r |Ψ0(t )⟩e−iSp(t

′,t ) (3.385)

where we used (3.305) and U (V)(t ′, t )|p+A(t )⟩= e−iSp(t′,t )|p+A(t ′)⟩. With

Sp,E0(t , t ′) =

∫ t

t ′dt ′′

1

2[p+A(t ′′)]2−E0

(3.386)

we can write (3.385) as

d (L)(t ) =−i∫ t

ti

dt ′∫

d3 p ⟨Ψ0|E (t ′) · r |p+A(t ′)⟩⟨p+A(t )|r |Ψ0⟩e−iSp,E0(t ′,t ). (3.387)

Introducing the dipole matrix element

µ(p) = ⟨p|r |Ψ0⟩=1

(2π)3/2

∫d3 r e−ip·r rΨ0(r ) (3.388)

we have

d (L)(t ) = −i∫ t

ti

dt ′∫

d3 p e−iSp,E0(t ′,t )µ[p+A(t )]E (t ′) ·µ∗[p+A(t ′)]+ c.c. (3.389)

= i∫ t

ti

dt ′∫

d3 p e−iSp,E0(t ,t ′)µ∗[p+A(t )]E ∗(t ′) ·µ[p+A(t ′)]+ c.c. (3.390)

The integration over momentum can be performed using the saddle-point approxima-tion again:

∇pSp,E0(t , t ′) != 0. (3.391)


For a linearly polarized laser pulse with a slowly varying envelope we have

E (t ) = Eez cosωt , A(t ) =− E

ωez sinωt , (3.392)

so that

α(t )−α(t ′) =∫ t

t ′dt ′′A(t ′′) =

E

ω2ez(cosωt − cosωt ′) (3.393)

and∂ Sp,E0

(t , t ′)

∂ pz

= pz(t − t ′)+E

ω2(cosωt − cosωt ′) != 0 (3.394)

⇒ pz,s(t ,τ) =− E[cosωt − cosω(t −τ)]ω2τ

, τ = t − t ′. (3.395)

The transverse stationary momentum vanishes, px,s = py,s = 0. Plugging ps into (3.386)and integrating yield the stationary action

Ss(t ,τ) = (Up−E0)τ− 2Up

1− cosωτ

ω2τ−Up

C (τ)

ωcos[(2t −τ)ω] (3.396)

with

C (τ) = sinωτ− 4

ωτsin2 ωτ

2. (3.397)

In the original Lewenstein-paper it is shown that the cut-off law 3.17Up + |E0| can bederived from the function C (τ). Setting ti = 0, the final result for the SFA dipole aftersaddle-point integration reads

d (L)(t ) = i∫ t

0dτ2π

iτ

3/2

µ∗z[pz ,s(t ,τ)+A(t )]

×µz[pz,s(t ,τ)+A(t −τ)]E cosω(t −τ)e−iSs(t ,τ)+ c.c.

(3.398)

from which, via Fourier-transformation, the harmonic spectra εrad,ω [Eq. (3.378)] canbe calculated. Figure 3.24 shows an example for a harmonic spectrum calculated usingthe Lewenstein model.

Harmonic generation selection rules

As the name “harmonics” suggests, the emission of laser-driven targets mainly occurs atmultiples of the fundamental, incoming laser frequency ω1,

ω = nω1 (3.399)


Figure 3.24: Harmonic spectrum obtained using the SFA for H(1s), 2×1014Wcm−2, and photonenergy 1.17 eV. The time integral in (3.398) was calculated either directly (labelled “exact”) orapplying the saddle-point approximation. The expected cut-off at harmonic order n = 72.3 isconfirmed. [From D.B. MILOŠEVIC and W. BECKER, Role of long quantum orbits in high-orderharmonic generation, Phys. Rev. A 66, 063417 (2002).]


with n the harmonic order. In the case of atoms in a linearly polarized laser field, forinstance, only odd harmonics are emitted, i.e., n = 1,3,5, . . .. One could think that this“quantized” emission is a quantum effect. However, this is not the case. Pure classicalsimulations also show harmonic generation and not just continuous spectra. The selec-tion rules governing which harmonic orders n are allowed and which are forbidden aredetermined by the symmetry of the combined system target + laser field, i.e., the sym-metry of the field dressed target, also called dynamical symmetry. We shall now employthe Floquet theory introduced in Sec. 3.5.1 to derive the selection rules for harmonicemission for a few exemplary systems.

Let H (t ) be the Hamiltonian of an electron in a linearly polarized monochromaticlaser field E cos(ω1t )ez of amplitude E and an ionic potential V (r ),

H (t ) =p2

2+ V (r )+ E z cosω1t . (3.400)

The Schrödinger equation reads i∂t |Ψ(t )⟩ = H (t )|Ψ(t )⟩, and since the Hamiltonian isperiodic in time,

H (t + 2π/ω1) = H (t ), (3.401)

from the Floquet theorem (cf. Section 3.5.1)

|Ψ(t )⟩= e−iεt |Φ(t )⟩, |Φ(t + 2π/ω1)⟩= |Φ(t )⟩ (3.402)

follows where ε is the quasi-energy, and |Φ(t )⟩ fulfills the Schrödinger equation

H (t )|Φ(t )⟩= ε|Φ(t )⟩, H (t ) = H (t )− i∂t (3.403)

which looks like a stationary Schrödinger equation in an extended Hilbert space withthe time as an additional dimension and with the scalar product

⟨⟨Φ|Φ′⟩⟩ := ω1

2π

2π/ω1∫

0

dt ⟨Φ(t )|Φ′(t )⟩. (3.404)

Hereafter, we refer to H and |Φ(t )⟩ already as Floquet-Hamiltonian and Floquet state,respectively (although usually this is done not before expanding them in Fourier series[see Section 3.5.1]).

Floquet states are field-dressed states. For the derivation of the harmonic generationselection rules we assume an infinitely long laser pulse. It is then reasonable to assumethat the system is well described by a single, nondegenerate Floquet state.

The only nonvanishing dipole expectation value is in field-direction and then reads

d (t ) =−⟨Ψ(t )|z |Ψ(t )⟩=−⟨Φ(t )|z |Φ(t )⟩. (3.405)


We define the dipole strength of the harmonic n as

|d (n)|2 =

2π/ω1∫

0

dt exp(−inω1t )d (t )

2

=⟨⟨Φ|exp(−inω1t ) z |Φ⟩⟩

2

(3.406)

which is proportional to the absolute square of the Fourier-transformed dipole. Thesquared extended Hilbert space matrix element on the right-hand side of (3.406) maybe also interpreted as the probability for a transition from a Floquet state to itself, gen-erated by the operator exp(−inω1t ) z, accompanied by the emission of radiation offrequency nω1.

The Floquet-Hamiltonian H (t ) is invariant under space inversion plus a translationin time by π/ω1,

Pinv = (r →−r , t → t +π/ω1) (3.407)

so that Pinv|Φ⟩ = σ |Φ⟩ and σ is a phase, i.e, |σ |2 = 1. Inserting the unity P−1inv

Pinv in thematrix element in (3.406) twice yields

⟨⟨Φ|exp(−inω1t ) z |Φ⟩⟩= ⟨⟨Φ|P−1inv︸︷︷︸

⟨⟨Φ|σ∗Pinv exp(−inω1t ) z P−1

inv Pinv|Φ⟩⟩︸︷︷︸σ |Φ⟩⟩

(3.408)

= ⟨⟨Φ|Pinv exp(−inω1t ) z P−1inv |Φ⟩⟩=−exp(−inπ) ⟨⟨Φ|exp(−inω1t ) z |Φ⟩⟩. (3.409)

It follows that n must be odd in order to fulfill

−exp(−inπ) = 1. (3.410)

Hence, only odd harmonics are emitted in the case of linearly polarized laser pulsesimpinging on spherically symmetric systems such as atoms.

In the case of monochromatic circularly polarized laser light (with the electric fieldvector lying in the xy-plane) the Floquet-Hamiltonian may be written (using the cylin-drical coordinates ρ and ϕ) as

H (t ) = Hkin+ V (r )+Ep

2ρcos(ϕ−ω1t )− i∂t . (3.411)

This expression is invariant under the continuous symmetry operation

Prot = (ϕ→ ϕ+θ, t → t −θ/ω1) (3.412)

with θ an arbitrary real number. Instead of (3.408) and (3.410) one has

⟨⟨Φ|exp(−inω1t )ρexp(∓iϕ)|Φ⟩⟩= exp(inθ∓ iθ) ⟨⟨Φ|exp(−inω1t )ρexp(∓iϕ)|Φ⟩⟩(3.413)


where ρexp(∓iϕ) is the dipole operator for circularly polarized light (with the samehelicity (−) and the opposite helicity (+) as the incident pulse, respectively). Equa-tion (3.413) requires for all θ

exp[iθ(n∓ 1)] = 1 (3.414)

to hold, which cannot be fulfilled for any n > 1 so that no harmonics are emitted.However, circularly polarized harmonics may be emitted if bichromatic incident

laser light is used. With the two lasers polarized in opposite directions and frequen-ciesω1 and mω1, respectively, the interaction Hamiltonian reads

W (t ) =E1p

2ρcos(ϕ−ω1t )+

E2p2ρcos(ϕ+mω1t ). (3.415)

E1 and E2 are the electric field amplitudes of the first and second laser, respectively. Thesymmetry operation under which the Floquet-Hamiltonian is invariant now reads

P (m+1)rot =

ϕ→ ϕ+

2π

m+ 1, t → t − 2π

ω1(m+ 1)

. (3.416)

In the same manner as in the two previous examples one arrives at the condition

1= exp

i2πn∓ 1

m+ 1

. (3.417)

Hence, harmonics of order

n = k(m+ 1)± 1, k = 1,2,3, . . . (3.418)

are expected. The harmonics with n = k(m+ 1) + 1 have the same polarization as theincident laser, whereas those with n = k(m + 1)− 1 are oppositely polarized. Withincreasing m more and more low order harmonics are suppressed. This might be apromising way to transfer laser energy efficiently to shorter wavelengths.

The same selection rule (3.418) is obtained for a target having a M -fold discrete rota-tional symmetry axis parallel to the laser propagation direction. An example for such atarget is the benzene molecule with M = 6. The Floquet-Hamiltonian in this case maybe written as

H (t ) = Hkin+ V (ρ,ϕ, z)+Ep

2ρcos(ϕ−ω1t )− i∂t . (3.419)

Owing to the discrete rotational symmetry CM of V (ρ,ϕ, z) the symmetry operationof interest now is

P (M )rot =ϕ→ ϕ+

2π

M, t → t − 2π

ω1M

, (3.420)


from which the selection rule

n = kM ± 1, k = 1,2,3, . . . (3.421)

follows which is indeed of the same form as in the bichromatic, atomic case (3.418).Note, that (3.419) is only a single active electron-Hamiltonian but sufficient for thepurposes here because the electron-electron interaction term is invariant under the op-eration (3.420) anyway.

For deducing these selection rules one assumes that the incident laser pulse is in-finitely long. This is required for (3.401) to be true. In finite laser pulses the simpleselection rules above may be violated and one has to consider not only a single Floquetstates but superpositions of them (see “Further reading”).

Further reading: The “classic” paper on the Lewenstein model is M. LEWENSTEIN,PH. BALCOU, M. YU. IVANOV, ANNE L’HUILLIER, and P.B. CORKUM, Theoryof high-harmonic generation by low frequency laser fields, Phys. Rev. A 49, 2117(1994). Perhaps the clearest account of HOHG [including a discussion of thefourth term in (3.383)] is presented in W. BECKER, A. LOHR, M. KLEBER, and M.LEWENSTEIN, A unified theory of high-harmonic generation: Application to polar-ization properties of the harmonics, Phys. Rev. A. 56, 645 (1997). A review is givenP. SALIÈRES, A. L’HUILLIER, P. ANTOINE, and M. LEWENSTEIN, Adv. At. Mol.Opt. Phys. 4, 83 (1999). Concerning the proper quantum mechanical calculationof the harmonic emission of an isolated atom see, for instance, B. SUNDARAMand P.W. MILONNI, Phys. Rev. Lett. 41, 6571 (1990); J.H. EBERLY and M.V. FE-DOROV, Phys. Rev. A 45, 4706 (1992); P.L. KNIGHT and P.W. MILONNI, Phys.Rep. 66, 21 (1980). Work where attosecond pulses (produced from harmonics)were used to follow atomic dynamics in real time or for measuring the incom-ing laser pulse itself is presented in M. DRESCHER et al., Time-resolved atomicinner-shell spectroscopy, Nature 419, 803 (2002) and R. KIENBERGER et al., Steer-ing attosecond electron wave packets with light, Science 297, 1144 (2002). For the re-altion of dynamical symmetries and harmonic generation selection rules see, e.g.,V. AVERBUKH, O.E. ALON, and N. MOISEYEV, Phys. Rev. A 60, 2585 (1999); F.CECCHERINI and D. BAUER, Phys. Rev. A 64, 033423 (2001); F. CECCHERINI,D. BAUER, and F. CORNOLTI, J. Phys. B: At. Mol. Opt. Phys. 34, 5017 (2001).

d. bauer- theory of intense laser-matter interaction

Documents