Lecture notes on the
Theory of intense laser-matter interaction
D. Bauer
Max-Planck-Institut für Kernphysik, Heidelberg, Germany
dieter.bauer@mpi-hd.mpg.de
June 22, 2006
2
Contents
1
Introduction
2
Classical motion in electromagnetic fields
2.1 The nonrelativistic ponderomotive force . . . . . . . . .
2.1.1 The ponderomotive force of a standing wave . .
2.2 Relativistic dynamics in an electromagnetic wave . . . .
2.2.1 The oscillation center frame . . . . . . . . . . . . .
2.2.2 The adiabatically ramped pulse in the lab frame
2.3 The relativistic ponderomotive force . . . . . . . . . . . .
2.3.1 Example: sin2 -pulse . . . . . . . . . . . . . . . . . .
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Atoms in external fields
3.1 Atomic units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Dyson series, S-matrix, Green’s functions, propagators, and resolvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Important models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Discrete level coupled to the continuum . . . . . . . . . . . . . . . .
3.3.2 Resonantly coupled discrete levels: Rabi-oscillations . . . . . . . .
3.4 Atoms in strong, static electric fields . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Tunneling ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Atoms in strong laser fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Floquet formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Non-Hermitian Floquet theory . . . . . . . . . . . . . . . . . . . . .
3.5.3 High-frequency Floquet theory and stabilization . . . . . . . . . .
3.6 Strong field approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Circular polarization and long pulses . . . . . . . . . . . . . . . . . .
3.6.2 Channel closing in above-threshold ionization . . . . . . . . . . . .
3.6.3 Linear polarization and long pulses . . . . . . . . . . . . . . . . . . .
3.6.4 Few-cycle above-threshold ionization . . . . . . . . . . . . . . . . . .
3.6.5 “Simple man’s theory” . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.6 High harmonic generation . . . . . . . . . . . . . . . . . . . . . . . .
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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 in
laser technology over the last twenty years, and there are good reasons to believe that
this 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., inertial
confinement fusion (i.e., nuclear fusion using lasers), “table top” particle acceleration,
or particle physics (e.g., particle creation in vacuum using lasers). Shorter pulses are
needed to study fast atomic processes directly in the time domain. Just as femtosecond
(1 fs = 10−15 = 0.000 000 000 000 001 s) laser technology revolutionized chemistry, the
hope is that attosecond (1 as = 10−18 s) laser pulses give a new twist on good old atomic
physics. First, successful “proof of principle”-like experiments have been already carried out. The shortness of these kind of laser pulses is impressively illustrated by the
comparison
8 min
1 fs
'
.
(1.1)
1s
age of the universe
5
CHAPTER 1. INTRODUCTION
6
Table 1.1: Comparison between a strong laser and sunlight.
strong laser
sun
ponderom.
[eV]
potential
(at 800 nm)
field strength [V/m]
intensity [W/cm2 ]
excursion [Å]
5.14 · 1011
3.51 · 1016
163
2169.2
1000
0.13
3.3 · 10−7
8 · 10−15
Finally, shorter wavelength lasers are desirable to complement (or to offer a cheaper
alternative to) synchrotron radiation sources with a vast amount of applications in material and biological sciences. Moreover, shorter wavelengths λ and shorter pulse durations are related since a lower limit for the pulse duration is the laser period 2π/ω
where ω = 2πc/λ is the laser frequency.
What do we mean by strong or intense when we talk about laser pulses? Whether a
laser pulse has to be considered strong or not depends on the force it exerts as compared
to other, competing forces. These other forces are typically the binding forces that
prevents the target from falling apart (even without laser). A good example is the force
that 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 and
proton. This electric field can be directly compared with the electric field of a laser pulse
E (r , t ) = Ê ei(ωt −k·r ) ,
(1.3)
and the latter is at latest to be considered strong if it equals the binding force. The laser
intensity is defined as the cycle-averaged modulus of the Poynting vector,
1
I = c 2 "0 |E (t ) × B(t )| = c"0 Ê Ê ∗ .
2
(1.4)
Here we use SI units and the relation between the electric field and the magnetic field B
for an electromagnetic wave in vacuum. The unit of the intensity I is that of an energy
flux, namely W/cm2 , which is energy passing through an area per time. Plugging in
EH yields I = 3.51 · 1016 Wcm−2 . This sounds a lot, especially if compared to sunlight
on 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 while
1022 Wcm−2 seems to be the maximum achieved so far. The other entries in Table 1.1
will be referred to later on.
The shortest pulses generated so far consist of only a few cycles in the “full width half
maximum” (FWHM) of the pulse at λ = 800 nm. The laser period for this TitaniumSapphire 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 ' 1.5 µm
in vacuum so that these pulses are not laser beams but rather laser light bullets of a
few wavelengths width. The pulse duration determines the time scale which can be
resolved directly in the time domain. Since atomic processes occur on the attosecond
time scale (except phenomena involving mainly Rydberg states), 5 fs pulses are still too
long do resolve them. However, in combination with shorter wavelength pulses of attosecond duration which can be generated by the 800 nm pulse itself via high harmonic
generation, femtosecond laser pulses have been already used to study experimentally
attosecond atomic processes such as the Auger decay.
So far, few-cycle laser pulses were only generated at rather modest laser intensities
below 1018 Wcm−2 . The most intense pulses with an intensity of 1020 Wcm−2 or higher
are of picosecond (1 ps = 10−12 s) duration or longer. It is instructive to calculate the
power and the energy of such a, e.g., 10 ps pulse if it is focused down to, say, (10 µm)2 :
1020 Wcm−2 · (10 µm)2 = 0.1 PW
(1.5)
where PW stands for petawatt (1015 W), and
0.1 PW · 10 ps = 1 kJ.
(1.6)
Currently, megajoule lasers are being constructed in several laboratories around the
world. The light pressures such pulses exert are enormous. From Eq. (1.7) below we
estimate pressures of the order of 10 Gigabar. Since dimensionally, pressure is an energy
density, another way of estimating the pressure is 1 kJ/[(10 µm)2 · c ·10 ps], which yields
the same answer. Such pressures even exceed those in the interior of stars and may offer
the opportunity to study the equation of state of various materials under extreme conditions. 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 underground
nuclear explosions. The newly constructed “national ignition facility” (NIF) in the US
aims at those aspects of intense laser-matter interaction.
Coming back from “boom-boom”-physics to theory, experience shows that standard, quantum mechanical, perturbative approaches start to fail already around
1013 Wcm−2 when applied to the ionization of atomic hydrogen in a laser field, for instance. The new analytical techniques that have been developed (as well as those still
lacking) will be topics of this lecture.
Many things become more complicated in strong laser fields, however, some others
are 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 or
harmonics spectra and nonsequential double ionization, although utterly unaccessible
in terms of standard perturbation theory, turned out to have rather simple, almost classical explanations. These effects and their theoretical explanation will be also part of
this lecture.
Finally, there is the relativistic domain which separates the strong laser fields from
the ultra-strong ones. In the case of a free electron in a laser pulse there is no binding
CHAPTER 1. INTRODUCTION
8
force to compete with. However, there are strong field effects as well. Already at moderate laser intensities the so-called ponderomotive force is appreciable, depending on the
pulse shape. Ponderomotive effects will be discussed in quite some detail and reappear
frequently during the course of this lecture; in brief: the ponderomotive force in general
pushes charged particles out of regions of high intensity (i.e., the laser focus) and exerts
pressure 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 to
the ponderomotive force. The ponderomotive force is related to the breakdown of the
so-called dipole approximation
E (r , t ) = Ê e−i(ωt −k·r ) ' Ê e−iωt = E (t )
(1.8)
where the spatial dependence of the field is neglected because the wavelength is so much
larger than the excursion r (t ) of, say, an electron in the field. The classical motion of
the electron is governed by the Lorentz force (SI units)
ṗ = −e[E (r , t ) + v × B(r , t )]
(1.9)
(where p, v are the electron momentum and velocity, respectively). Making the dipole
approximation implies the omittance of the magnetic field so that there is no v × Bforce. Hence, the electron is assumed to just oscillate in laser polarization direction (or
in 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 the
v × B-force is a v/c-effect, from a certain laser intensity on there is no excuse anymore
for neglecting it. This threshold intensity depends on the laser wavelength, and for the
commonly used 800 nm it turns out to be of the order of 1018 Wcm−2 . The v × B-force
is a v/c effect and as such not yet relativistic. Truly relativistic effects are of order (v/c)2
or higher and will be discussed in the last part of this lecture.
Chapter 2
Classical motion in electromagnetic
fields
In this Chapter we shall derive the non-relativistic ponderomotive force and the relativistic 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 force
The 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 making
up the electromagnetic field of a laser pulse, that is,
E (r , t ) = Ê (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. The
derivation of the ponderomotive force relies on the possibility to separate the relevant
time 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 Ê (r , t ), apart from containing e−ik·r , describes the laser field envelope, having only a “slow” time dependence
so that ∂ t B = iωB to high accuracy.
In lowest order the particle just oscillates around its current position r0 due to the
electric field (the possible slow time-dependence in Ê is suppressed for notational convenience):
m r̈1 = q Ê (r0 )eiωt .
(2.4)
9
10
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
The corresponding velocity and position are
r˙1 = −
iq
mω
Ê (r0 )eiωt ,
r1 = −
q
mω 2
Ê (r0 )eiωt .
(2.5)
In the next higher order one has
m r̈2 = q[(r1 · ∇)E (r0 , t ) + ṙ1 × B(r0 , t )]
(2.6)
where the electric field has been expanded around r0 . So far we used complex fields and
implicitly understood that the real part has to be taken. Now the calculation becomes
nonlinear and this trick is not applicable anymore. Hence we write

q
(Ê e+ + Ê ∗ e− ) · ∇(Ê e+ + Ê ∗ e− )
(2.7)
m r̈2 = q −
2
4mω
‹
q
+
(−iÊ e+ + iÊ ∗ e− ) × ∇ × (iÊ e+ − iÊ ∗ e− )
2
4mω
q2
[Ê · ∇Ê ∗ + Ê × ∇ × Ê ∗ + c.c. + Ω2ω ]
(2.8)
= −
2
4mω
with e± = e±iωt and Ω2ω collecting all terms ∼ e±i2ωt which disappear upon averaging
over a laser period,
m r̈2 = −
q2
4mω 2
[Ê · ∇Ê ∗ + Ê × ∇ × Ê ∗ + 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 r̈2 = −
q2
4mω
2
∇|Ê (r , t )|2 .
(2.11)
In Ê (r , t ) the position r now refers to the so-called oscillation center. We remind that the
time dependence of the envelope must be “slow” as compared to the laser period because
otherwise the separation of time-scales used in the derivation makes no sense. The
ponderomotive force (2.11) can obviously be derived from the ponderomotive potential
Φp (r , t ) =
q2
4mω
2
|Ê (r , t )|2
(2.12)
which is proportional to the laser intensity and independent of the sign of the particle’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 the
denominator the immediate effect on electrons is much larger than on ions. The ponderomotive force is inverse proportional to the square of the laser frequency, meaning
that 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
|ṙ1 | c,
k · r1 1
(2.13)
for an electromagnetic wave and a longitudinal wave, respectively. The condition for
the electromagnetic wave is exactly the same indicating the breakdown of the dipole
approximation (1.8). As a consequence, (2.12) is not applicable to relativistically intense
laser pulses. We shall derive in the subsequent Section the relativistic expression for the
ponderomotive force of a laser pulse.
The ponderomotive potential (at a certain position r ) equals the average quiver energy of the charged particle in the laser field (at that position). In fact, taking r˙1 from
(2.5) yields
1
q2
Up = m r˙1 2 =
|Ê |2 .
(2.14)
2
2
4mω
Is there a connection between this mere number Up and the potential Φp ? Consider a
spatially finite field structure which leads to Φp . Now, think of a particle injected into
this structure with some kinetic energy W0 . While strolling through the field structure
the particle will encounter regions of varying Φp . In regions of high Φp it will oscillate
with larger amplitude (high Up ) than in regions of low Φp (low Up ). After averaging out
the fast time scales, the whole system is conservative so that W + Up = W0 . This means
that the local Up indeed serves as a potential and is just the Φp derived above.
2.1.1 The ponderomotive force of a standing wave
Let us consider the standing wave
E (r , t ) = Ê eiωt (e−i k·r + ei k·r ) = 2Ê eiωt cos(k · r ).
(2.15)
Comparing this with (2.2) we identify
Ê (r , t ) = 2Ê cos(k · r )
(2.16)
and can immediately use expressions (2.11) and (2.12) to give
Φp (r , t ) =
q 2 |Ê |2
mω 2
2
cos (k · r ),
Fp (r , t ) =
q 2 |Ê |2
mω 2
k sin(2k · r ).
(2.17)
The ponderomotive force will push the particle towards the nodes of the standing wave
where the field vanishes. Depending on its initial velocity, the particles may be trapped
12
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
Figure 2.1: Trajectories of an electron in a standing laser wave (2.15) (polarized along ey , propagation directions along e x ) that starts at t = 0 from y = 0 and k x = π/40. The amplitudes were
e Â/mc = −e Ê/mωc = −0.1 (upper left), −0.2 (upper right), and −0.5 (lower). With increasing
laser intensity, the electron is able to escape from the “valley” in which it is released. It can then
be trapped temporarily in other valleys (upper right and lower plot), and the dynamics can be
shown to become chaotic.
inside the “valleys” of the standing wave. If we assume that the propagation direction
is e x and expand the force around the position of a node, we obtain a ponderomotive
force of the form
2q 2 |Ê|2
00
.
(2.18)
Fp“valley ' −Ω2 x,
Ω2 =
mc 2
The trapped particle will thus undergo harmonic secular motion with a frequency Ω
that is proportional to the electric field (which reminds of the Rabi-frequency in resonant laser-atom interaction, to be discussed later on). However, since the frequency increases with the field strength, the whole idea of separation of time scales breaks down
at some point. In fact, the numerical solution of the (relativistic) equations of motion
shows that for sufficiently high field strengths the particle may leave the valley in which
it was trapped. The motion can be shown to be chaotic then. The particle may be temporarily 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 be
used 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 laser
beams will be deflected. If the de Broglie wavelength is chosen properly, a detector on
the opposite side will measure an interference pattern.
Some remarks
In the physics of laser-matter interaction, the importance of the ponderomotive potential is hard to overestimate. In many effects it sets the relevant energy scale, as we shall
see later on. In laser-atom interaction, for instance, Φp (r , t ) determines not only the
cut-offs of photoelectron and harmonics spectra but also equals the AC Stark-shift of
the continuum. In laser-plasma physics, instabilities can be understood in terms of the
ponderomotive force.
As a historical side remark it should be mentioned that the ponderomotive force has
been 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 the
ponderomotive force on two charged particles bound together by a linear force (i.e.,
a harmonic oscillator). Such particles with internal degrees of freedom may display
counter-intuitive behavior, for instance, acceleration into regions of high |Ê |2 rather
than repulsion from them. Such kind of ponderomotive forces are used to trap neutral 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 center
is done in the physics of magnetized plasmas for the so-called guiding center. In the
same way we got rid of the fast but uninteresting oscillations on the time scale of a laser
period (and twice the laser period), the fast cyclotron motion around the field lines of a
strong 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 of
celestial mechanics. This is also where the term secular motion comes from. The effect of
Jupiter on the earth’s motion around the sun, for instance, is negligible on the time scale
of a few years but may have to be taken into account on longer time scales. One practical
advantage of the separation of time scales is for numerical simulations. Imagine you
want to simulate one second of the laser plasma dynamics using a fluid code. All effects
you 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 time
scale. 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 ponderomotive
potential comes to your rescue: there only the laser pulse envelope enters while the time
scales of the laser period (and faster) are eliminated.
14
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
Problem 1.1 Calculate the ponderomotive force of an elliptically polarized laser pulse.
2.2
Relativistic dynamics of a charged particle in an
electromagnetic wave
Let us consider the motion of a particle of charge q and mass m in an electromagnetic
field of the form
A(r , t ) = Âσ(η)P (η),
η = ωt − k · r = k µ xµ ,
A · k = 0.
(2.19)
Here, A(r , t ) is the vector potential of amplitude Â, σ(η) carries the fast timedependence, and P (η) is the slowly varying envelope. The phase η is a relativistic
invariant, as it is evident from η = k µ xµ where we use common relativistic notation (µ = 0, 1, 2, 3), x µ = (c t , r ), k µ = (ω/c, k), the sum convention, and the metric
g = diag(1, −1, −1, −1). The fields are
E = −∂ t A,
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 of
Fields, based on the relativistic Hamilton-Jacobi equation.
The relativistic Hamiltonian governing the motion of a charged particle in an electromagnetic field given by the scalar potential φ and the vector potential A reads
Æ
H = c 2 m 2 c 2 + (P − qA)2 + qφ.
(2.21)
Here, P is the canonical momentum (while p = P − qA is the kinetic momentum). As
in the derivation of the Klein-Gordon equation we square (2.21) and obtain
1
c2
(H − qφ)2 = m 2 c 2 + (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 = m 2 c 2 + (∇S − qA)2 .
(2.24)
This expression can be written in a covariant manner. Introducing the four vectors
∂ µ S = (∂c t S, −∇S) = (−E /c, −P) =: −P µ
(2.25)
Aµ = (φ/c,A),
(2.26)
gµν (∂ µ S + qAµ )(∂ ν S + qAν ) = m 2 c 2 .
(2.27)
and
eq. (2.24) can be written as
We are interested in the dynamics of a charged particle in an electromagnetic wave
and therefore have
Aµ = (0,A),
Aµ kµ = 0.
(2.28)
With the Ansatz
S = Sfree + Sfield (η) = βµ x µ + Sfield (η)
(2.29)
where the βµ play the role of the new, constant momenta (+ energy), we obtain
∂µ S = βµ + ∂µ Sfield = βµ + ∂µ η
∂ Sfield
∂η
0
= βµ + kµ Sfield
(2.30)
0
where Sfield
= ∂η Sfield . The Ansatz with Sfield depending only on the invariant phase η but
not on space and time separately is crucial. The fact that it works makes the problem
soluble at all.
Plugging (2.30) into (2.27) leads to
0
0
(βµ + kµ Sfield
+ qAµ )(βµ + k µ Sfield
+ qAµ ) = m 2 c 2 .
(2.31)
Making use of kµ k µ = 0 and Aµ k µ = 0, this equation can be solved for Sfield :
Sfield (η) =
1
2βµ k µ
Z
η
η0
€
Š
dη0 m 2 c 2 − [βµ + qAµ (η0 )][βµ + qAµ (η0 )]
(2.32)
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
16
and the total action thus is
S = βµ x +
= βµ x µ +
Z
1
µ
η
2βµ k µ η0
Zη
1
2βµ k µ
€
Š
dη0 m 2 c 2 − [βµ + qAµ (η0 )][βµ + qAµ (η0 )]
(2.33)
€
Š
dη0 m 2 c 2 − βµ βµ − 2qβµ Aµ (η0 ) − q 2 Aµ (η0 )Aµ (η0 ) .
η0
The new, constant four-momentum fulfills
βµ βµ = m 2 c 2
(2.34)
and is given through the initial conditions. The canonical four-momentum is given by
[making use also of (2.34)]
µ
µ
µ
P = −∂ S = −β +
2qβ · A(η) + q 2 A(η) · A(η)
2β · k
kµ
(2.35)
with a·b = aµ b µ . The derivatives of the action (2.33) with respect to the βs are constant
and give us the trajectory:
∂S
∂β
µ
= xiniµ = xµ −
−
⇒
xµ
Z
kµ
2(β · k)
Zη
1
2
η
dη0 m 2 c 2 − β · β − 2qβ · A(η0 ) − q 2 A(η0 ) · A(η0 )
η0
€
Š
dη0 βµ + qAµ (η0 )
β · k η0
Zη
Š
€
1
= xiniµ +
dη0 βµ + qAµ (η0 )
β · k η0
kµ Z η
0
0
2
0
0
−
dη
2qβ
·
A(η
)
+
q
A(η
)
·
A(η
)
.
2(β · k)2 η0
(2.36)
(2.37)
In the last line we used (2.34) again.
We shall now specialize on an electromagnetic wave
A(η) = Âey sin η,
k = ke x ,
(2.38)
i.e.,
Aµ = (0, 0, Âsin η, 0),
k µ = (ω/c, k, 0, 0) = (k, k, 0, 0).
(2.39)
In this case we have
β · A = −β2 Âsin η,
A · A = −Â2 sin2 η,
β · k = k(β0 − β1 ),
(2.40)
2.2. RELATIVISTIC DYNAMICS IN AN ELECTROMAGNETIC WAVE
17
(the reader should be careful not to mix up upper indices with powers!) and (2.35)
becomes explicitly
0
P =
E
c
0
= −β −
P 1 = p x = −β1 −
2qβ2 Âsin η + q 2 Â2 sin2 η
2(β0 − β1 )
2qβ2 Âsin η + q 2 Â2 sin2 η
2(β0 − β1 )
P 2 = py + q Âsin η = −β2 ,
3
,
(2.41)
,
(2.42)
(2.43)
3
P = p z = −β .
(2.44)
Clearly, the canonical momenta in y- and z-direction are conserved, and the relation
between canonical and kinetic momentum is included in (2.43).
2.2.1 The oscillation center frame
We are still free to choose the initial conditions βµ . This is equivalent to choose a
certain reference frame in which we want to study the dynamics. The only restriction
is β · β = m 2 c 2 . Let us choose the βs in such a way that the oscillation center of the
particle is at rest:
p x = py = p z = 0.
!
(2.45)
β2 = β3 = 0
(2.46)
From eqs. (2.43) and (2.44) follows
so that with
β · β = m2 c 2
⇒
Æ
β0 = − m 2 c 2 + (β1 )2
(2.47)
(minus sign because E in (2.41) must be positive in the free particle case where  = 0).
From p x = 0 and (2.42) we have

‹
Æ
1
1
2 2
1 2
2β β + m c + (β ) = 2qβ2 Âsin η + q 2 Â2 sin2 η.
(2.48)
The cycle average of sin η vanishes while sin2 η = 1/2. Equation (2.48) can then be solved
for β1 ,
q 2 Â2
1
.
(2.49)
|β1 | = Æ
4 m 2 c 2 + q 2 Â2 /2
The sign will be chosen later. The trajectory in the oscillation center frame can be
calculated from (2.37) and reads
Zη
β0
q 2 Â2
c t = c tini +
(η − η0 ) +
dη0 sin2 η0 ,
(2.50)
0
1
0
1 2
k(β − β )
2k(β − β ) η0
18
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
x
x
1
1
= xini +
2
2
= xini +
β1
k(β0 − β1 )
(η − η0 ) +
Z
q Â
k(β0 − β1 )
Z
q 2 Â2
2k(β0 − β1 )2
η
dη0 sin2 η0 ,
(2.51)
η0
η
dη0 sin η0 ,
(2.52)
η0
x 3 = xini 3 .
(2.53)
The difference of (2.50) and (2.51) yields
c t − x 1 = c tini − xini 1 +
η − η0
(2.54)
k
so that
η = ωt − k x,
η0 = ωtini − k xini ,
(2.55)
1
as it should. Since in the oscillation center x should only oscillate but not drift, the
term ∼ (η − η0 ) has to be canceled by the first term of the integral
Z
1
1
(2.56)
dη sin2 η = η − sin 2η.
2
4
This is the case if we choose the positive sign in (2.49),
q 2 Â2
1
1
.
β = Æ
4 m 2 c 2 + q 2 Â2 /2
(2.57)
Explicitly, we have
q
m 2 c 2 + q 2 Â2 /4
β = −Æ
,
m 2 c 2 + q 2 Â2 /2
0
0
1
β −β =−
m 2 c 2 + q 2 Â2 /2,
(2.58)
and the trajectory is given by [setting rini appropriately and x µ = (c t , x, y, z)]
x = −
y =
q 2 Â2
8k(m 2 c 2 + q 2 Â2 /2)
Æ
k
z = 0.
q Â
sin 2η,
cos η,
(2.59)
(2.60)
m 2 c 2 + q 2 Â2 /2
(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 (e x in our case), as shown in Fig. 2.2. If
we define the dimensionless vector potential amplitude as
a=
q Â
mc
(2.62)
2.2. RELATIVISTIC DYNAMICS IN AN ELECTROMAGNETIC WAVE
19
Figure 2.2: Figure-eight dynamics of a charged particle in an electromagnetic wave, as seen in
the oscillation center frame. The trajectory in the xy-plane, where x is the propagation direction
of 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 x̂ and k ŷ approach
p
the calculated values 1/4 and 2, respectively.
we have
kx = −
a2
8(1 + a 2 /2)
| {z }
k x̂
sin 2η,
ky = Æ
a
1 + a 2 /2
| {z }
cos η,
k z = 0.
(2.63)
k ŷ
The size of the figure-eight does not increase infinitely as the laser intensity goes to
infinity:
p
1
x̂
1
(2.64)
lim k x̂ = ,
lim k ŷ = 2,
lim = p ' 0.177.
a→∞
a→∞
a→∞ ŷ
4
4 2
Note that we only have the orbit parameterized with the invariant phase η (which is
proportional to the proper time). We do not know the explicit expressions for x and y
as functions of t . Parameterized with η, the trajectory looks extremely simple. In fact,
only η and 2η shows up. If η were just ωt we could talk about the fundamental and the
second harmonic and nothing else. However, since η = ωt − k x and x itself depends
on t (or η), all frequencies, that is, all multiples of the laser frequency enter. This
has consequences for the radiation emitted by such a particle. We neglect the emitted
radiation in this Chapter but will come back to it later on.
2.2.2 The adiabatically ramped pulse in the lab frame
We shall now investigate the same situation in the lab frame. We assume that the laser
pulse has been ramped up adiabatically from P = 0 at η → −∞ to P = 1, and that the
20
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
Figure 2.3: Particle motion in the lab frame for a = 1. While oscillating in polarization direction, the particle is drifting in propagation direction. The laser pulse is adiabatically ramped up
and of constant amplitude afterwards. The particle was initially at rest. Note that there is no
backward 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 k µ change if we switch from one frame to the other (relativistic Doppler effect). Since it should be clear from the context in which frame we are
working, we suppress explicit indices indicating the frame.
The initial condition p = 0 for η → −∞ implies that β1 = β2 = β3 = 0 and
β0 = −mc. For the trajectory one finds
x =
a2
1
η − sin 2η ,
4k
2
a
y =
cos η,
k
z = 0,
(2.65)
(2.66)
(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 before
while in laser propagation direction it is drifting. An example of such an orbit is shown
in Fig. 2.3.
There seems to be a contradiction as far as the excursion of the particle in polarization direction is concerned: in the lab frame this excursion is just proportional to a and
thus
pis, in principle, unlimited. In the oscillation center frame the excursion was limited
to 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 center
frame.
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
P0
mc
=
E
mc
2
=1+
a2
2
P1
sin2 η,
mc
=
px
mc
=
a2
2
sin2 η,
P 2 = P 3 = p z = 0.
(2.68)
It is obvious that for our choice of the vector potential, the Lorentz transformation has
to be performed parallel to e x so that


γoc
−voc γoc /c 0 0


γoc
0 0 
 −voc γoc /c
Λµν = 
(2.69)

0
0
1 0 

0
0
0 1
with voc the oscillation center velocity we are looking for and γoc = (1 − voc 2 /c 2 )−1/2 .
Applying this transformation to (2.68), i.e., P 0 µ = Λµν P ν gives us
P 01
γoc mc
=−
voc
c
‚
1+
a2
2
Œ
2
sin η +
a2
2
sin2 η.
(2.70)
1
The condition P 0 = p x0 = 0 yields
voc =
a2
4 + a2
c.
(2.71)
This is the velocity of the oscillation center of the particle in the lab frame if it was at
rest before the pulse arrived.
2.3 The relativistic ponderomotive force
We shall now derive the relativistic orbit of the oscillation center for a vector potential of the form (2.19). It is desirable to consider the oscillation center as a relativistic
pseudo particle. However, it is a priori not given for granted that the oscillation center
coordinates and momenta automatically “behave” in a relativistic, proper way. The correct averaging over a laser period is crucial here. The situation is somewhat similar to
the relativistic center of mass, which
Pif one naively extends the2 nonrelaP is ill-defined
2 −1/2
,
tivistic expression and writes R =
i γi mi , where γi = (1 − vi /c )
i ri γi mi /
without specifying in which system this expression should be evaluated. This is easily
illustrated by the following example: Imagine two particles of equal rest mass moving
with velocities ±ve x in the lab frame, and v very close to c. The observer in the lab
frame will come to the conclusion that the center of mass (as defined above) is half way
between both particles and stationary. Let us now transform to the reference frame in
which one of the two particles is at rest. In this frame the other particle will have a
so much higher “effective mass” γ m m that the center of mass will be effectively at
22
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
the position of this second particle. The two centers of mass measured by the two observers will not transform properly into each other by a Lorentz transformation. The
ambiguity can be circumvented by defining the center of mass in the system in which it
is at rest. Once determined it can then be transformed to any other system by a Lorentz
transformation.
In the case of the relativistic oscillation center of a charged particle in a laser field
such as (2.19) we are in the fortunate situation to have an invariant phase η = k · k
over which we can average in an invariant manner. Already in a standing wave there is
not such a single invariant η anymore. In the lab frame, for instance, the two phases
η1 = ωt − k · r and η2 = ωt + k · r show up. In a reference frame moving along k
one of the two waves will be red-shifted while the other will be blue-shifted, i.e., in this
frame there will be no standing wave at all. As in the center of mass-problem, in such a
situation the oscillation center — if existing — has to be defined in the system where it
is at rest, and averaging has to be done over the proper time.
In this lecture we will restrict ourselves to the relativistic ponderomotive force in
the travelling laser pulse where averaging over the invariant phase η can be done in any
frame.
Averaging of (2.35) and (2.37) yields [with βµ = (β0 , β)]
roc = r (η) = rini + β
η − η0
β·k
poc = P(η) = −β + k
Eoc
c
=
E
c
= −β0 + k
−k
q 2A · A
2β · k
q 2A · A
2β · k
Z
q2
η
dη0 A · A,
2
2(β · k)
(2.72)
η0
,
(2.73)
.
(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
poc µ = (Eoc /c, poc ).
poc µ poc µ = M 2 c 2 ,
(2.76)
For consistency, this mass has also to fulfill
‚
poc = γoc M voc ,
voc = ṙoc ,
γoc = 1 −
voc 2
Œ−1/2
c2
.
(2.77)
For the velocity one finds
voc = (ω − k · voc )
droc
dη
= (ω − k · voc )
β − kΓ
β·k
(2.78)
2.3. THE RELATIVISTIC PONDEROMOTIVE FORCE
23
where the abbreviation
Γ=
q 2A · A
(2.79)
2β · k
has been introduced. In order to solve (2.78) for voc we decompose the oscillation center
velocity and β in components parallel and perpendicular to k,
voc = voc k
k
k
+ voc ⊥ ,
β = βk
k
k
+ β⊥ .
(2.80)
This yields
voc k = ω
βk − kΓ
β0 k − k 2 Γ
,
voc ⊥ = ω
and
γoc = − Æ
β⊥
β0 k − k 2 Γ
β0 − kΓ
m 2 c 2 − 2kΓ(β0 − βk )
,
.
(2.81)
(2.82)
In the same way we decompose poc ,
poc = poc k
k
k
+ poc ⊥
(2.83)
with (2.73) leading to
poc k = −βk + kΓ,
poc ⊥ = −β⊥ .
We then obtain
!
poc µ poc µ = β · β − q 2 A · A = M 2 c 2
(2.84)
(2.85)
so that, using β · β = m 2 c 2 , the mass of the oscillation center is given by
M=
1
q
c
m 2 c 2 − q 2 A · A.
(2.86)
It is left as an exercise to the interested reader to check that this mass is indeed consistent
with poc = γoc M voc .
Another check is the following: if we transform to the system moving with voc , the
0
energy Eoc
in this system should be simply M c 2 . The transformation reads
0
Eoc
c
‚
= γoc
Eoc
c
−
voc · poc
c
Œ
.
(2.87)
Using (2.74), (2.82), and the decompositions for voc and poc above, one readily checks
that indeed
0
Eoc
= M c 2.
(2.88)
24
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
The relativistic oscillation center pseudo particle possesses a rest mass which is not a
constant 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 = ṗoc = (ω − k voc k )
d
dη
poc .
(2.89)
Using (2.73) and (2.79) one obtains
Fp =
ωq 2
k
d
k 2(β0 − kΓ) dη
The Minkowski force
FpM =
A · A.
(2.90)
d
poc = γoc Fp
(2.91)
dτ
(where τ is the proper time) assumes a particularly simple form. Using (2.82) (note that
the denominator of γoc is proportional to M ) and
d
1
=− ∇
dη
k
(2.92)
0
FpM = −∇Eoc
= −c 2 ∇M .
(2.93)
one obtains
The phase-dependent rest mass is the origin of the oscillation center motion. In the
nonrelativistic case we saw already that the local quiver motion gives rise to the ponderomotive potential. Relativistically, this quiver motion is included in the rest mass of
the oscillation center.
2.3.1 Example: sin2 -pulse
Let us consider a pulse envelope of the form
P (η) = sin2 (εη),
0<ε1
(2.94)
which may describe a pulsed laser beam in which one pulse lasts from η = nπ/ε to
η = (n + 1)π/ε with n = 0, 1, 2, . . .. At time t = 0 the particle is assumed to be at rest at
r = 0 so that
β = 0,
β0 = −mc.
(2.95)
Equations (2.72) and (2.73) become in this case
3
1
1
q 2 Â2
k
η − sin(2εη) +
sin(4εη)
,
roc =
2 2
4ε
32ε
k
4m c k 8
q 2 Â2 4
k
poc =
sin (εη) .
4mc
k
(2.96)
(2.97)
2.3. THE RELATIVISTIC PONDEROMOTIVE FORCE
25
Figure 2.4: Electron trajectory for a = −0.5 and ε = 0.01. Polarization and propagation directions are ey and e x , respectively. The displacement after the first pulse agrees with formula (2.99)
(which yields k · roc 1 = 7.36 in this case).
The relativistic ponderomotive force (2.90) is
‚
Œ
ωmc a 2
d
sin2
Fp =
sin4 (εη)
4
2
a sin (εη) + 4 dη
(2.98)
where a = q Â/mc has been used again. In the nonrelativistic limit |a| 1 the nonrelativistic ponderomotive force (2.11) is recovered (using (2.92) and Â2 = Ê 2 /ω 2 ). From
(2.96) we can immediately infer the displacement of the particle after the first pulse:
a2 3
k
π
η1 ,
η1 = .
(2.99)
4k 8 k
ε
Figure 2.4 shows the trajectory of an electron for a = −0.5 and ε = 0.01 during the first
pulse, confirming formula (2.99).
roc 1 =
Problem 2.1 Check that poc = γoc M voc .
Further reading: An excellent paper on the subject (including references to important
previous work) is E. S. SARACHIK and G. T. SCHAPPERT, “Classical Theory of
the Scattering of Intense Laser Radiation by Free Electrons”, Phys. Rev. D 1, 2738
(1970).
26
CHAPTER 2. CLASSICAL MOTION IN ELECTROMAGNETIC FIELDS
Chapter 3
Atoms in external fields
In this Chapter we shall first introduce atomic units and remind the reader of basic
quantum mechanical techniques and atomic phenomena such as time-dependent perturbation theory, resonant interaction (Rabi flopping), Stark-effect etc., before we come
to the actual topic of this lecture, i.e., the strong field effects and the new theoretical
approaches that have been developed in this context.
3.1 Atomic units
We will introduce the atomic units in a somewhat formal way. For beginners, the use
of atomic units is sometimes confusing because dimensional checks are not possible in
a straightforward way once ħh , me , e have been set “equal to unity”. Another practical
problem for beginners is to convert the result of a calculation performed using atomic
units 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 be
as well 42 ħh , 42 e, 42 me , 42 · 4π"0 , 42 ħh me /e, . . .. A good example is the expression for
the nonrelativistic eigenenergies of hydrogen-like ions, which in atomic units is simply
given by
Z2
En = − 2 ,
n = 0, 1, 2, . . .
(atomic units)
(3.1)
2n
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 to
SI units. If one performs the entire calculation (i.e., the solution of the Schrödinger
equation) in SI units, one obtains
En = −
Z2
me e 4
2n 2 ħh 2 (4π"0 )2
,
n = 0, 1, 2, . . .
(SI units).
(3.2)
Even without knowing the left-hand-side we can recover from the right-hand-side alone
that the result is an energy, provided we know the SI units of ħh , me , e, and "0 . This
27
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
28
seeming asymmetry between the unit systems arise from the fact that “setting ħh , me , e,
and 4π"0 equal to unity” is more than a change of units. It is like setting kg, m, s, and
C to unity.
Let us denote mass, length, time, and charge by M, L, T, and C, respectively. In
atomic units we wish to use the action ħh , the electron mass me , the modulus of the
electron charge e, and 4π times the permittivity of vacuum 4π"0 as the basic units. The
relation between both system of units is established by (“[...]” meaning “units of ...”)
[ħh ]
[me ]
[e]
[4π"0 ]
=
=
=
=
Ma11 La12 Ta13 Ca14 ,
Ma21 La22 Ta23 Ca24 ,
Ma31 La32 Ta33 Ca34 ,
Ma41 La42 Ta43 Ca44 .
(3.3)
(3.4)
(3.5)
(3.6)
The exponents a41 , . . . , a44 , for instance, are determined by noticing that in SI units
the dimension of "0 is Coulomb per Volt and meter. Volt is a derived unit. Because
Coulomb times Volt is an energy one has
V=
ML2
CT2
,
(3.7)
and hence
[4π"0 ] = M−1 L−3 T2C2 .
(3.8)
Similarly, the remainder of the dimensional matrix is easily calculated and reads

a
 11
 a
A =  21
 a31
a41
a12
a22
a32
a42
a13
a23
a33
a43
a14
a24
a34
a44
 
 
 
=
 

1
2 −1 0

1
0
0 0 
.
0
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, and
charge unit, denoted by M , L , T , and C , respectively. Those are later needed for the
transformation 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 meb22 e b23 (4π"0 ) b24 .
(3.11)
Plugging in (3.3)–(3.6)and using the values for the as in (3.9) lead to
L = M b21 +b22 −b24 L2b21 −3b24 T−b21 +2b24 C b23 +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, and
the exponents of mass, time, and charge must be zero. One finds b21 = 2, b22 = −1,
b23 = −2, b24 = 1 so that
L=
ħh 2 4π"0
= 0.5292 · 10−10 m,
2
me e
(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 meb12 e b13 (4π"0 ) b14 ,
(3.14)
L
= ħh b21 meb22 e b23 (4π"0 ) b24 ,
(3.15)
T
= ħh b31 meb32 e b33 (4π"0 ) b34 ,
(3.16)
C
with

b
 11
 b
B =  21
 b31
b41
= ħh
b12
b22
b32
b42
b13
b23
b33
b43
b41
meb42 e b43 (4π"0 ) b44
b14
b24
b34
b44
(3.17)
 
0
1
0
 
  2 −1 −2
=
  3 −1 −4
0
0
1
0
1
2
0



.

(3.18)
It is readily checked that
A · B = B · A = 1.
(3.19)
As an example for the transition from SI to atomic units, let us now formally rewrite
the time-dependent Schrödinger equation in position space representation
iħh
∂
∂t
Ψ(r , t ) = H Ψ(r , t )
(3.20)
in atomic units. We have that (with the values of primed symbols in atomic units)
ħh = ħh
0ML
2
= ħh 0 ħh
T
0
t = t T =t
0
ħh 3 (4π"0 )2
me e 4
Ψ = Ψ0 L −3/2 = Ψ0
H = H0
ML2
T2
= H0
ħh 0 = 1,
⇒
,
ħh 2 4π"0
(3.21)
(3.22)
!−3/2
,
me e 2
me e 4
ħh 2 (4π"0 )2
.
(3.23)
(3.24)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
30
Plugging this into (3.20) (and taking the primed wavefunction as a function of the
primed variables) simply yields
iħh 0
∂
∂ t0
Ψ0 (r 0 , t 0 ) = H 0 Ψ0 (r 0 , t 0 ),
(3.25)
i.e., the same equation primed. However, since ħh 0 = 1 [cf. (3.21)] this simplifies to
i
∂
∂t
0
Ψ0 (r 0 , t 0 ) = H 0 Ψ0 (r 0 , t 0 ).
(3.26)
The ħh effectively disappeared or, more precisely, is hidden in the units. This is what
is actually meant by “setting ħh = me = e = 1”. In the same way ħh disappeared in this
example, me , e, and 4π"0 are effectively removed. If, for instance,
H=
p2
2me
−
Z e2
+ eE · r ,
4π"0 r
(3.27)
that is, the Hamiltonian governing the electron of a hydrogen-like ion in an electric field
E , in atomic units
p 02 Z
(3.28)
H0 =
− 0 + E0 · r0
2
r
holds.
Finally, we give an example how to return to SI units after a busy calculation employing atomic units. Let us suppose the result of this calculation is a an electric field
strength
0 0
0 0
E 0 = f (α0 , β0 , . . .) ei(ω t −p ·r )
(3.29)
and f is some function of the entities α0 , β0 , . . . which can be converted to the SI values
α, β, . . ., giving rise to some overall factor K = ħh σ meη e ξ (4π"0 )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
me2 e 5
ML
0
E = E0
=
E
.
(3.30)
CT 2
ħh 4 (4π"0 )3
Hence
me2 e 5
me2 e 5
0 0
0 0
0
E =E 4
=K 4
f (α, β, . . .) ei(ω t −p ·r ) .
(3.31)
3
3
ħh (4π"0 )
ħh (4π"0 )
Obviously ω 0 t 0 − p 0 · r 0 is dimensionless in atomic units, as is ωt in SI units, so that
ωt = ω 0 t 0
(3.32)
but what about p 0 · r 0 ? In SI units p · r is not dimensionless but an action. Well, we have
p=p
0
ML
T
=p
0
me e 2
ħh 4π"0
,
0
r=r L =r
0
ħh 2 4π"0
me e 2
(3.33)
3.1. ATOMIC UNITS
31
so that
p0 · r 0 =
p·r
ħh
,
(3.34)
which, in fact, is dimensionless. Hence the expression (3.29) reads in SI units
E=
me2+η e 5+ξ
ħh 4−σ (4π"0 )3−ζ
f (α, β, . . .) ei(ωt −p·r /ħh ) .
(3.35)
We conclude this section by giving the explicit expressions and values for frequently
occurring entities in laser-atom interaction:
atomic mass unit M = me = 9.1094 · 10−31 kg,
(3.36)
2
atomic length unit L
=
atomic time unit T
=
atomic charge unit C
atomic action unit
atomic permittivity unit
=
=
=
atomic energy unit
=
atomic velocity unit
=
atomic el. field strength unit
=
atomic intensity unit
=
ħh 4π"0
e 2 me
= a0 = 0.5292 · 10−10 m,
ħh 3 (4π"0 )2
4
(3.37)
= 2.4189 · 10−17 s = 0.024 fs, (3.38)
me e
e = 1.6022 · 10−19C,
ħh = 1.0546 · 10−34 J,
4π"0 = 4π · 8.8542 · 10−12CV−1 m−1 ,
me e 4
= 4.3598 · 10−19 J = 27.21 eV,
2
2
ħh (4π"0 )
e2
= 2.1877 · 106 ms−1 = αc,
ħh 4π"0
me2 e 5
= 5.1422 · 1011Vm−1 ,
4
3
ħh (4π"0 )
e 12 me4
= 3.5095 · 1020Wm−2
9
6
8παħh (4π"0 )
= 3.5095 · 1016Wcm−2 .
(3.39)
(3.40)
(3.41)
(3.42)
(3.43)
(3.44)
(3.45)
Here, the fine structure constant
α=
e2
ħh 4π"0 c
=
1
137.04
(3.46)
has been introduced. The value of the light velocity in vacuum in atomic units equals
the inverse fine structure constant. Note that with ħh , e, me , and "0 alone it is not
possible 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.
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
32
A useful formula to convert the field strength E in atomic units (a.u.) to a laser
intensity I in the commonly used Wcm−2 is
(I in Wcm−2 ) = 3.51 · 1016 × (E 2 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 theory
In this Section we briefly review time-dependent perturbation theory. We consider
a quantum mechanical system such as an atom, for instance, interacting with a timedependent field, e.g., an electromagnetic field. In lowest order perturbation theory we
expect transitions between the unperturbed states to occur. The Hamiltonians of interest
are of the form
Ĥ (t ) = Ĥ0 + Ŵ (t ),
Ŵ (t ≤ 0) = 0.
(3.48)
Here we assume that the perturbation W (t ) is switched off for times t ≤ 0. We further
assume that the solution of the unperturbed problem is known, that is,
Ĥ0 |φn ⟩ = En |φn ⟩
(3.49)
with given eigenenergies En and eigenstates |φn ⟩. The unperturbed states (including
the continuum states, if present) form a complete basis so that the full solution of the
time-dependent Schrödinger equation (TDSE)
i∂ t |ψ(t )⟩ = Ĥ (t )|ψ(t )⟩
can be expanded as
|ψ(t )⟩ =
Z
X
cn (t )|φn ⟩ e−iEn t .
(3.50)
(3.51)
n
Here we use atomic units and factor the unperturbed time-evolution ∼ e−iEn t out of
cn (t ) (interaction picture). If at time t = 0 the system is in a single, unperturbed state
we have
|ψ(0)⟩ = |φi ⟩
⇒
cn (0) = δni .
(3.52)
At later times, the probability for finding the system in the state |φf ⟩ of Ĥ0 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 ⟨φ m |
yields
Z
dc m X
i
=
W mn (t )cn (t ) ei(Em −En )t ,
W mn (t ) = ⟨φ m |Ŵ (t )|φn ⟩.
(3.54)
dt
n
Formal integration of (3.54) gives
c m (t ) = c m (0) +
1
Z
t
dt
i
0
Z
X
0
0
W mn (t 0 ) ei(Em −En )t cn (t 0 ),
(3.55)
n
which is an integral equation because the coefficient c m for a certain m also appears under the integral within the sum over n. Such integral equations may be solved iteratively
by replacing the cn s under the integral by the expression (3.55) itself. If the perturbation
is small (i.e., the W mn s are small on the relevant energy scale), the series converges and
one can stop at some sufficiently high order. In first order one simply has
Z
Z
1 t 0X
0
W mn (t 0 ) eiωmn t cn (0),
ω mn = E m − En .
(3.56)
dt
c m (t ) = c m (0) +
i 0
n
In second order we have
Z
Z
1 t 0X
0
c m (t ) = c m (0) +
dt
W mn (t 0 ) eiωmn t
i 0
n
× cn (0) +
1
i
(3.57)
Z
t0
dt 00
0
Z
X
!
00
Wnu (t 00 ) eiωnu t c u (0) .
u
If the final state of interest was not populated at time t = 0 and ci = 1, the first order
expression (3.56) becomes
Z
1 t 0
0
dt Wfi (t 0 ) eiωfi t .
(3.58)
cf (t ) =
i 0
Now let us assume that the perturbation is of the form
Ŵ (t ) = V̂ sin ωt
or
Ŵ (t ) = V̂ cos ωt .
(3.59)
In the sin-case, for instance, one has
Wfi (t ) = Vfi sin ωt =
so that
cfsin (t ) =
Vfi €
2i
Š
eiωt − e−iωt ,
Vfi
1 − ei(ωfi +ω)t
2i
ωfi + ω
−
Vfi = ⟨φf |V̂ |φi ⟩
1 − ei(ωfi −ω)t
ωfi − ω
(3.60)
!
,
(3.61)
34
and
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
2
i(ωfi +ω)t
i(ωfi −ω)t 2
|V
|
1
−
e
1
−
e
fi sin
wi→f
(t ) =
−
.
4
ωfi + ω
ωfi − ω In the same way one finds for the cos-expression in (3.59)
i(ωfi +ω)t
i(ωfi −ω)t 2
2
|V
|
1
−
e
1
−
e
fi
cos
wi→f
+
(t ) =
.
4 ωfi + ω
ωfi − ω (3.62)
(3.63)
The case of a constant perturbation can be obtained by setting ω = 0:
‚
Œ
2
|Vfi |2 sin(ωfi t /2) 2
2
const
iωfi t F (t , ωfi ) =
wi→f (t ) =
. (3.64)
1 − e = |Vfi | F (t , ωfi ),
ωfi /2
ωfi2
The results for the sin and cos-like perturbations can be written as (upper sign for sin,
lower for cos)
|Vfi |2 sin,cos
A ∓ A 2
(3.65)
wi→f
(t ) =
+
−
4
where
1 − ei(ωfi ±ω)t
sin[(ωfi ± ω)t /2]
A± =
= −i ei(ωfi ±ω)t /2
.
(3.66)
ωfi ± ω
(ωfi ± ω)/2
If we assume that ω, ωfi > 0 and t ω −1 , only the resonant term A− is important
and the highly oscillatory and small anti-resonant term A+ can be neglected. This is the
so-called rotating wave approximation (RWA). We then obtain
RWA
wi→f
(t ) =
|Vfi |2
4
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 different
times. 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 we
can 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 origin
compared to uncertainty relations for noncommuting operators (such as the positionmomentum uncertainty relation, for instance).
Let us now discuss the validity of our first order perturbation theory results. On
one hand, the time t must not be too big, because at resonance one has
res
wi→f
(t ) =
|Vfi |2
4
t 2.
(3.70)
3.2. TIME-DEPENDENT PERTURBATION THEORY
(a)
35
(c)
(b)
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 .
res
Since wi→f
(t ) is a probability (i.e., a number between zero and one) and the perturbation
is supposed to be small
|Vfi |2
4
t2 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
1
t
⇒
|ωfi | |Vfi |
(3.73)
|ωfi |
|Vfi |
which confirms our statement above, that “small perturbation” means that the values
of the relevant matrix elements W mn are small compared to the “relevant energy scale”,
which here is the transition energy ωfi .
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 over
all the possible final states to obtain the transition probability,
Z
dα |⟨α|ψ(t )⟩|2 .
wi→f (t ) =
α∈Df
(3.74)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
36
We now choose the energy E as one of these parameters describing the final state, and
the set β comprises the remaining parameters. Introducing the density of states ρ(β, E )
we can write
dα = ρ(β, E ) dβ dE ,
(3.75)
and (3.74) turns into
Z
dβ dE ρ(β, E ) |⟨β, E |ψ(t )⟩|2 .
wi→f (t ) =
(3.76)
(β,E )∈Df
In the case of a constant perturbation, for instance, we obtain
Z
2
wi→f (t ) =
dβ dE ρ(β, E ) ⟨β, E |V̂ |φi ⟩ F (t , E − Ei ).
(3.77)
(β,E )∈Df
2
If t is sufficiently large, and ρ(β, E ) ⟨β, E |V̂ |φi ⟩ varies much slower over an energy
interval 4π/t than F , the integration over energy in (3.77) can be performed using
lim F (t , E − Ei ) = πt δ[(E − Ei )/2] = 2πt δ(E − Ei )
t →∞
so that
Z
wi→f (t ) = 2πt
2
dβ ⟨β, Ef = Ei |V̂ |φi ⟩ ρ(β, Ef = Ei )
if Ei ∈ Df
(3.78)
(3.79)
β∈Df
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 other
parameters β to be integrated over, we simply have
2
Γconst.
=
2π
⟨E
=
E
|
V̂
|φ
⟩
f
i
i ρ(Ef = Ei ).
i→f
In the case of a sinusoidal perturbation we obtain
2
sin,cos
Γi→f = 2π ⟨Ef = Ei + ω|V̂ |φi ⟩ ρ(Ef = Ei + ω).
(3.81)
(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, single
photon ionization or the photo effect, respectively.
3.2. TIME-DEPENDENT PERTURBATION THEORY
37
In the case of a laser field, V̂ is proportional to r · Ê where Ê is the electric field
amplitude (not an operator!). Hence
2
(3.83)
⟨Ef = Ei + ω|V̂ |φi ⟩ ∼ |E |2 ∼ I ,
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 W mn appear n times in the expression for cf , the nth order
ionization rate is proportional to I n and thus fulfills
(n)
Γi→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, and
resolvents
Let us consider the time-evolution operator Û corresponding to the TDSE (3.50),
|ψ(t )⟩ = Û (t , t 0 )|ψ(t 0 )⟩.
(3.85)
The operator Û fulfills the TDSE,
i∂ t Û (t , t 0 ) = Ĥ (t )Û (t , t 0 ),
and formal “solutions” of this equation are given by
Zt
0
0
Û (t , t ) = Û0 (t , t ) − i dt 00 Û (t , t 00 )Ŵ (t 00 )Û0 (t 00 , t 0 )
0
Zt t
= Û0 (t , t 0 ) − i dt 00 Û0 (t , t 00 )Ŵ (t 00 )Û (t 00 , t 0 )
t0
(3.86)
(3.87)
(3.88)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
38
where Û0 is the time-evolution operator associated to the unperturbed Hamiltonian Ĥ0 .
Equations (3.87) and (3.88) are integral equations since Û appears also on the right hand
side under the time integrals [cf. (3.55)]. Iteration yields
Zt
0
0
Û (t , t ) = Û0 (t , t ) + (−i) dt 00 Û0 (t , t 00 )Ŵ (t 00 )Û0 (t 00 , t 0 )
(3.89)
Z
t0
t 00
Z
t
dt 00
+(−i)2
t
0
t
0
dt 000 Û0 (t , t 00 )Ŵ (t 00 )Û0 (t 00 , t 000 )Ŵ (t 000 )Û0 (t 000 , t 0 ) + · · · .
Introducing the perturbation ŴIP in the interaction picture
ŴIP (t , t 0 ) = Û † (t , t 0 )Ŵ (t )Û (t , t 0 ) = Û (t 0 , t )Ŵ (t )Û (t , t 0 ),
(3.90)
Û0 (t 00 , t 000 ) = Û0 (t 00 , t 0 )Û0 (t 0 , t 000 )
(3.91)
making use of
and multiplying (3.89) by Û0 (t 0 , t ) from the left yields
Z
Zt
Zt
00
00
00
2
0
0
dt
Û0 (t , t )Û (t , t ) = 1 + (−i) dt ŴIP (t ) + (−i)
t
0
t
0
t 00
t
dt 000 ŴIP (t 00 )ŴIP (t 000 ) + · · · .
0
(3.92)
Here we suppressed the second time index of ŴIP (which is always t ). We now rewrite
this series in the following way:
Zt
0
0
(3.93)
Û0 (t , t )Û (t , t ) = 1 + (−i) dt 00 ŴIP (t 00 )
t0
Zt
Zt
2
00
+(−i)
dt
dt 000 θ(t 00 − t 000 )ŴIP (t 00 )ŴIP (t 000 ) + · · ·
(3.94)
0
0
t
t
Zt
(3.95)
= 1 + (−i) dt 00 ŴIP (t 00 )
0
2
+
(−i)
2
Z
t0
t
Z
dt
t0
t
00
t0
¦
dt 000 θ(t 00 − t 000 )ŴIP (t 00 )ŴIP (t 000 )
(3.96)
©
+θ(t 000 − t 00 )ŴIP (t 000 )ŴIP (t 00 ) + · · ·
Z
Zt
∞
X
¦
©
(−i)n t
dt1 · · · dtn T̂ ŴIP (t1 ) · · · ŴIP (tn ) .
=
(3.97)
t0
t0
n=0 n!
In the last line we introduced the time ordering operator T̂. The expansion of the time
evolution 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
Û0 (t 0 , t )Û (t , t 0 ) = T̂e−i
Rt
t0
dt 00 ŴIP (t 00 )
.
(3.98)
3.2. TIME-DEPENDENT PERTURBATION THEORY
39
If the ŴIP s commute at different times this expression reduces to
– Zt
™
exp −i dt 00 ŴIP (t 00 ) ,
t0
Rt
i.e., the problem is boiled down to the solution of the single integral t 0 dt 00 ŴIP (t 00 ).
Unfortunately, in general [ŴIP (t1 ), ŴIP (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 Ŵ is absent for t → ±∞. The probability for a
transition from the initial to the final state is
wi→f = |Mif |2
(3.100)
with
Mif
= ⟨ψf (∞)|Û (∞, −∞)|ψi (−∞)⟩
=
⟨ψf (−∞)|Û0† (∞, −∞))Û (∞, −∞)|ψi (−∞)⟩
(3.101)
(3.102)
= ⟨f|Û0 (−∞, ∞)Û (∞, −∞)|i⟩
(3.103)
=: ⟨f|Ŝ|i⟩ = Sfi .
(3.104)
Sfi is the so-called S-Matrix. Note that Ŝ is the left hand side of (3.98) for t 0 → −∞ and
t → ∞.
Let us come back to the integral equation for Û (3.88). Multiplication by θ(t − t 0 )
and insertion of θ functions under the integral so that the integration limits can be
extended to ±∞ yields
Z∞
0
0
0
0
Û (t , t )θ(t − t ) = Û0 (t , t )θ(t − t )−i dt 00 Û0 (t , t 00 )θ(t − t 00 )Ŵ (t 00 )Û (t 00 , t 0 )θ(t 00 − t 0 ).
−∞
(3.105)
Defining the retarded Green’s functions
K̂+ (t , t 0 ) = Û (t , t 0 )θ(t − t 0 ) ,
K̂0+ (t , t 0 ) = Û0 (t , t 0 )θ(t − t 0 ),
(3.106)
dt 00 K̂0+ (t , t 00 )Ŵ (t 00 )K̂+ (t 00 , t 0 ).
(3.107)
turns (3.105) into
Z
0
∞
0
K̂+ (t , t ) = K̂0+ (t , t ) − i
−∞
It is easy to show [using (3.86), (3.106), and ∂ t θ(t ) = δ(t )] that K̂+ fulfills the inhomogeneous TDSE
(i∂ t − Ĥ )K̂+ (t , t 0 ) = iδ(t − t 0 ).
(3.108)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
40
The advanced Green’s function is defined through
K̂− (t , t 0 ) = −Û (t , t 0 )θ(t 0 − t ).
(3.109)
It fulfills the same Eq. (3.108) (but for different boundary conditions).
Let us now assume that Ĥ is time-independent so that
K̂+ (τ) = e−iĤ τ θ(τ),
τ = t − t 0.
(3.110)
This assumption may appear strange, especially if we have in mind interactions with
strong laser fields (which are time-dependent electromagnetic fields). However, if we
quantize the electromagnetic field, the system atom + field becomes conservative, and
Ĥ would have no explicit time-dependence. It is always—at least in principle—possible
to extend the system under study in such a way that the perturbing driver becomes part
of the system, making the total system conservative. In this sense the TDSE is a special
case of the time-independent Schrödinger equation and not the other way round. The
TDSE arises when we divide bigger, conservative systems into smaller non-conservative
systems. The ultimate system would be the entire universe (including parallel ones, of
course) with no room for “external” time-dependent drivers whatsoever.
We introduce a new operator Ĝ+ which is proportional to the Fourier-transformed
Green’s function,
Z∞
1
dE e−iE τ Ĝ+ (E ),
K̂+ (τ) = −
(3.111)
2πi −∞
Z∞
dτ eiE τ K̂+ (τ).
(3.112)
Ĝ+ (E ) = −i
−∞
We can then switch from the time-domain to the energy-domain (or frequency-domain)
and back by working either with K̂+ (τ) or Ĝ+ (E ), respectively. Using (3.110) we obtain
Z∞
dτ ei(E −Ĥ )τ
(3.113)
Ĝ+ (E ) = −i
0
Z∞
= −i lim
dτ ei(E −Ĥ +iη)τ ,
η≥0
(3.114)
η→0+
⇒
0
Ĝ+ (E ) = lim
η→0+
and in the same way
Ĝ− (E ) = lim
η→0+
Ĝ± are called propagators.
1
E − Ĥ + iη
1
E − Ĥ − iη
.
(3.115)
(3.116)
3.2. TIME-DEPENDENT PERTURBATION THEORY
The Fourier-transform of (3.107) for a time-independent Ŵ gives us
Z∞
Z∞
iE τ
Ĝ+ (E ) = Ĝ0+ (E ) −
dτ e
dt 00 K̂0+ (t , t 00 )Ŵ K̂+ (t 00 , t 0 )
Z−∞
Z ∞ −∞
∞
= Ĝ0+ (E ) −
dτ
dt 00 eiE τ1 K̂0+ (τ1 )Ŵ eiE τ2 K̂+ (τ2 )
Z−∞
Z−∞
∞
∞
= Ĝ0+ (E ) −
dτ1
dτ2 eiE τ1 K̂0+ (τ1 )Ŵ eiE τ2 K̂+ (τ2 ).
−∞
41
(3.117)
(3.118)
(3.119)
−∞
The last step follows from
00
τ1 = t − t ,
00
τ2 = t − t
0
⇒
∂ (τ, t 00 ) = 1.
∂ (τ1 , τ2 ) (3.120)
We thus have
Ĝ+ (E ) = Ĝ0+ (E ) + Ĝ0+ (E )Ŵ Ĝ+ (E )
(3.121)
which could have been immediately derived from the identity
1
A
=
1
B
+
1
1
(B − A) .
B
A
(3.122)
Equation (3.121) is an algebraic equation as compared to the integral equations for K̂+
and Û .
Finally, we introduce the resolvent
Ĝ(z) =
1
z − Ĥ
,
z ∈ C.
(3.123)
Obviously, as z → E
Ĝ± (E ) = lim Ĝ(E ± iη).
η→0+
(3.124)
Since θ(x) + θ(−x) = 1 we can write
Û (τ) = K̂+ (τ) − K̂− (τ)
Z∞
1
=
dE e−iE τ [Ĝ− (E ) − Ĝ+ (E )]
2πi −∞
Z
1
=
dz e−izτ Ĝ(z).
2πi C+ +C−
(3.125)
(3.126)
(3.127)
The integration contours are depicted in Fig. 3.2. Discrete eigenenergies appear as poles
on the real z-axis, continua correspond to cuts.
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
42
z
C+
poles
cuts
C−
Figure 3.2: Illustration of the integration contours C± in (3.127) in the z-plane. For τ > 0 the
contribution from C− is zero, for τ < 0 the contribution from C+ is zero. Eigenenergies of Ĥ
are located along the real axis (discrete ones indicated by crosses, a continuum by a thick line.
For the resolvent (3.121) becomes
Ĝ(z) = Ĝ0 (z) + Ĝ0 (z)Ŵ Ĝ(z)
(3.128)
which can be iterated,
Ĝ(z) = Ĝ0 (z) + Ĝ0 (z)Ŵ Ĝ0 (z) + Ĝ0 (z)Ŵ Ĝ0 (z)Ŵ Ĝ0 (z) + · · · .
3.3
(3.129)
Important models
In this Section we will go through two important, exactly soluble models. The first
one shows how discrete states cease to exist if they are coupled to a continuum. The
second one discusses two resonantly (or almost resonantly) coupled discrete states and
the population transfer between them (Rabi-oscillations). It is a nice example for a
soluble problem where the previously introduced perturbation theory is not applicable.
3.3.1 Discrete level coupled to the continuum
Starting point is the Hamiltonian
Ĥ = Ĥ0 + Ŵ .
(3.130)
We allow for a single discrete state |ϕ⟩ and the continuum states. We now discretize the
continuum by introducing a box of length L. The continuum states |k⟩ will then be
spaced by δ, and the density of states is 1/δ so that Fermi’s Golden Rule (3.81) yields
1
Γ = 2πw 2 ,
δ
lim
δ→0
w2
δ
= const.
(3.131)
Here
wk := ⟨k|Ŵ |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|Ĥ0 |k⟩ = Ek = kδ,
Eϕ = 0
−∞ < k < ∞,
⇒
k ∈ Z,
(3.133)
Eϕ = Ek=0 ,
(3.134)
⟨ϕ|Ŵ |ϕ⟩ = ⟨k|Ŵ |k 0 ⟩ = 0,
(3.135)
wk = ⟨k|Ŵ |ϕ⟩ = ⟨ϕ|Ŵ |k⟩ = w,
(3.136)
and that
that is, all these transition matrix elements are equal and real. We are now looking for
the new energies Eµ and states |ψµ ⟩ fulfilling
Ĥ |ψµ ⟩ = Eµ |ψµ ⟩.
(3.137)
By multiplying from the left with ⟨k| and ⟨ϕ|, and making use of Eϕ = 0 and
1 = |ϕ⟩⟨ϕ| +
X
|k⟩⟨k|,
(3.138)
k
we obtain
Ek ⟨k|ψµ ⟩ + w⟨ϕ|ψµ ⟩ = Eµ ⟨k|ψµ ⟩,
X
w
⟨k|ψµ ⟩ = Eµ ⟨ϕ|ψµ ⟩
(3.139)
(3.140)
k
so that
⟨k|ψµ ⟩ = w
⟨ϕ|ψµ ⟩
Eµ − E k
and
Eµ =
,
Eµ − Ek 6= 0
X
w2
k
Eµ − E k
(3.141)
,
(3.142)
which is the desired (although implicit) equation for the new eigenenergies Eµ . It can
be rewritten
Eµ =
X
w2
k
Eµ − δk
=
w2 X
δ
k
1
z−k
=
w2
π
δ tan πz
,
z=
Eµ
δ
.
(3.143)
Hence,
Eµ =
πw 2
δ tan(πEµ /δ)
⇒
1
tan(πEµ /δ)
=
2Eµ
Γ
,
Γ=
2πw 2
δ
.
(3.144)
44
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
Figure 3.3: Graphical determination of the eigenvalues Eµ of Ĥ . The abscissa of each intersection point M k is an eigenvalue. Let M k be the point of intersection whose abscissa is between
kδ and (k +1)δ (the abscissa of M−k being between −(k +1)δ and −kδ). The associated eigenvalue is denoted Eκ = Eµ where the Greek index κ corresponds to the Roman index k of the
unperturbed 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)].
3.3. IMPORTANT MODELS
45
Finding the new eigenenergies Eµ amounts to find the intersections of y = a x with
y = 1/ tan(b x) where a = 2/Γ and b = π/δ, and x = Eµ . This is shown in Fig. 3.3. The
new eigenvalues Eµ are interspersed between the old ones. The larger |k| the closer are
the new eigenvalues to the old ones. If Eµ Γ one has Eµ ' Ek .
Using the normalization condition
X
!
|⟨k|ψµ ⟩|2 + |⟨ϕ|ψµ ⟩|2 = 1
(3.145)
k
we find for the projections of the new states on the unperturbed states

!2 −1/2
X
w


⟨ϕ|ψµ ⟩ = 1 +
,

E
−
E
0
0
µ
k
k

!2 −1/2
X
w
w 

.
⟨k|ψµ ⟩ =
1 +

Eµ − E k
E
−
E
0
0
µ
k
k
(3.146)
(3.147)
The square bracket can be rewritten using
X
(z − k)
−2
=
k
and yields
1+
!2
X
w
k0
Eµ − E k 0
=1+
π2
(3.148)
sin2 πz
π2 w 2
–
‚
1 + tan
δ2
−2
πEµ
Ϊ
δ
With the help of (3.144) this can be recast in the form
!2
–
™
2
X
w
1
Γ
2
2
= 2 w +
+ Eµ
1+
Eµ − E k 0
2
w
k0
so that
⟨ϕ|ψµ ⟩ = h
2
w +
w
€ Š2
Γ
2
+ Eµ2
i1/2 .
.
(3.149)
(3.150)
(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 =
X
Eµ ∈[E ,E +dE ]
|⟨ϕ|ψµ ⟩|2 '
dE
δ
|⟨ϕ|ψµ ⟩|2 .
(3.153)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
46
We thus have
dNϕ
Γ/2π
.
€ Š2
w 2 + Γ2 + E 2
=
dE
(3.154)
In the continuum limit δ → 0 the probability w 2 has vanishing measure and
dNϕ
Γ/2π
= € Š2
Γ
dE
+ E2
2
(3.155)
which is a Lorentzian of width Γ around the energy E = Eϕ = 0. We learn from this
study that discrete levels cease to exist when they are coupled to a continuum and that
the continuum states in the vicinity ' Γ around the energy of the unperturbed, discrete
state |ϕ⟩ retain memory of |ϕ⟩.
We finally show that the decay in this simple model system in exponential. Let us
assume that at time t = 0 the system is prepared to be in the state |ϕ⟩,
|ψ(0)⟩ = |ϕ⟩ =
X
h
2
w +
µ
w
€ Š2
Γ
2
+ Eµ2
i1/2 |ψµ ⟩.
(3.156)
As a consequence, the state at a later time will be
|ψ(t )⟩ =
X
µ
w e−iEµ t
h
i1/2 |ψµ ⟩.
€ Š2
Γ
2
2
w + 2 + Eµ
(3.157)
Using (3.151) yields
⟨ϕ|ψ(t )⟩
=
X
µ
=
δ
w 2 e−iEµ t
€ Š2
w 2 + Γ2 + Eµ2
Γ X
2π
µ
Z
∞
2
w +
e−iEµ t
€ Š2
Γ
2
(3.158)
+ Eµ2
(3.159)
=
e−iE t
dE € Š2
Γ
2π −∞
+ E2
2
(3.160)
=
e−Γt /2 ,
(3.161)
δ→0
Γ
t ≥ 0.
For the last step we used the method of residues. The population of the formerly discrete level thus decays exponentially according
|⟨ϕ|ψ(t )⟩|2 = e−Γt ,
t ≥ 0.
(3.162)
3.3. IMPORTANT MODELS
47
We stop the discussion of the model at this point. We only mention that it can
be extended toward more than a single discrete state. If, for instance, the population
of the formerly discrete state can either reach the continuum directly or via transition
to another (formerly) discrete state, an interference effect arises which leads to spectral
shapes (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 very
simple nature of this model system. In fact, it can be shown that real systems cannot
decay exponentially on a time scale shorter than the half-time or much longer than the
half-time. The deviation from exponential decay at short time scales offers the possibility to alter the dynamics of the quantum system by repeated measurements (Zeno and
anti-Zeno effect).
Further reading: Time-dependent perturbation theory is covered in every reasonable
quantum mechanics text book. Dyson series, S-matrix, Green’s functions, propagators etc. are usually covered in books on quantum field theory or many-body
theory. Models and techniques relevant to (weak) laser-atom interaction are discussed in CLAUDE COHEN-TANNOUDJI, JACQUES DUPONT-ROC, GILBERT
GRYNBERG, Atom-Photon Interactions, (Wiley, New-York, 1998) and in FARHAD
H. M. FAISAL, Theory of Multiphoton Processes, (Plenum, New York, 1987).
3.3.2 Resonantly coupled discrete levels: Rabi-oscillations
Let us consider the Hamiltonian of a two-level system coupled to a laser field in dipole
approximation,
Ĥ (t ) = Ĥ0 + Ŵ (t ) = ωa |a⟩⟨a| + ω b |b ⟩⟨b | − q Ê z cos ωt .
(3.163)
The charge of the particle in atomic units is q (i.e., q = −1 for an electron), and ωa , ω b
are the energies of the states |a⟩ and |b ⟩, respectively. Plugging the Ansatz
|ψ(t )⟩ = ca (t )e−iωa t |a⟩ + c b (t )e−iωb t |b ⟩
(3.164)
i∂ t |ψ(t )⟩ = Ĥ (t )|ψ(t )⟩
(3.165)
into the TDSE
and multiplication from the left by ⟨a| and ⟨b |, respectively, leads to
1
iċa (t ) = − Ê q c b (t ) ei∆t ⟨a|z|b ⟩,
2
1
iċ b (t ) = − Ê q ca (t ) e−i∆t ⟨b |z|a⟩.
2
(3.166)
(3.167)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
48
Here, we introduced the detuning
∆ = ωa − ω b − ω = ωab − ω,
(3.168)
made use of ⟨a|z|a⟩ = ⟨b |z|b ⟩ = 0 (assuming that |a⟩ and |b ⟩ have well-defined parity), and neglected the anti-resonant terms ∼ e±i(ωab +ω)t (RWA, see Sec. 3.2) assuming
without loss of generality that ωab > 0 and ω > 0. Setting
q Ê⟨a|z|b ⟩ = Ê|q⟨a|z|b ⟩| e−iϕ = ΩR e−iϕ ,
| {z }
ΩR = Ê |q⟨a|z|b ⟩|
(3.169)
ΩR
(3.166) and (3.167) become
1
ċa (t ) = i ΩR ei∆t −iϕ c b (t ),
2
1
ċ b (t ) = i ΩR e−i∆t +iϕ ca (t ).
2
The solutions are
€
Š
a1 eiΩt /2 + a2 e−iΩt /2 ei∆t /2 ,
€
Š
c b (t ) = b1 eiΩt /2 + b2 e−iΩt /2 e−i∆t /2
ca (t ) =
where
(3.170)
(3.171)
(3.172)
(3.173)
q
Ω=
Ω2R + ∆2 .
(3.174)
The constants a1 , a2 , b1 , b2 are determined through the initial conditions,
—
1 ”
a1 =
(Ω − ∆)ca (0) + ΩR e−iϕ c b (0) ,
(3.175)
2Ω
—
1 ”
(Ω + ∆)ca (0) − ΩR e−iϕ c b (0) ,
(3.176)
a2 =
2Ω
—
1 ”
b1 =
(Ω + ∆)c b (0) + ΩR eiϕ ca (0) ,
(3.177)
2Ω
—
1 ”
b2 =
(Ω − ∆)c b (0) − ΩR eiϕ ca (0) .
(3.178)
2Ω
We thus have
– ™
i∆
Ωt
Ωt
−
sin
ca (t ) =
ca (0) cos
(3.179)
2
Ω
2
ΩR −iϕ
Ωt
+i e c b (0) sin
ei∆t /2 ,
Ω
2
– ™
i∆
Ωt
Ωt
+
sin
c b (t ) =
c b (0) cos
(3.180)
2
Ω
2
ΩR iϕ
Ωt
+i e ca (0) sin
e−i∆t /2 ,
Ω
2
3.3. IMPORTANT MODELS
49
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 c b (0) = 1 and ca (0) = 0, Eqs. (3.179) and (3.180)
simplify to
ΩR t
−iϕ
ca (t ) = ie sin
,
(3.181)
2
ΩR t
c b (t ) = cos
(3.182)
2
so that the probability to find the system, e.g., in the state |b ⟩ is
ΩR t
1
2
2
w b (t ) = |c b (t )| = cos
= [1 + cos ΩR t ] ,
2
2
(3.183)
i.e., it oscillates with the frequency ΩR. These oscillations are called Rabi oscillations (or
Rabi floppings).
In first order time-dependent perturbation theory we saw that the depopulation
of the initial state at resonance goes ∼ t 2 which corresponds to the first term of the
expansion of cos ΩR t . Obviously, perturbation theory is not adequate to describe the
population transfer between the two states.
Rabi floppings are employed experimentally to prepare atoms in a certain desired
state 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 is
transferred completely to the desired state. As long as the detuning is small compared to
transition energies to other states, the two-level approximation is adequate. In the case
of nonvanishing detuning ∆ 6= 0 the population transfer is not complete (i.e., |ca (t )|2
never 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 the
states. However, the so-called AC Stark effect modifies the energies of the two states so
that a laser frequency that equals the unperturbed level spacing ωab will be detuned.
As an example, we shall finally calculate the Rabi-frequency ΩR for the resonantly
driven 1s→2p transition in atomic hydrogen for a driving laser of field amplitude Ê =
9 · 108 V/m. We have for the dipole matrix element
Z
⟨a|z|b ⟩ =
Z
dr r
2
dΩ
φ∗2p
φ1s
Y10∗ (Ω) |r cos
Y00
ϑ}
{z
r
r
| {z }
| {z }
z
0
2p0
(3.184)
1s
where Y`m (Ω) are the spherical harmonics and Ω is the solid angle (in this context).
Using
1
r2
Y00 = p ,
(3.185)
φ1s = 2r e−r ,
φ2p0 = p e−r /2 ,
2 6
4π
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
50
and
È
z = r cos ϑ = r
we obtain
1
⟨a|z|b ⟩ = p
3
Z
4π
3
Y10 (Ω),
r 4 −3r /2
256
dr p e
=p
' 0.7449.
6
2 243
(3.186)
(3.187)
In SI units one has
256 a0 e Ê
ΩR = p
' 5.4 · 1013 s−1
ħ
h
2 243
(3.188)
(with a0 the Bohr radius). For comparison: one atomic frequency unit corresponds to
4.1 · 1016 s−1 , and the laser frequency at 800 nm is 2.3 × 1015 s−1 . Hence, Rabi-flopping is
still “slow” for the considered field strength both on the inner-atomic time scale and on
the time scale of the laser period.
An atom driven resonantly emits at the frequencies ω and ω±ΩR (Mollow-spectrum,
fluorescence). Corrections stemming from the anti-resonant terms can be taken into
account in a perturbative manner (Bloch-Siegert shifts). It is common to introduce in the
context of Rabi-floppings so called dressed states. These states form a new basis in which
the Hamiltonian becomes diagonal at resonance. We postpone the introduction of the
dressed 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 subject (employing an interesting application of continued fractions) is S.H. AUTLER 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 fields
Although we assume the external electric field to be static in this Section, the following
analysis is useful for atoms in laser fields too, as will become clear below. The Hamiltonian governing an electron moving in a Coulomb potential −Z/r and a static electric
field E = Ee z reads
Z
p̂ 2
+ Veff ,
Veff = − + E z.
(3.189)
Ĥ =
2
r
3.4. ATOMS IN STRONG, STATIC ELECTRIC FIELDS
51
V
Ez
z
"over barrier"
−Z/r
"tunneling"
Veff
Figure 3.4: Effective potential Veff in field direction. The unperturbed Coulomb potential and
the 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 perturbative treatment of the problem can be found in almost all quantum mechanics or
atomic physics text books (Stark effect). In first order (linear Stark effect) the degeneracy with respect to the angular quantum number ` is removed while the degeneracy in
the magnetic quantum number m is maintained. The non-degenerate ground state is
only affected in second order (quadratic Stark effect). It is down-shifted since the potential widens in the presence of the field. In the case of the hydrogen atom (Z = 1) this
down-shift is given by ∆E = −9E 2 /4.
Let us first point out that, strictly speaking, there exist no discrete, bound states
anymore even for the tiniest electric field. This is because even a very small field gives
rise to a potential barrier (see Fig. 3.4) through which the initially bound electron may
tunnel. The electric field couples all bound states to the continuum and thus, as we have
learnt 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 from
below. However, since the barrier for small fields is far out, the probability for tunneling
is extremely low (note that the tunneling probability goes exponentially down with the
distance to be tunneled) and the states are “quasi-discrete”.
A strong increase in the ionization probability is expected when the electron can
even escape classically, that is, when it does not have to tunnel. In a zeroth order approximation (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 energy of the electron does not change (i.e., Stark effect negligible), classical over-barrier
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
52
ionization sets in when the barrier maximum coincides with the energy level of the
electron. The position of the barrier is (for E > 0) located at
È
Z
zbarr = −
(3.190)
E
and the energy at the barrier maximum is
p
Vbarr = −2 Z E.
(3.191)
Hence, if we restrict ourselves to the ground state of hydrogen-like ions, we require that
E =−
Z2
2
Æ
= −2 Z Ecrit
!
⇒
Ecrit =
Z3
16
.
(3.192)
Because of the strong Z-dependence of the critical field even with the most intense lasers
available 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 a
factor of two, as will be shown soon.
The Schrödinger equation with the Hamiltonian (3.189) separates in parabolic coordinates (ξ , η, ϕ),
ξ = r + z,
η = r − z,
1
r = (ξ + η),
2
1
z = (ξ − η),
2
0 ≤ ξ , η.
(3.193)
Here, r is the radial coordinate, and ϕ is the usual azimuthal angle (as in spherical, polar
or cylindrical coordinates). Cuts of contours of constant ξ and η in the x z-plane are
shown in Fig. 3.5. The Hamiltonian in parabolic coordinates reads
Ĥ = −
—
1 2
2Z
ξ −η
∂ξ (ξ ∂ξ ) + ∂η (η∂η ) −
∂ϕ −
+E
.
ξ +η
2ξ η
ξ +η
2
2
”
(3.194)
Plugging the Ansatz
Ψ = f1 (ξ ) f2 (η)eimϕ
(3.195)
into the Schrödinger equation E Ψ = Ĥ Ψ and multiplying by (ξ + η)/2 leads to an
equation that can be decoupled into
‚
Œ ‚
Œ
d
m2 E 2
d f1
E
ξ−
− ξ
ξ
+
f1 + Z1 f1 = 0,
(3.196)
dξ
dξ
2
4ξ
4
‚
Œ ‚
Œ
m2 E 2
d
d f2
E
η−
+ η f2 + Z2 f2 = 0
η
+
(3.197)
dη
dη
2
4η
4
where Z1 , Z2 are separation constants fulfilling
Z1 + Z2 = Z.
(3.198)
3.4. ATOMS IN STRONG, STATIC ELECTRIC FIELDS
53
9
8
7
6
5
4
3
0.1
2
1
1
2
0.1
3
4
5
6
7
8
9
Figure 3.5: Illustration of parabolic coordinates. Cuts of contours ξ = const. (dashed, values
given next to the lines) and η = const. (solid) in the x z-plane (azimuthal symmetry with respect
to the z-axis!).
Division by 2ξ and 2η, respectively, yields the two Schrödinger equations
–
‚
Œ
™
1 d2
1 d
m2
Z1 E
E
−
+
−
−
+
ξ
f
=
f1 ,
1
2 dξ 2 ξ dξ 4ξ 2
2ξ
8
4
–
‚
Œ
™
1 d2
1 d
m2
Z2 E
E
−
+
−
−
−
η
f
=
f2
2
2 dη2 η dη 4η2
2η 8
4
(3.199)
(3.200)
which have the same shape like two Schrödinger equations in cylindrical coordinates
and the potentials
Z1 E
Z2 E
Vξ = −
+ ξ,
Vη = − − η.
(3.201)
2ξ
8
2η 8
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 ionization
happens with respect to the η coordinate while the electron is expected to remain rather
confined in ξ . The consequences of this in Cartesian coordinates can be understood
with the help of Fig. 3.5: confinement to a region ξ < ξmax implies preferred electron
emission toward negative z with a lateral spread that can be estimated by the confining
contour 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 sequence
of Z1 for which the solution of the Schrödinger equation in ξ , Eq. (3.199), leads to
normalizable bound states. This sequence can be labelled by the number of nodes in
f1 for ξ > 0, n1 = 0, 1, 2, . . . (note that Z1 < 0 is also possible). The second equation
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
54
V
Vξ
"uphill"
(bound motion)
ξ,η
"downhill"
(ionization)
Vη
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 barrier
through which the electron may tunnel.
(3.200) has to be solved for Z2 = Z − Z1 and the same energy E . This is possible because
the corresponding Hamiltonian Ĥη [i.e., the square bracket in (3.200)] is neither bound
from below nor from above.
In the field-free case E = 0 the two Schrödinger equations (3.199) and (3.200) are
identical, and the relation between the “usual” principal quantum number n and the
parabolic 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 perform
an additional, simple coordinate transformation
u=
p
2ξ ,
v=
p
2η
(3.203)
which, after multiplication of the new Schrödinger equation by (u 2 + v 2 )/4, and plugging in the Ansatz Ψ = Φ u (u)Φv (v)eimϕ yields
–
1
‚
1
m2
Œ
1
2
2
1
™
4
∂ u (u∂ u ) − 2 + Ω u − g u Φ u = Z1 Φ u ,
2 u
2
4
u
–
‚
Œ
™
2
m
1 1
1 2 2 1 4
∂v (v∂v ) − 2 + Ω v + g v Φv = Z2 Φv ,
−
2 v
2
4
v
−
(3.204)
(3.205)
3.4. ATOMS IN STRONG, STATIC ELECTRIC FIELDS
55
where, again, Z = Z1 + Z2 is used, and Ω and g are defined as
1
E
Ω2 = − ,
2
4
E ≤ 0,
E
g =− .
4
(3.206)
The Schrödinger equations (3.204) and (3.205) have the shape of two-dimensional oscillators (with radial coordinates u and v, respectively) of frequency Ω and with a quartic
perturbation that is proportional to the electric field. In the field-free case, the Coulomb
problem is mapped to two two-dimensional oscillators where, however, the energy assumes the role of the oscillator frequency, and the nuclear charge (splitted into Z1 and
Z2 ) assumes the role of the energy. The transformation to the coordinates (u, v, ϕ) corresponds to the Kustaanheimo-Stiefel transformation.
Let us now evaluate an improved critical field for the case of hydrogen-like ions. As
mentioned above, formula (3.192) underestimates the critical field by more than a factor
of 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
r
a u −a u 2 /2
Φ u (u) =
e u
(3.207)
π
(and analogous for v) as trial functions with parameters a u and av . The “energy” then is
!
Z∞
1 Ω2 a u2
g
∗
Z1 ( g ) = 2π
du uΦ u Ĥ u Φ u =
−
+ au − 2 .
(3.208)
au
2
2
2a u
0
Minimizing this energy yields up to first order in g


g ‹
g ‹
av = Ω 1 + 3 .
au = Ω 1 − 3 ,
Ω
Ω
(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 the
ground 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 = −Z 2 /2, as it should for the ground states of hydrogen-like ions.
In physical coordinates the variationally determined wave function for, e.g., hydrogen (Z = 1) reads
1 − 4E 2 −r −2E ·r
e e
,
(3.212)
ΨH (r ) = p
π
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
56
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
p
displays a barrier while the v-oscillator does not. The barrier is located at ubarr = Ω/ g
4
2
and the energy at the barrier maximum is Ω2 ubarr
/2− g ubarr
/4 = Ω4 /(4 g ). Claiming that
at the critical field strength the energy of the u-state coincides with the barrier-energy
gives us
Z1 ( g ) = Ω −
g
2Ω
2
=
Ω4
gcrit = Ω3 (1 − 2−1/2 ) ' 0.3 Ω3
⇒
4g
(3.213)
which translates [using (3.206)] to
p
H−like
Ecrit
= ( 2 − 1) E 3/2 .
(3.214)
H−like
= 0.147 instead of the 0.0625 preIn the case of atomic hydrogen one obtains Ecrit
dicted by (3.192). The wrong prediction of (3.192) is due to the erroneous assumption
that one can consider the electron motion in z-direction and in lateral direction as being
independent. Instead, the problem separates in parabolic coordinates! However, since
the “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)] is
useful and quite accurate for many practical applications.
3.4.1 Tunneling ionization
Going one step beyond a classical over-barrier analysis amounts to take tunneling into
account. This can be done in a semi-classical way. Let us consider the tunneling of the
electron 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
–
‚
Œ
™
1 d2
1 d
1
E
1
−
+
−
− η f2 (η) = − f2 (η).
(3.215)
2
2 dη
η dη
4η 8
8
Substituting
χ (η) =
p
η f2 (η)
yields the Schrödinger equation
Œ
‚
1
1
∂ 2χ
1
1
+
+ − +
+ Eη χ = 0.
4 2η 4η2 4
∂ η2
(3.216)
(3.217)
Comparison with
−
1 ∂ 2χ
2 ∂ η2
+ V (η)χ = εχ
(3.218)
3.4. ATOMS IN STRONG, STATIC ELECTRIC FIELDS
57
1/ E
η0
Figure 3.7: Plot of the potential V (η) [Eq. (3.219)] for E = 0.0075. The tunnel “exit” η1 for
sufficiently low fields E is in good approximation given by η1 ' 1/E. The matching point η0 is
located inside the barrier where 1 η0 1/E holds.
shows that we effectively deal with one-dimensional motion of an electron in the potential
‚
Œ
1
1
1 1
+
+ Eη
(3.219)
V (η) = −
2 2η 4η2 4
with total energy
1
ε=− .
8
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
the “left” quasi-classical wave function
iC
χleft (η) = − p exp
| p|
Z
!
η
p(η0 ) dη0 η
(3.220)
(3.221)
(3.222)
0
with the “right” quasi-classical outgoing wave function
!
Zη
C
χright (η) = p exp i
p(η0 ) dη0 − iπ/4
p
η0
(3.223)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
58
where
p(η) =
Æ
s
2[ε − V (η)] =
1
1
1
1
− +
+ 2 + Eη.
4 2η 4η
4
(3.224)
The semi-classical approximation breaks down at the classical turning point η1 ' 1/E
since p(η1 ) = 0. In general, semi-classical wave functions are accurate as long as the de
Broglie wave length
2πħh
λdB =
(3.225)
p
is small compared to the length scale characterizing changes in the potential (i.e., the
potential should be sufficiently “flat”). For vanishing momentum p the de Broglie
wave 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 disagreement between the semi-classical wave function and the exact wave function in a narrow
region around the classical turning point η1 plays no role.
For the determination of the normalization constant C we set the left wave function
at position η0 equal to the unperturbed wave function so that
p
1
iC
η0 p e−(ξ +η0 )/2 = − p
π
| p0 |
(3.226)
with p0 = p(η0 ). The “uphill” coordinate ξ appears as a parameter here which will
be integrated out later on; in other words: the wave function is assumed to retain its
ground state shape with respect to ξ (i.e., the Stark effect is neglected). We obtain for
the right wave function
s
!
Zη
η0 | p0 | −(ξ +η )/2
0
χright (η, ξ ) = i
p(η0 ) dη0 − iπ/4
(3.227)
e
exp i
π p(η)
η0
so that
|χright (η, ξ )|2 =
η 0 | p0 |
π p(η)
Z
!
η
exp −ξ − η0 + 2ℜ i
p(η0 ) dη0 
(3.228)
η0
where ℜ denotes the real part. Because of (3.221) we can expand p(η) in " = 1/η,

!
p
1
1


+ ···
outside barrier, η > η1
Eη − 1 + p

 2
η Eη − 1
!
. (3.229)
p(η) =

1 p
1



i 1 − Eη + p
+ ···
inside barrier, η < η1
2
iη 1 − Eη
Since Eη0 1 it is sufficient to take | p0 | = 1/2. In order to keep the leading terms
dependent on E in the prefactor as well as in the exponent, we set in the denominator
3.4. ATOMS IN STRONG, STATIC ELECTRIC FIELDS
59
p
of the prefactor in (3.228) p(η) = ( Eη − 1)/2. In the exponent we integrate inside the
barrier and have to keep both terms in the expansion (3.229). We thus obtain
 !
Z η1 
p
η
1
0
 Eη − 1 − p
 dη .
|χright (η, ξ )|2 = p
e−(ξ +η0 ) exp −
π Eη − 1
η Eη − 1
η0
(3.230)
The integral can be solved. Using η1 ' 1/E and η0 E 1 we obtain for the probability
density outside the barrier
4 e−ξ
|χright (η, ξ )| =
e−2/(3E) .
p
πE Eη − 1
2
(3.231)
The total probability current through a plane perpendicular to the z-axis is
Z∞
| f1 (ξ ) f2 (η)|2 v z 2πρ dρ
(3.232)
w=
0
where f1 (ξ ), f2 (η) are the wave functions introduced in (3.195), v z is the velocity in
z-direction and ρ is the radial cylindrical coordinate. The ξ -dependent part | f1 |2 is
included in our |χright (η, ξ )|2 so that
Z
∞
w=
|χright (η, ξ )|2
η
0
v z 2πρ dρ.
(3.233)
With z = (ξ − η)/2 ' −η/2 for small ξ and large η, we estimate for v z
1
1
1
v z2 + E z ' v z2 − Eη
2
2
2
so that
Z
w=
∞
⇒
|χright (η, ξ )|2 p
η
0
vz '
p
Eη − 1
(3.234)
Eη − 1 2πρ dρ.
(3.235)
Finally, with
ρ=
Æ
ξη
⇒
Æ
1
η
1 ξ
1
dρ = d ξ η = p dξ + p dη '
2 ξη
2 ξη
2
È
η
ξ
dξ
(where the last step again follows from η ξ ) we arrive at
È
Z∞
Z ∞ |χ (η, ξ )|2
Æ 1 η
p
4 −2/(3E) −ξ
right
dξ =
e
e dξ
w=
Eη − 1 2π ξ η
η
2 ξ
0 E
0
⇒
w=
4
E
e−2/(3E) .
(3.236)
(3.237)
(3.238)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
60
1/16
0.147
Figure 3.8: The LANDAU-rate (3.238) vs field strength E. The vertical lines indicate the (here
wrong) over-barrier field strength (3.192) (1/16, dashed) and the correct (3.214) 0.147 (solid),
respectively.
This is the L ANDAU-rate for tunneling ionization of atomic hydrogen from the ground
state. Its derivation is given as an exercise in the Quantum Mechanics volume of L ANDAU & L IFSHITZ ! It has been checked numerically that (3.238) is exact in the limit of
low field strengths E while it overestimates ionization as the over-barrier field strength
is 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 more
complex atoms, and to higher field strengths. All directions have been pursued, and
there exists a vast amount of literature on tunneling ionization (see “Further reading”
below). Here we have restricted ourselves to the “generic” case of atomic hydrogen
where the typical tunneling exponent ∼ E −1 already arises, as derived above.
The most commonly used tunneling formula is the so-called ADK-formula (named
after AMMOSOV, DELONE, AND KRAINOV although it was, even more generally, derived much earlier by POPOV):
s
w = Cn2∗ f (`, m)
Z
2
2n ∗2
3E n ∗3
πZ 3
‚
2Z 3
E n ∗3
Œ2n ∗ −|m|−1
‚
exp −
2Z 3
3n ∗3 E
Œ
(3.239)
with
n ∗
(2` + 1)(` + |m|)!
Z
∗
,
n
=
Æ
n∗
2|m| |m|! (` − |m|)!
2Eip
(3.240)
∗
and e = 2.71828. n , `, and m are the (effective) quantum numbers of the initial state
with ionization potential Eip . The behavior of w as a function of E is again dominated
Cn ∗ =
2e
(2πn ∗ )−1/2 ,
f (`, m) =
,
3.5. ATOMS IN STRONG LASER FIELDS
61
by the exponent ∼ E −1 . The varying laser field is taken into account by cycle-averaging
E(t ). Hence, by E in (3.239) the laser electric field amplitude is meant. The expression
for circular polarization is obtained by multiplying (3.239) with (πZ 3 /3E n ∗3 )1/2 . The
expression is supposed to be increasingly accurate as E Z 3 /16n 4 , n ∗ 1, and ` n ∗ .
Experimentalists are used to compare their measured results for ion yields with the
corresponding ADK-prediction. Often, the agreement is satisfactory if one allows for
an 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 appearance intensity Iapp,Z for a charge state Z of an atom with ionization potential
Eip,Z is given by
Iapp,Z =
4
Eip,Z
16Z 2
.
(3.241)
Further reading: The hydrogen atom in parabolic coordinates is treated in many
Atomic Physics and Quantum Mechanics textbooks (i.e., L ANDAU & LIF SHITZ ’ S Quantum Mechanics volume). The correct over-barrier field strength of
hydrogen-like ions is calculated in D. BAUER, Ejection energy of photoelectrons in
strong field ionization, Phys. Rev. A 55, 2180 (1997). A review of tunneling ionization is given by V.S. POPOV in Tunnel and multiphoton ionization of atoms and
ions in a strong laser field (Keldysh theory), Physics Uspekhi 47, 855 (2004).
3.5 Atoms in strong laser fields
There are at least three different energy scales (and the related time scales) in the interaction physics of atoms in strong laser fields: (i) the ionization potential Eip = |E |, (ii)
the photon energy ω, and (iii) the ponderomotive energy Up [see Eq. (2.14)]. The pulse
duration may introduce an additional laser-related time-scale while the energy spectrum
of the atom, through typical transitions, could introduce an additional species-related
time-scale. If one ignores the two latter parameters, the atomic species enter through
|E | only.
In case ω > |E | Up or |E | > ω Up perturbation theory in lowest nonvanishing
order (LOPT) can be applied. In contrast, when with the increasing laser intensity
the regime |E | > Up > ω is reached, non-perturbative effects such as above-threshold
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
62
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 timedomain this implies that both the inner-atomic time-scale and the ionization dynamics
are “fast” with respect to a laser period. If this is the case, a quasi static field ionization
picture may be applied where, at the instant of ionization t 0 , the electron “feels” an
effective potential which is the sum of the Coulomb (or effective core) potential and the
instantaneous potential of the laser, as depicted in Fig. 3.4. If the field reaches the critical
field estimated above, the electron may escape classically over the barrier [over-barrier
or barrier suppression ionization (OBI) and (BSI), respectively]. Below the critical field
strength the electron can escape via tunneling through the barrier (tunneling ionization).
MPI and tunneling are defined by
Up
Up
< 1 (MPI),
> 1 (tunneling).
(3.242)
|E |
|E |
KELDYSH introduced a parameter γ as the ratio of “tunneling time” and laser period,
or, expressed
p in frequencies, γ = ω/ω t with the tunneling frequency ω t estimated by
ω t = E/ 2|E | where E is the electric field |E |. Hence, one has the Keldysh parameter
v
u
ω u
|E |
γ=
=t
,
(3.243)
ωt
2Up
p
and conditions (3.242) can be also stated as (neglecting a prefactor 2 which does not
matter here)
γ > 1 (MPI),
γ < 1 (tunneling).
(3.244)
Numerous strong laser-atom experiments operate around γ ' 1 or at γ > 1 and are thus
not in the tunneling domain. Taking, for instance, the case of atomic hydrogen in an
800 nm and 1014 W/cm2 laser pulse one finds γ ' 1.1. This is a typical value for ATI
measurements.
What are the differences between the ionization dynamics in the MPI and in the
tunneling domain? Since in the tunneling regime the process is fast compared to a
laser period, significant ionization occurs during a single half laser cycle, predominantly
around the electric field maximum because the barrier is lowest then. Furthermore, in
tunneling ionization the quiver amplitude E/ω 2 of the freed electron in the laser field
is large compared to the atomic dimension while in MPI this is not the case. This has
consequences for the rescattering dynamics which is responsible for various effects, such
as the ATI plateau, high-harmonic generation, and nonsequential ionization, as will be
discussed below.
3.5.1
Floquet formalism
When we discussed atoms in strong, static electric fields in Sec. 3.4 we made use of the
fact that in the tunneling regime the laser field may be considered “quasi-static” since
3.5. ATOMS IN STRONG LASER FIELDS
63
the inner-atomic dynamics is much faster than a laser period. What else could we take
advantage of in the treatment of atoms in laser fields in order to render it tractable? If
the laser pulse duration is long, i.e., if the pulse contains many laser cycles, we may in
good approximation consider it infinitely long. As a consequence the Hamiltonian will
then be periodic,
Ĥ (t ) = Ĥ0 + Ŵ (t ),
Ŵ (t + T ) = Ŵ (t ),
T=
2π
ω
,
(3.245)
and the TDSE will be a partial differential equations with periodic coefficients. This
type of problem has been studied by FLOQUET more than 120 years ago (see “Further
reading” below). The Floquet theorem tells us that the TDSE
Hˆ(t )|Ψ(t )⟩ = 0,
Hˆ(t ) = Ĥ (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 Bloch
theorem used in solid state physics to treat particle motion in periodic potentials is the
Floquet theorem applied to spatially periodic systems. Inserting (3.247) into (3.246)
leads to the eigenvalue equation
Hˆ(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
ε0 = ε + mω,
|Φ(t )⟩0 = eimωt |Φ(t )⟩,
m∈Z
(3.249)
do so. Let |α⟩ be the solution of the unperturbed problem
Ĥ0 |α⟩ = Eα0 |α⟩.
(3.250)
Because of the periodicity of |Φ(t )⟩ we can expand
|Ψ(t )⟩ = e
−iεt
|Φ(t )⟩ = e
−iεt
∞ X
X
n=−∞ α
Φ(n)
|α⟩e−inωt
α
(3.251)
are time-independent. Inserting (3.251) into
where the expansion coefficients Φ(n)
α
(3.246) gives
X
[Ĥ (t ) − ε − nω]Φ(n)
|α⟩e−inωt = 0.
(3.252)
α
nα
RT
Multiplying from the left with ⟨β|, eimωt , and integrating T −1 0 dt yields
X
[⟨β|Ĥ (m−n) |α⟩ − (ε + mω)δn m δαβ ]Φ(n)
=0
α
nα
(3.253)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
64
with the time-independent Hamiltonian
Z
1 T
(m−n)
Ĥ
=
Ĥ (t ) ei(m−n)ωt dt .
T 0
(3.254)
Introducing the Floquet state
⟨t |n⟩ = einωt
|αn⟩ = |α⟩ ⊗ |n⟩,
we can recast (3.253) into
X
nα
(3.255)
[⟨βm|ĤF |αn⟩ − ε⟨βm|αn⟩]Φ(n)
=0
α
(3.256)
where ĤF is the Floquet Hamiltonian whose matrix elements read
⟨βm|ĤF |αn⟩ = ⟨β|Ĥ (m−n) |α⟩ − mωδ mn δαβ .
(3.257)
Hence, we obtain the eigenvalue equation
X
nα
(m)
[⟨βm|ĤF |αn⟩Φ(n)
= εΦβ
α
(3.258)
or, in matrix notation,
HF Φ = 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 of
Sec. 3.3.2 where the Hamiltonian reads
Ĥ (t ) = ωa |a⟩⟨a| + ω b |b ⟩⟨b | − q Ê z cos ωt ,
|
{z
} | {z }
ωa > ω b .
(3.260)
−Ŵ (t )
Ĥ0
We thus have
Ĥ
(n)
=
1
T
Z
T
dt Ĥ (t ) einωt
0
= Ĥ0 δn0 −
= Ĥ0 δn0 −
q Ê z
2T
q Ê z
2
Z
T
€
Š
dt einωt eiωt + e−iωt
0
(δn,−1 + δn1 ),
(3.261)
i.e., a tridiagonal Hamiltonian in the “photon subspace”. For (3.257) we obtain
€
Š
1
⟨βm|ĤF |αn⟩ = ⟨β|Ĥ0 |α⟩ − mωδαβ δnm − q Ê⟨β|z|α⟩(δ m,n−1 + δ m,n+1 ) (3.262)
2
3.5. ATOMS IN STRONG LASER FIELDS
65
so that (using ⟨β|Ĥ0 |α⟩ = ωα δαβ ) (3.258) reads
∞
X
|b ⟩ X
n=−∞ |α⟩=|a⟩
(m)
(ωα − mω)δαβ δ mn − q Ê⟨β|z|α⟩(δ m,n−1 + δ m,n+1 ) Φ(n)
= εΦβ .
α
2
(3.263)
1
Defining
1
1
A = − q Ê⟨a|z|b ⟩ = − ΩRe−iϕ ,
2
2
Eα(m) = ωα − mω,
(3.264)
the corresponding matrix equation has the following structure (e.g., around m = 0):
 (−1)
E
A
 a
(−1)
∗

Eb
A


A Ea(0)
A


(0)
∗
∗
 A
Eb
A


A Ea(1)

(1)
A∗
Eb











Φ(−1)
a
(−1)
Φb
Φ(0)
a
(0)
Φb
Φ(1)
a
(1)
Φb












 = ε








Φ(−1)
a
(−1)
Φb
Φ(0)
a
(0)
Φb
Φ(1)
a
(1)
Φb






.




(3.265)
In Sec. 3.3.2 we applied the rotating wave approximation (RWA). In the Floquet framework this corresponds to allow only for transitions between Floquet states |αn⟩ and
(m)
|βm⟩ of equal energies Eα(n) = Eβ . Hence,
!
(m)
Ea(n) = ωa − nω = ω b − mω = E b
(m)
⇒ Ea(n) − E b
(3.266)
= ∆ = ωa − ω b − (n − m) ω
| {z } | {z }
ωab
(3.267)
!
=1
where ∆ is the detuning (3.168), and we are only interested in one-photon transitions
between |a⟩ and |b ⟩ (so that n − m = 1). It would have been smarter to choose the
opposite 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 the
electromagnetic field). However, in most of the literature on Floquet theory the sign
convention is like in (3.251). Equation (3.265) now reduces to the 2 × 2 equation
‚
(n−1)
Eb
A
A∗
Ea(n)
Œ‚
(n−1)
Φb
Φ(n)
a
Œ
‚
=ε
(n−1)
Φb
Φ(n)
a
Œ
.
(3.268)
The eigenvalues are
(n)
ε1,2
1
1
1q
= (ωa + ω b ) + ω
−n ±
2
2
2
Ω2R + ∆2 .
(3.269)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
66
Ω
n= −1
ωa
Ω
(ωa + ωb )/2
n= 0
n= 1
ωb
Ω
n= 2
Ω
Figure 3.9: Illustration of Eq. (3.269) for ∆ = 0. The coupling of the two unperturbed levels b
and a to the laser field gives rise to an infinite manifold of pairs of field-dressed levels (labelled
by n). Because of (3.249) all the different ns are equivalent. The field dressed levels are separated
by 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 energy
of the “photon field”, and the third term gives rise to two levels, separated by the frequency (3.174). Figure 3.9 illustrates the situation for vanishing detuning ∆ = 0. The
energy levels (3.269) are called field dressed energy levels.
Let us finally calculate the field dressed states for our two-state problem in RWA and
vanishing detuning. Because of (3.249) we are free to choose, say, n = 1. We also can,
without loss of generality, set ω b = 0. Vanishing detuning ∆ = 0 then implies ωa = ω.
Moreover we assume A to be real. Then
1
ε1,2 = ± ΩR
(3.270)
2
and the eigenvalue problem reduces to
‚ (0) Œ
‚ (0) Œ
0 A
Φb
Φb
.
= ε1,2
(1)
A 0
Φ(1)
Φa
a
(3.271)
The matrix
1
−1 1
M= p
= M−1
1
1
2
diagonalizes (3.271) and yields
− − 1 ΩR
0
Ψ
Ψ
= ε1,2
+
0 −ΩR
Ψ
Ψ+
2
where
1
(0)
Ψ− = p (Φ(1)
− Φ b ),
a
2
are the field-dressed states or Floquet states.
1
(0)
Ψ+ = p (Φ(1)
+ Φb )
a
2
(3.272)
(3.273)
(3.274)
3.5. ATOMS IN STRONG LASER FIELDS
67
3.5.2 Non-Hermitian Floquet theory
As long as we are dealing only with discrete states the quasi energy ε is real. If, on
the other hand, we allow for transitions into the continuum, e.g., via (multiphoton)
ionization, the quasi-energies become complex,
Γ
ε = ε0 + ∆ε − i .
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 yield
complex eigenvalues. The reason for complex quasi-energies lies in the boundary conditions. Decaying dressed bound states or dressed resonances must fulfill the so-called
Siegert boundary conditions. Instead of explicitly taking these boundary conditions into
account 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−iεt Φ(r , t ) = e−iεt
∞
X
n=−∞
e−inωt
X
(n) 1 (κ)
ΦN LM SN L (r )YLM (Ω)
r
N LM
(3.276)
(κ)
where SN L are Sturmian functions, YLM (Ω) are spherical harmonics, N LM are principal,
angular momentum, and magnetic quantum number, respectively, and κ is a scaling
parameter which may be complex. Sturmian functions proved useful in calculations
involving Coulomb potentials because they have the following properties (independent
of κ):
Z∞
1
SN L SN 0 L dr = 0
for
N 0 6= N
(3.277)
r
0
Z∞
SN L SN L dr = 1,
(3.278)
0
and their overlap matrix is tridiagonal. Note that none of the Sturmian functions under
the integrals in (3.277), (3.278) have to be taken complex conjugated.
Sturmian functions can be expressed in terms of Laguerre polynomials (note that
the definition of the Sturmians is not unique in the literature),
s
|κ|(N − 1)!
(|κ|)
SN L (r ) =
(2|κ|r )L+1 e−|κ|r L2L+1
(2|κ|r )
(3.279)
N −1
(N + L)(N + 2L)!
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 → r eiθ
(3.280)
68
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
where θ is a real angle. Imagine we express the matrix elements of the non-Hermitian
Floquet Hamiltonian in (3.258) in position space representation. The complex scaled
Floquet Hamiltonian is obtained as
HF (θ) (r ) = HF (eiθ r ).
(3.281)
Transforming the Floquet Hamiltonian in this manner turns out to be equivalent to
calculate the matrix elements of ĤF using the complex Sturmians
s
−iκ(N − 1)!
(κ)
SN L (r ) =
(−2iκr )L+1 eiκr L2L+1
(−2iκr )
(3.282)
N −1
(N + L)(N + 2L)!
if the rotation angle θ and the complex scaling parameter κ are related through
θ=
π
2
− arg κ.
(3.283)
The Siegert boundary condition will be automatically fulfilled if κ lies within the first
quadrant 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, dimensionally, a momentum. From (3.283) can be inferred that the angle of κ with the imaginary
axis of the complex κ-plane is θ. Because energy is ∼ κ2 we see from (3.283) that energy
eigenstates (previously all on the real energy-axis, of course) will be rotated by 2θ into
the lower half of the complex energy plane. In the absence of the perturbation and if κ
is chosen properly, only the continuum states are rotated (about the continuum threshold) while the bound states remain on the real axis. However, if the perturbation (the
laser field) is present, also the formerly discrete, real levels acquire an imaginary contribution to their quasi-energy ε, namely the −iΓ/2 in Eq. (3.275). This is illustrated in
Fig. 3.10.
We mentioned above that κ has to be chosen “properly”. What do we mean by
that? The modulus |κ| should be chosen such that the states of interest are well represented with as few as possible Sturmian functions in the numerical basis set. Since
(|κ|)
SN L (r ) ∼ e−|κ|r , |κ| is obviously related to the width of the states in position space (if
we are interested in the shift of the H(1s) state, for instance, |κ| close to one would be
a reasonable choice). arg κ is directly related to θ [cf. Eq. (3.283)]. It turns out that the
quasi-energies become quasi-stationary up to high accuracy (i.e., up to 6 or more decimal 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 transition energy) are related to avoided crossings of the dressed energies. Let us consider an
experiment where we drive atomic hydrogen with a laser frequency around the 1s ↔
2p-resonance at ω = −1/8 − (−1/2) = 0.375. Let us first keep the driving field strength
very low so that the Rabi-frequency is small. From Fig. 3.9 we then expect at resonance the two dressed states being very close to one of the unperturbed states. In fact,
3.5. ATOMS IN STRONG LASER FIELDS
Im ε
69
Im ε
(a)
(b)
discrete states
...
x
x x x xx
continuum
threshold
2θ
Re ε
continuum
x
x
x x x
x
Re ε
new
continuum
threshold
Figure 3.10: Illustration of the effect of complex scaling on the eigenenergies. Without perturbation (a) the discrete states remain on the real energy-axis while the continuum is rotated
by 2θ about the continuum threshold into the lower half of the complex energy plane. With
perturbation (b) the discrete states acquire a negative imaginary contribution −iΓ/2 where Γ is
the 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
(Ê = F = 0.1, ω = 0) in the complex energy-plane. In the case of N = 10 basis functions (per
atomic 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-coordinate
approach, Phys. Rev. A 27, 2946 (1983).]
70
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
Figure 3.12: Real part of the quasi-energy ε for laser-driven atomic hydrogen as a function of
the laser frequency ω and for three different driver amplitudes Ê = F , as given in the plot.
Avoided crossings occur at the 1s ↔ 2p-resonance frequency ω = 0.375. The separations of
the quasi-energies at resonance (indicated by the colored arrows) equal the corresponding Rabifrequencies. [From A. MAQUET, SHIH-I CHU, and WILLIAM P. REINHARDT, Stark ionization
in dc and ac fields: An L2 complex-coordinate approach, Phys. Rev. A 27, 2946 (1983), colored
arrows added by the author of these Lecture Notes.]
Fig. 3.12 explains how this comes about. For a small driving field far off any resonance
the dressed energies are close to the unperturbed ones. As the resonance at 0.375 is approached the dressed energy (which once was the 2p-energy −1/8) “collides” with the
1s-energy. However, the energies do not cross. Instead, as ω becomes > 0.375 the former 2p state assumes the role of the 1s state while the former 1s state continues like the
2p state for ω < 0.375. The separation of the two quasi-energy levels at resonance is
given by Ω. As expected, the separation at resonance increases with increasing driver
strength since Ω ∼ |Ê| = F . However, the resonance frequency will also shift due to the
AC 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 function of the laser intensity, calculated using the non-Hermitian Floquet method. At this
wavelength 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 overestimate
the ionization rate. The exact Floquet-result displays an interesting structure which
can be explained in terms of so-called Freeman resonances. As the AC Stark up-shift of
the continuum (which is given by the ponderomotive potential Up ) increases, the min-
3.5. ATOMS IN STRONG LASER FIELDS
71
Figure 3.13: Ionization rate vs laser intensity for H(1s) irradiated by linearly polarized light
of wavelength λ = 1064 nm. Dashed curves are partial rates for (12 + S)-photon ionization obtained within LOPT. The arrows indicate the intensities at which the real part of the 1s Floquet
eigenvalue crosses the 13- and 14-photon ionization thresholds. [From R.M. POTVLIEGE and
ROBIN SHAKESHAFT, Multiphoton processes in an intense laser field: Harmonic generation and
total 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 laser
intensities. 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 Ŵ (t ) to the laser field in (3.245) reads in dipole approximation and velocity gauge
1
(3.284)
Ŵ (t ) = p̂ · A(t ) + A2 (t )
2
with A(t ) the vector potential. The transformation of the wave function
i
|Ψ(t )⟩ = e− 2
Rt
−∞
A2 (t 0 ) dt 0 −iα(t )· p̂
e
|ΨKH (t )⟩
(3.285)
removes the A2 -term and transforms to the system of an electron oscillating with an
excursion
Zt
α(t ) =
A(t 0 ) dt 0
(3.286)
−∞
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
72
in the laser field (Kramers-Henneberger transformation). In this system the nuclear potential appears to oscillate. The TDSE then reads
‚
i∂ t |ΨKH (t )⟩ =
p̂ 2
2
Œ
+ V [r + α(t )] |ΨKH (t )⟩.
(3.287)
The Kramers-Henneberger Hamiltonian
ĤKH (t ) =
p̂ 2
2
+ V [r + α(t )]
(3.288)
is, for an infinitely long laser pulse, periodic as well so that we may apply the Floquet
theorem. Introducing the time-averaged Kramers-Henneberger potential
VKH (α̂, r ) =
1
T
Z
T
V [r + α(t )] dt ,
α̂ = max |α(t )|,
(3.289)
0
which is the zero-frequency contribution in a Fourier-expansion of the potential,
Eq. (3.253) can be written as
(m)
−(ε + mω)Φβ +
X
α
⟨β|
p̂ 2
2
+
+ VKH (α̂, r )|α⟩Φ(m)
α
X
αn
n6= m
= 0. (3.290)
⟨β|V (m−n) |α⟩Φ(n)
α
If the laser frequency is large compared to the relevant inner-atomic transitions, we may
neglect 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
‚ 2
Œ
p̂
ε|ΨKH ⟩ =
+ VKH (α̂, r ) |ΨKH ⟩.
(3.291)
2
If it is possible to transfer the entire electron population to the bound states of VKH (α̂, r )
there will be no ionization whatsoever. For intense fields where α̂ 1 Bohr radius the
potential VKH (α̂, r ) looks very different from the unperturbed nuclear potential since
it has a double-well structure with the minima close to the classical turning points ±α̂.
If electronic probability density is trapped in this potential the ionization rate decreases
despite increasing laser field strength. This has indeed been observed in numerical simulations 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 a
soft-core binding potential V (x) = −(x 2 + ")−1/2 with " = 1.9 (leading to a binding
energy of −0.5). The full TDSE was solved. The laser pulse of frequency ω = 2.5 was
ramped over 10 cycles up to Ê = 62.5 and thereafter held constant. The excursion of a
free electron in this field is α(t ) = 10 sin ωt . The cycle-averaged Kramers-Henneberger
3.5. ATOMS IN STRONG LASER FIELDS
73
Energy (a.u.)
(a)
Time (cycles)
(b)
x (a.u.)
Figure 3.14: Adiabatic stabilization in a one-dimensional model atom. The cycle-averaged
ionic 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 corresponding
probability densities are also plotted. In (b) a shadowgraph of the probability density, obtained
from the full solution of the time-dependent Schrödinger equation, is shown. The probability
density remains trapped in the effective potential. Low-frequency Rabi-floppings are responsible
for the oscillatory pattern. Note that the time scale of these oscillations is small compared to the
laser period.
74
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
Figure 3.15: Lifetime of the H atom in the ground state according to the high-frequency Floquet theory, vs intensity, at various laser frequencies ω; circular polarization. Numbers adjacent
to points on the curves are the corresponding values of α̂. The descending branches of the curves
correspond 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 circular
polarization, Phys. Rev. Lett. 65, 2362 (1990).]
potential, its three lowest levels and the corresponding probability densities are indicated in Fig. 3.14a. The potential has the above mentioned double-well shape with the
minima close to the classical turning points ±α̂. In Fig. 3.14b a shadowgraph of the
probability density obtained from the TDSE solution is shown. The probability density remains confined within the two turning points. Only during the up-ramping of
the pulse some density escapes. Obviously, not only a single dressed state is occupied
since the probability density distribution oscillates in time. It can be shown that this
oscillation is due to Rabi floppings between several of the dressed states. Note that the
time-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 in
circularly laser pulses of various frequencies and intensities as predicted by the highfrequency 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 was
observed in numerical simulations only. This is because there are no sufficiently strong
3.5. ATOMS IN STRONG LASER FIELDS
75
lasers available yet at short wavelengths. The photon energy ω has to exceed the ionization potential, and the laser intensity must be strong enough in order to lead to the
two minima in the time-averaged potential so that the probability density is kept far
away from the nucleus (where absorption of laser energy is efficient) most of the time.
Since the elongation α is inversely proportional to ω 2 , the laser intensity necessary for
adiabatic stabilization increases with ω. However, with the free electron lasers (FELs)
under development, for example at DESY in Hamburg, the regime of adiabatic stabilization should become accessible experimentally. Stabilization of Rydberg atoms by
the 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 edge
of a strong laser pulse field intensities in which violent ionization may occur and hence
no more electrons are left to be stabilized once the optimal stabilization condition is
reached. Secondly, inner electrons might be excited or removed by few photon processes
induced by the incident high frequency radiation. Then the question arises whether an
outer 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, thus
enhancing ionization. These three effects will probably make the experimental verification 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 matter
is 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 and
DIMITRY A. TELNOV, Beyond the Floquet theorem: generalized Floquet formalisms
and quasienergy methods for atomic and molecular multiphoton processes in intense
laser fields, Phys. Rep. 390, 1 (2004). A Floquet-code can be downloaded from the
Computer Physics Communications-archive: R.M. POTVLIEGE, STRFLO: a program for time-dependent calculations of multiphoton processes in one-electron atomic
systems I. Quasienergy spectra and angular distributions, Comput. Phys. Comm.
114, 42 (1998). The original paper on what later became the Floquet theorem
is G. FLOQUET, Sur les equations differentielles lineares à coefficients periodique,
Ann. Ecol. Norm. Sup. 12, 47 (1883). The fact that Floquet states can be interpreted as the quantum field states of the electromagnetic field was pointed out in
J.H. SHIRLEY, Solution of the Schrödinger Equation with a Hamiltonian Periodic
in Time, Phys. Rev. 138, B979 (1965).
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
76
A relatively recent review on stabilization (by one of the eminent persons in the
field) is MIHAI GAVRILA, Atomic stabilization in superintense laser fields, J. Phys.
B: At. Mol. Opt. Phys. 35, R147 (2002). Dynamic stabilization of a two-electron
model system has been studied in D. BAUER and F. CECCHERINI, Electron correlation 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 fields
was investigated in D. BAUER and F. CECCHERINI, Two-color stabilization of
atomic hydrogen in circularly polarized laser pulses, Phys. Rev. A 66, 053411 (2002).
Very recently, two-electron stabilization beyond the dipole approximation has
been studied in ANDREAS STAUDT and CHRISTOPH H. KEITEL, Two-electron
ionization 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 workhorse
in strong field laser-atom and laser-molecule interaction. It is sometimes referred to as
Keldysh-Faisal-Reiss (KFR) theory because of the papers of these authors (see “Further
reading” below).
As the laser field is far from being a small perturbation, “conventional” perturbation
theory is not applicable. The SFA does neither consider the laser field being small
compared to the binding forces nor does it assume the contrary at all times during the
interaction. Instead, the SFA’s assumptions consist basically of considering the binding
potential dominant until ionization whereas the laser field takes over after ionization.
The SFA has been applied to ionization, harmonic generation, and non-sequential
ionization. The beauty of the SFA lies, besides in its predictive power, in the possibility
to 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 electron
may at time ti be in the ground state |Ψ0 (ti )⟩ with energy E0 < 0, for instance, and the
laser field is not yet switched on. Now let us consider the matrix element
M p (tf , ti ) = ⟨Ψ p (tf )|Û (tf , ti )|Ψ0 (ti )⟩,
(3.292)
which governs the probability wi→f = |M p (tf , ti )|2 to find the electron at time tf in the
scattering state |Ψ p (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 off
again. Û (t , t 0 ) = Û † (t 0 , t ) is the time-evolution operator associated with the TDSE
i
∂
∂t
|Ψ(t )⟩ = Ĥ (t )|Ψ(t )⟩,
Ĥ (t ) =
1
2
[ p̂ + A(t )]2 + V̂ (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 Ĥ (t ) can be splitted in various ways:
i
∂
∂t
|Ψ(t )⟩ = [Ĥ0 + Ŵ (t )]|Ψ(t )⟩ = [Ĥ (V) (t ) + V̂ (r )]|Ψ(t )⟩,
with
Ĥ0 =
p̂ 2
2
Ĥ (V) (t ) =
+ V̂ (r ),
p̂ 2
2
+ Ŵ (t ),
(3.294)
(3.295)
and Ŵ (t ) the interaction with the laser field,
1
Ŵ (t ) = p̂ · A(t ) + A2 (t )
2
(velocity gauge).
(3.296)
The gauge transformation of the potentials (both scalar potential φ and vector potential A) and the wave function |Ψ(t )⟩
A0 = A + ∇χ (r , t ),
φ0 = φ −
∂ χ (r , t )
∂t
|Ψ0 (t )⟩ = e−iχ (r ,t ) |Ψ(t )⟩
,
(3.297)
where χ (r , t ) is an arbitrary differential scalar function, leaves the electric and the magnetic field unchanged:
E = −∂ t A − ∇φ = E 0 ,
B = ∇ × A = B 0.
(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 holds
true as long as all approximations we make do not destroy the gauge invariance. Unfortunately, the standard SFA does break the gauge invariance. Transformation to the
so-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 φ0 = −∂ t χ =
−E · r . The Hamiltonian in length gauge reads
0
Ĥ (t ) =
p̂ 2
2
0
+ V̂ (r ) − φ (r , t ) =
p̂ 2
2
+ V̂ (r ) + E (t ) · r̂
(3.300)
(one could also absorb V̂ (r ) in φ and φ0 ). Note that the transformation of the wave
function
|Ψ0 (t )⟩ = eir̂ ·A(t ) |Ψ(t )⟩
(3.301)
can be interpreted as a translation in momentum space. In fact, while in velocity gauge
the quiver momentum is effectively subtracted from the kinetic momentum, leading to
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
78
a canonical momentum different from the kinetic momentum, in length gauge kinetic
and canonical momentum are equal.
From (3.300) we can infer
Ŵ 0 (t ) = E (t ) · r̂
(length gauge)
(3.302)
with E (t ) = −∂ t A(t ).
Ĥ0 describes the unperturbed atom and seemingly does not depend on the gauge
chosen, i.e., Ĥ0 = Ĥ00 . However, one should bear in mind that the momentum p̂ in Ĥ0
is not the kinetic momentum (which, in atomic units, equals the velocity) while in Ĥ00
(length gauge) it is.
The Volkov-Hamiltonian Ĥ (V) (t ) governs the free motion of the electron in the laser
field. It fulfills in velocity gauge
i
∂
1
|Ψ(V) (t )⟩ = Ĥ (V) (t )|Ψ(V) (t )⟩ = [ p̂ + A(t )]2 |Ψ(V) (t )⟩.
∂t
2
(3.303)
Thanks to the dipole approximation the Volkov-Hamiltonian is diagonal in momentum
space. The solution of (3.303) is thus readily written down:
−iS p (t ,ti )
(V)
|Ψ (t )⟩ = e
|p⟩,
S p (t , ti ) =
1
2
Z
t
dt 0 [p + A(t 0 )]2
(3.304)
ti
where |p⟩ are momentum eigenstates, ⟨r |p⟩ = eip·r /(2π)3/2 . Note that the lower integration limit ti affects the overall phase of the Volkov solution only. As mentioned
above, the transition to the length gauge corresponds to a translation in momentum
space. It is thus easy to check that in length gauge one has
0
|Ψ(V) (t )⟩ = e−iS p (t ,ti ) |p + A(t )⟩
(length gauge)
(3.305)
with the same action S p (t , ti ) as in (3.304).
Let us now continue to derive the SFA transition matrix element. The time evolution operator Û (t , t 0 ) satisfies the TDSE (3.294),
i∂ t Û (t , t 0 ) = [Ĥ0 + Ŵ (t )]Û (t , t 0 ).
(3.306)
Its formal solution is given by the integral equations [cf. (3.87) and (3.88)]
Z
0
t
0
Û (t , t ) = Û0 (t , t ) − i
Z
t
0
t
0
= Û0 (t , t ) − i
dt 00 Û (t , t 00 )Ŵ (t 00 )Û0 (t 00 , t 0 ),
t0
dt 00 Û0 (t , t 00 )Ŵ (t 00 )Û (t 00 , t 0 ),
(3.307)
3.6. STRONG FIELD APPROXIMATION
79
where Û0 (t , t 0 ) is the evolution operator corresponding to the TDSE with Ĥ0 only.
Inserting (3.307) in the matrix element (3.292) leads to
Z tf
M p (tf , ti ) = −i
dt 0 ⟨Ψ p (tf )|Û (tf , t 0 )Ŵ (t 0 )|Ψ0 (t 0 )⟩
(3.308)
ti
where use of ⟨Ψ p (tf )|Û0 (tf , ti )|Ψ0 (ti )⟩ = ⟨Ψ p (tf )|Ψ0 (tf )⟩ = 0 was made because |Ψ p (tf )⟩
is a scattering state perpendicular to |Ψ0 (tf )⟩, and Û0 (t 0 , ti )|Ψ0 (ti )⟩ = |Ψ0 (t 0 )⟩. Since the
propagator Û (t , t 0 ) also satisfies the integral equations
Zt
0
(V)
0
Û (t , t ) = Û (t , t ) − i dt 00 Û (V) (t , t 00 )V̂ Û (t 00 , t 0 ),
(3.309)
t0
Zt
(V)
0
= Û (t , t ) − i dt 00 Û (t , t 00 )V̂ Û (V) (t 00 , t 0 ),
t0
where Û (V) (t , t 0 ) is the evolution operator corresponding to the TDSE (3.303), one obtains, upon inserting (3.309) in (3.308),
–Z t
f
M p (tf , ti ) = −i
dt 0 ⟨Ψ p (tf )|Û (V) (tf , t 0 )Ŵ (t 0 )|Ψ0 (t 0 )⟩
(3.310)
ti
Z
Z
tf
dt
−i
00
t 00
ti
™
tf
0
dt ⟨Ψ p (tf )|Û
(V)
0
0
00
00
00
(tf , t )V̂ Û (t , t )Ŵ (t )|Ψ0 (t )⟩ .
Rt
Rt
Rt
Rt
Rt
R t0
Using t f dt 00 t 00f dt 0 = t f dt 0 t f dt 00 Θ(t 0 − t 00 ) = t f dt 0 t dt 00 expression (3.310) may
i
i
i
i
i
be recast in the form
Z tf
•
M p (tf , ti ) = −i
dt 0 ⟨Ψ p (tf )|Û (V) (tf , t 0 ) Ŵ (t 0 )|Ψ0 (t 0 )⟩
(3.311)
ti
Z
t0
˜
dt V̂ Û (t , t )Ŵ (t )|Ψ0 (t )⟩ .
00
−i
0
00
00
00
ti
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 operator Û (t 0 , t 00 ) appears.
Neglecting the second term and replacing the final state |Ψ p (tf )⟩ with a plane wave
|p⟩ yields the SFA or so-called Keldysh-amplitude
Z
M p(SFA) (tf , ti ) =
tf
−i
ti
dt ⟨Ψ(V)
(t )|Ŵ (t )|Ψ0 (t )⟩
p
(3.312)
where in velocity gauge the Volkov wave |Ψ(V) (t )⟩ = Û (V) (t , ti )|p⟩ is given by (3.304) and
in length gauge by (3.305). The SFA transition amplitude integrates over all ionization
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
80
(V)
times t where the transition from the bound state |Ψ0 (t )⟩ to the Volkov state |Ψ p (t )⟩,
mediated by the interaction with the laser field Ŵ (t ), may take place. It is thus a bit
surprising, at least at first sight, that the Keldysh amplitude can be recast in a form
where Ŵ (t ) is replaced by V̂ (r ):
Z tf
(SFA)
M p (tf , ti ) = −i
dt ⟨Ψ(V)
(t )| Ĥ (V) (t ) − Ĥ0 + V̂ |Ψ0 (t )⟩
p
{z
}
|
t
i
Z
= −i
Z
ti
tf
= −
ti
= −
Ŵ (t )
tf
→
←
dt ⟨Ψ(V)
(t )| − i ∂ t +V̂ − i ∂ t |Ψ0 (t )⟩
p
n
o
(V)
dt ∂ t ⟨Ψ(V)
(t
)|Ψ
(t
)⟩
+
i⟨Ψ
(t
)|
V̂
|Ψ
(t
)⟩
0
0
p
p
tf
⟨Ψ(V)
(t
)|Ψ
(t
)⟩
0
p
t
i
Z
tf
−i
ti
dt ⟨Ψ(V)
(t )|V̂ |Ψ0 (t )⟩.
p
(3.313)
Since the laser is off at the times ti and tf , the first term is just the difference of the
Fourier-transformed initial state at the two times ti and tf . The latter can be always
chosen such that the term vanishes. The fact that the boundary term shows up at all is
due to the non-orthogonality of plane waves (introduced within the SFA) and the initial
bound state. However, the contribution of the first term vanishes at latest when the
asymptotic rate is calculated. Hence, the SFA-amplitude (3.312) may be also written as
Z
M p(SFA) (tf , ti ) =
tf
−i
ti
(t )|V̂ (r )|Ψ0 (t )⟩.
dt ⟨Ψ(V)
p
(3.314)
Following the above interpretation it now seems that ionization is mediated by the
binding potential V̂ (r ), which appears to be absurd. However, upon time-reversal ionization turns into (re)combination, for which indeed the nuclear potential is responsible. Moreover, in the above derivation of (3.312) Ŵ and V̂ 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 Ŵ (t ) = p̂ · A(t ) + A2 (t )/2 one can write (3.312) as
ΠZf
‚ 2
t
p
iS p,E (t ,ti ) f
(SFA)
M p (tf , ti ) = −Ψ0 (p) e 0 + iΨ0 (p)
− E0
dt eiS p,E0 (t ,ti ) (3.315)
ti
2
t
ti
with the classical action
‚
Zt
S p,E0 (t , ti ) =
dt
ti
0
p2
2
Œ
0
− E0 + Ŵ (t ) ,
(3.316)
3.6. STRONG FIELD APPROXIMATION
81
Ψ0 (p, t ) = ⟨p|Ψ0 (t )⟩, and Ψ0 (p, t ) = exp(−iE0 t )Ψ0 (p), that is, Ψ0 (p) is the Fouriertransformed initial state wave function
Z
1
Ψ0 (p) =
d3 r e−ip·r Ψ0 (r )
(3.317)
3/2
(2π)
and E0 is the initial energy. The first term in (3.315) vanishes when the asymptotic rate
Γ p = lim
2
M p T
T →∞
is calculated:
,
(3.318)
i
tf →∞
‚
M p(SFA) = iΨ0 (p)
M p = tlim
M p (tf , ti )
→−∞
p2
2
Œ Z∞
dt eiS p,E0 (t ,−∞) .
− E0
(3.319)
−∞
3.6.1 Circular polarization and long pulses
In this case one may write the vector potential in dipole approximation as
1
A(t ) = Â[" exp(iωt ) + "∗ exp(−iωt )]
2
(3.320)
where the polarization vectors ", "∗ fulfill "2 = "∗2 = 0 and " · "∗ = 1, e.g., " = (e x +
p
iey )/ 2. The factor 1/2 is introduced in order to obtain Up = Â2 /4, as in the linearly
polarized case. With
1
[p + A(t )]2 = p 2 + Â2 + 2Â|p · "| cos(ωt − ϕ),
2
(3.321)
where p · " = |p · "| exp(−iϕ), the action (3.316) reads
‚
S p,E0 (t , −∞) =
p2
2
Œ
− E0 + Up t +
Â
ω
|p · "| 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 then
is
!
∞
X
−Â|p · "|
(SFA)
M p, circ. = 2πiΨ0 (p)
(nω − Up ) exp(inϕ) Jn
δ( p 2 /2 − E0 + Up − nω),
ω
n=−∞
(3.323)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
82
where use of the Bessel functions Jn (ζ ), obeying
exp[−iζ sin(ωt − ϕ)] =
∞
X
Jn (ζ ) exp[−in(ωt − ϕ)],
(3.324)
n=−∞
was made. The time integration in (3.319) leads to the energy-conserving δ-function.
Employing (3.318) one obtains the ionization rate
!
∞
X
−Â|p · "|
(SFA)
2
2 2
Γ p, circ. = 2π|Ψ0 (p)|
(nω − Up ) Jn
δ( p 2 /2 − E0 + Up − nω). (3.325)
ω
n=−∞
(SFA)
Γ p, circ. has the dimension of a density in momentum space per time. To evaluate the
square of the δ-function we used the “working formula”
Z T /2
T
1
exp(iΩt ) dt = lim
lim
for Ω = 0
(3.326)
δ(Ω) =
T →∞ 2π
2π T →∞ −T /2
(and zero otherwise). For obtaining the total rate Γ the partial rate Γ p has to be integrated over all final momenta p,
Z
Z
Z
dΓ
3
2
Γ = d p Γ p = d p dΩ p Γ p = dΩ
(3.327)
dΩ
where dΩ = sin ϑ dϑ dϕ is the solid angle element. The final rate for ionization with
the electron ejected into the solid angle dΩ is given by
!
(SFA)
∞
p
X
dΓcirc.
p
−
Âp
sin
ϑ
n
n
= 2π 8ω 5
(n − Up /ω)2 p
|Ψ0 ( pn )|2 Jn2
,
(3.328)
p
dΩ
2ω
2ω
n=nmin
with
pn =
Æ
2(nω − Up + E0 ).
(3.329)
The sum in (3.328) runs over all n which yield real pn , starting with the minimum
number of absorbed photons nmin .
3.6.2
Channel closing in above-threshold ionization
The increase of nmin with increasing Up is the channel-closing phenomenon. This is
illustrated in Fig. 3.16. We expect peaks in the photoelectron spectra at the positions
En =
1
2
pn2 = nω − (Up + |E0 |),
n ≥ nmin
(3.330)
where |E0 | is the ionization potential and Up + |E0 | is the “effective” ionization potential
due to the AC Stark shift of the continuum threshold with respect to the ground state
3.6. STRONG FIELD APPROXIMATION
(a)
83
short pulse
(b)
short pulse
long pulse
long pulse
Up
ε0
Figure 3.16: Channel closing in the short and long pulse regime. In (a) the laser intensity is
small so that the Up -shift of the continuum threshold is less than a photon energy. In (b) the
channels 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) since
the released electron gains Up in the latter case upon leaving the laser focus.
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
84
level. Due to the fact that peaks n > nmin are present—even with higher probability
than n = nmin —the name above-threshold ionization has been coined for this strong field
ionization phenomenon. In actual measurements the positions of the peaks depend on
the laser pulse duration. In short-pulse experiments the released electron has no time to
leave the focus before the laser pulse is over. In long pulses, instead, the electron has time
to leave the focus and, by doing so, gains the energy Up . As a consequence the Up -term
in (3.330) is cancelled and the peak positions in the long-pulse regime are determined by
En(long pulse) =
1
2
pn2 = 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 continuum
threshold is small, and the channel n = 5 is responsible for the first peak in the photoelectron spectra. At higher laser intensity, in Fig. 3.16b, the channels n = 5 and n = 6
are closed since 6 photons are not sufficient to overcome the effective ionization potential |E0 | + Up . In the long pulse regime peaks corresponding to a certain channel are
always at the same positions in the energy spectra whereas in the short pulse regime the
peaks move with Up . As a consequence, focal averaging reduces the contrast in the short
pulse 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 where
lower 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
"2 = 1.
A(t ) = Â" cos(ωt ),
(3.332)
The action (3.316) reads in this case
‚
S p,E0 (t , −∞) =
p2
2
Œ
− E0 + Up t −
Â
ω
p · " sin(ωt ) +
Â2
8ω
sin(2ωt ).
(3.333)
Proceeding as in the circular field-case one arrives at
(SFA)
M p,lin. = 2πiΨ0 (p)
(SFA)
Γ p,lin.
∞
X
(nω−Up )J˜n
−Âp · "
n=−∞
= 2π|Ψ0 (p)|2
∞
X
n=−∞
(nω − Up )2 J˜n2
ω
,−
Up
2ω
−Âp · "
ω
,−
!
δ( p 2 /2−E0 +Up −nω), (3.334)
Up
2ω
!
δ( p 2 /2 − E0 + Up − nω),
(3.335)
3.6. STRONG FIELD APPROXIMATION
85
Up
Figure 3.17: Channel closing in the long-pulse regime. The experimental spectrum was taken
from R.R. FREEMAN & P.H. BUCKSBAUM, Investigation of above-threshold ionization using subpicosecond laser pulses, J. Phys. B: At. Mol. Opt. Phys. 24, 325 (1991).
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
86
(SFA)
dΓlin.
dΩ
p
= 2π
∞
X
8ω 5
(n − Up /ω)2 p
n=−∞
pn
2mω
|Ψ0 ( pn )|2 J˜n2
−Âp · "
ω
,−
J˜n is the generalized Bessel function of integer order defined by
Z
1 π
˜
Jn (u, v) =
dθ exp[i(u sin θ + v sin(2θ) − nθ)].
2π −π
Up
2ω
!
. (3.336)
(3.337)
The relation to the ordinary Bessel functions is
J˜n (u, v) =
∞
X
Jn−2k (u)Jk (v).
(3.338)
k=−∞
The equation used in the derivation of (3.334) is
∞
X
exp(inθ)J˜n (u, v) = exp[iu sin θ + iv sin(2θ)].
(3.339)
n=−∞
Further properties of the generalized Bessel functions may be found in the Appendix B
of the original SFA paper by REISS. In the same article it is also demonstrated how the
typical exponential behavior ∼ exp[−2(2|E0 |)3/2 /(3|E|)] arises in the case of tunneling
ionization through the asymptotic behavior of the generalized Bessel functions J˜n .
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 Â(t ) varies on a time scale not much slower than
the period T = 2π/ω corresponding to the carrier frequency ω. Moreover, the carrier
envelope 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 ) = Â(t )" sin(ωt + φ)
with Â(t ) a sin2 or a Gaussian envelope, for instance.
(3.340)
3.6. STRONG FIELD APPROXIMATION
87
In the case of few-cycle pulses it would be very cumbersome to deal with Bessel
functions, as we did above for constant (or slowly varying) Â. Instead, we start off with
the still exact matrix element (3.311) and replace once again the final state by a plane
wave and the full propagator Û (t 0 , t ) by Û (V) (t 0 , t ). In that way we obtain the extended
SFA transition matrix element
M p(SFA) (tf , ti ) = M p(SFA,dir) (tf , ti ) + M p(SFA,resc) (tf , ti ),
Z
M p(SFA,dir) (tf , ti )
M p(SFA) (tf , ti ) =
=
Z
M p(SFA,resc) (tf , ti )
Z
tf
= −
−i
ti
dt ⟨Ψ(V)
(t )|Ŵ (t )|Ψ0 (t )⟩,
p
(3.342)
t
dt
ti
tf
(3.341)
ti
dt 0 ⟨Ψ(V)
(t )|V̂ Û (V) (t , t 0 )Ŵ (t 0 )|Ψ0 (t 0 )⟩. (3.343)
p
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 making
use of the fact that ionization can only happen while the laser is on, (3.342) and (3.343)
can be recast in the form
Z Tp
(SFA,dir)
dtion ⟨Ψ(V)
(tion )|Ŵ (tion )|Ψ0 (tion )⟩,
(3.345)
Mp
= −i
p
Z
M p(SFA,resc)
0
Tp
= −
Z
∞
dtresc ⟨Ψ(V)
(tresc )|V̂ Û (V) (tresc , tion )Ŵ (tion )|Ψ0 (tion )⟩. (3.346)
p
dtion
0
tion
The interpretation of the second term is straightforward: ionization due to interaction
with the laser (Ŵ ) 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 Û (V) ) before
it rescatters at time tresc with the ionic core (V̂ ). Note that tresc may be > Tp . After the
(V)
rescattering event, the electron ends up in the Volkov state |Ψ p ⟩, meaning that it will
finally arrive with a (field free and thus kinetic) momentum p at the detector.
In length gauge the matrix elements read
Z
M p(SFA,dir)
= −i
Z
M p(SFA,resc)
Tp
dtion ⟨p + A(tion )|r · E (tion )|Ψ0 ⟩ eiS p,E0 (tion ) ,
0
Tp
= −
Z
dtion
0
(3.347)
Z
∞
dtresc
d3 k eiS p (tresc ) ⟨p + A(tresc )|V̂ |k + A(tresc )⟩
tion
×e
−iSk (tresc )
⟨k + A(tion )|r · E (tion )|Ψ0 ⟩eiSk,E0 (tion )
(3.348)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
88
where
Zt
dt 0
S p,E0 (t ) =
1
2
[p + A(t 0 )]2 − E0 ,
S p (t ) =
1
Zt
dt 0 [p + A(t 0 )]2 .
2
0
(3.349)
0
Introducing the new variables
t = tresc ,
τ = tresc − tion
(3.350)
the rescattering SFA matrix element (3.348) can be written as
Z
M p(SFA,resc) = −
Z
∞
dt
}|
{
z
d3 k eiS p (t ) ⟨p + A(t )|V̂ |k + A(t )⟩
dτ
0
Z
V p−k =⟨p|V̂ |k⟩
Z
t
(3.351)
0
∞
= −
Z
t
dt eiS p,E0 (t )
d3 k V p−k e−iSk,E0 (t ,t −τ)
dτ
0
×e−iSk (t ) ⟨k + A(t − τ)|r · E (t − τ)|Ψ0 ⟩
Z
(3.352)
0
×⟨k + A(t − τ)|r · E (t − τ)|Ψ0 ⟩
where
Z
Sk,E0 (t , t − τ) =
t
dt
0
1
2
t −τ
2
k + A(t ) − E0 .
0
(3.353)
The time τ is the time the electron spends in the continuum between ionization and
rescattering. 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 to
energy absorption. As a consequence, the final energy will be within a region strongly
dominated by the more probable direct ionization process.
The integration over the intermediate momentum k can be performed using the
saddle-point approximation (stationary phase method) where we seek the stationary momentum ks (t , τ) contributing most to the k-integration:
!
∇k Sk,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,
Z
M p(SFA,resc)
= −
Z
Tp
dt e
0
iS p,E (t )
t
dτ
0
2π
3/2
V p−ks (t ,τ) e−iSs,E0 (t ,t −τ)
iτ
×⟨ks (t , τ) + A(t − τ)|r · E (t − τ)|Ψ0 ⟩
0
(3.355)
3.6. STRONG FIELD APPROXIMATION
89
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 numerical
evaluations the denominator iτ can be either regularized by adding a real, positive ε, or
the 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 |)5/4
k · E (t )
π(k 2 + 2|E0 |)3
.
(3.357)
For rescattering at potentials of the form

V (r ) = − b +
a‹
r
e−λr
(3.358)
the matrix element V p−k is given by
V p−k = −
2b λ + aC
2
2π C
2
C = (p − k)2 + λ2 .
,
(3.359)
In few-cycle laser pulses the concept of an ionization rate is not useful since the latter 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 differential ionization probability w p , which is the probability to find an electron of final
energy E p = p 2 /2 emitted in a certain direction, given by the solid angle element dΩ p
that is covered by the measuring device. The probability w p is related to the transition
matrix element M p through
w p dE p dΩ p = |M p |2 d3 p = |M p |2 p 2 d p dΩ p
|{z}
(3.360)
p dp
so that
w p = p |M p |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 or
approximately 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 transition
matrix element can be approximated by a sum over the stationary contributions,
X
M p(SFA) =
a s ,p eiSs,p .
(3.362)
s
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
90
As it turns out the saddle-point equations define quantum orbits that are close to the
classical orbits of the so-called simple man’s theory. In the following we will use simple
man’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 ionic
potential anymore, its momentum and position at times t > tion are given by
Zt
p(t ) = −
dt 0 E (t 0 ) = A(t ) − A(tion ),
(3.363)
Z
tion
t
dt 0 A(t 0 ) − A(tion )(t − tion )
r (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 determined by the value of the vector potential at the time of ionization, p(Tp ) = −A(tion ),
so that the final energy is
1
Edir,p (tion ) = A2 (tion ) ≤ 2Up
2
(3.365)
because the ponderomotive potential is Up = Â2 /4. The fact that the direct electrons are
classically restricted to energies up to 2Up is one of the celebrated cut-off laws in strong
field physics.
Let us now allow for one rescattering event, i.e., at the time tresc the electron returns
to 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 case
of 180◦ back-reflection where the electron changes the sign of its momentum so that
immediately after the scattering event
p(tresc + ) = −[A(tresc ) − A(tion )].
(3.367)
At later times we have
Z
t
dt 0 E (t 0 ) − [A(tresc ) − A(tion )]
p(t > tresc ) = −
tresc
= A(t ) − 2A(tresc ) + A(tion )
(3.368)
so that presc (Tp ) = −2A(tresc ) + A(tion ) and
1
Eresc,p (tresc , tion ) = [A(tion ) − 2A(tresc )]2 ≤ 10Up .
2
(3.369)
3.6. STRONG FIELD APPROXIMATION
91
final energy contours / Up
"direct" electrons
rescattered
electrons
Figure 3.18: Final photoelectron energy Eresc,p vs ionization time tion and rescattering time
tresc > tion (black: high value of Eresc,p , white: small value of Eresc,p , contours 2, 10 and 16Up
labelled explicitly). The red branch labelled “rescattered electrons” indicate times tion and tresc
where (3.366) is fulfilled. The highest energy those rescattered electrons can have is 10Up (see
red 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 not
so 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 that
fulfill (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 solution
with the corresponding SFA predictions. Apart from the dipole approximation (which
is well applicable for the peak intensities used) the TDSE result is exact. Comparisons
with TDSE results serve as a much more demanding testing ground for approximate
theories such as the SFA than comparison with experiments, mainly because of focal
averaging effects present in experiments, the uncertainties in laser intensity, pulse dura-
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
92
2 Up
2 Up
Yield (arb.u.)
(a) φ=0
θ=π
(b) φ=0
θ=0
10 Up
10 Up
Photoelectron energy (a.u.)
Figure 3.19: Photoelectron spectra of the H(1s) electron after irradiation with a 4-cycle laser
pulse (ω = 0.056, φ = 0, Ê = 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 polarization
e z ), panel (b) the “right-going” electrons (in e z )-direction. The spectra were adjusted vertically
by multiplication with a single factor in such a way that agreement is best in the cut-off region
for 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 the
yields.
In Fig. 3.19 the results for a n = 4-cycle pulse of the form E (t ) =
Êe z sin2 [ωt /(2n)] cos(ωt + φ) for 0 < t < n2π/ω, ω = 0.056, Ê = 0.0834, φ = 0
are shown. The agreement between TDSE and SFA results improve with increasing
photoelectron energy. A possible explanation for that might be that slow electrons
spend more time in the vicinity of the atomic potential where Coulomb corrections are
expected to be important. As can be seen in Fig. 3.19, the transition regime between
the cut-off for the direct electrons at E = 2Up = 1.1 up to energies where rescattered
electrons start to take over at E ≈ 2.5 is quite smooth in the TDSE spectra whereas
pronounced patterns are visible in the SFA results. Moreover, at very low energies the
positions 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 shortrange by cutting it at certain distances. This is expected since the crucial assumption
in SFA is that the electron is not affected by the ionic potential anymore once ionization has occurred. This assumption is well justified for short-range potentials but less
so for long range Coulombic ones as in the H(1s) case. However, the pronounced pat-
3.6. STRONG FIELD APPROXIMATION
same final energy
93
(a)
rescattered electrons
"direct" electrons
(b)
Figure 3.20: (a) Final energy of direct electrons (black line) vs the ionization time tion . If
rescattered is allowed, higher energies may occur. The color coding for the rescattered electrons
indicates 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 rescattered
electrons are 2 and 10Up , respectively. Panel (b) shows the course of the laser field. Ionization is
improbable for small |E(t )|.
tern showing a spiky, downward-pointing structure in the rescattering plateaux between
energies ≈ 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 momentum p the sum (3.362) is a sum over all quantum orbits that end up with the same
momentum at the detector. It turns out that, most of the time, there are two dominating
contributions. 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 in
Fig. 3.20. In the lower panel of Fig. 3.20 the course of the electric field is indicated. Let
us focus on the time t = 0.5 cycles where the electric field has a maximum and ionization is therefore most likely. The upper panel shows that a “direct” electron emitted at
that time will have vanishing final energy. In order for a direct, classical electron to have
the 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
94
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
emitted around the maximum of the electric field and rescatters once, its final energy
may be close to 10Up . It is clearly seen that two emission times very close to each other
lead to the same final energy (as indicated in the upper panel). These are the two trajectories that interfere. Exactly at the cut-off 10Up those two solutions merge to a single
one. The “travel times” tresc − tion between rescattering and ionization are color-coded
from 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 pulse
case. Since the ionization probability is strongly weighted with the modulus of the electric field, only a few “time-windows” may remain “open”, thus affecting the number of
interfering quantum orbits for a given p. The interference pattern is then very sensitive
to 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 fewcycle pulse, one may view the setup as a “double slit experiment in time” (see “Further
reading” below).
Problem 3.9 Write down the SFA amplitude for the direct electrons emitted during
a 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 of
the relevant references is D.B. MILOŠEVI Ć, 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. Recent work on the subject, showing that in the case of atoms the velocity gauge may
lead to (even qualitatively) wrong results, is D. BAUER, D.B. MILOŠEVI Ć, and W.
BECKER, Strong-field approximation for intense laser-atom processes: the choice of
gauge, Phys. Rev. A 72, 023415 (2005). A comparison of SFA with TDSE results
focussing on the effect of the range of the binding potential has been undertaken
in D. BAUER, D.B. MILOŠEVI Ć, and W. BECKER, On the validity of the strong
field approximation and simple man’s theory, J. Mod. Opt. 53, 135 (2006). The
emergence of the simple man’s theory-orbits in ab initio TDSE results has been
demonstrated in D. BAUER, Emergence of Classical Orbits in Few-Cycle AboveThreshold Ionization of Atomic Hydrogen, Phys. Rev. Lett. 94, 113001 (2005). The
above-mentioned “double slit in time”-experiment has been published in F. LINDNER , M.G. S CHÄTZEL , H. WALTHER , A. BALTUŠKA , E. G OULIELMAKIS , F.
KRAUSZ, D.B. MILOŠEVI Ć, D. BAUER, W. BECKER, and G.G. PAULUS, Attosecond Double-Slit Experiment, Phys. Rev. Lett. 95, 040401 (2005).
3.6. STRONG FIELD APPROXIMATION
95
3.6.6 High harmonic generation
When an intense laser pulse of frequency ω1 impinges on any kind of sample usually
harmonics of ω1 are emitted. A typical signature of the emission spectrum in the case of
strongly driven atoms, molecules or clusters is that the harmonic yield does not simply
roll-off exponentially with increasing harmonic order. Instead, a plateau is observed.
This is a prerequisite for high order harmonic generation (HOHG) being of practical
relevance as an efficient short wavelength source. As targets for HOHG one may think
of single atoms, dilute gases of atoms, molecules, clusters, crystals, or the surface of a
solid (which is rapidly transformed into a plasma by the laser). In fact, for all those
targets HOHG has been observed experimentally. Even a strongly driven two-level
system displays nonperturbative HOHG. The mechanism generating the harmonics
and its efficiency, of course, vary with the target-type. In the case of atoms the socalled 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 0 , subsequently it oscillates in the
laser field, and eventually recombines with its parent ion upon emitting a photon of
frequency
ω = n ω1 ,
n ≥ 1.
(3.370)
This process is illustrated in Fig. 3.21. If the energy of the returning electron is E , the
energy of the emitted photon is ω = E + |Ef | where Ef is the energy of the level which
is finally occupied by the electron, for example, the groundstate.
Energy
Space
Figure 3.21: Illustration of the three step model for high harmonic generation. An electron
is (i) released, (ii) accelerated in the laser field, and (iii) driven back to the ion. There it may
recombine upon emitting a single photon which corresponds to a multiple of the photon energy
of the incident laser light.
From this simple considerations we conclude that the maximum photon energy one
can expect is ωmax = Emax + |Ef |. Using (3.363) we obtain at the recombination time trec
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
96
for the return energy Eret
Eret =
1
1
p 2 (trec ) = [A(trec ) − A(tion )]2
2
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 infer
that the maximum return energy is around ' 3.2. Closer inspection shows that the
number 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 reads
ωmax = 3.17Up + 1.32|Ef |.
Figure 3.22: Return energy as a function of the ionization time. The color coding indicates
the time between recombination and ionization (the longer this time the lighter the color). The
maximum 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 much
less efficient in elliptically polarized laser fields since, classically, the freed electron never
comes back to its parent ion so that recombination can be considered unlikely. This
indeed was confirmed experimentally.
3.6. STRONG FIELD APPROXIMATION
97
Besides the academic interest, harmonic generation in atoms, molecules and clusters
has huge practical relevance as an efficient source of intense XUV radiation. This is because (i) the ponderomotive scaling of the cut-off allows to achieve high values of Emax
and thus high harmonic orders n, and (ii), fortunately, the strength of the harmonics
does not decay exponentially with the order n but displays, after a decrease over the first
few harmonics, an overall plateau (at least on the logarithmic scale) up to the cut-off energy 3.17 Up +|Ef |, allowing relatively high intensities at short wavelengths. In fact, high
order harmonics below λ = 4.4 nm, the so-called water-window (between the K-edges
of carbon and oxygen) have been observed experimentally. The intensity of the emitted radiation in the plateau region is about 10−6 of the incident laser intensity which
is typically 1015 –1018 Wcm−2 in rare gas experiments. Therefore, the intensity of the
high order harmonics is sufficient for various kinds of applications, such as interferometry, for dense plasma diagnostics, holography, high-contrast microscopy of biological
materials, and attosecond spectroscopy or metrology. Attosecond pulses are generated
via harmonic generation. If the incoming pulse is already short (i.e., consists only of
a few cycles) the harmonic emission is restricted to a narrow time window similar to
the “double slit in time”-experiment mentioned above. As a consequence, the harmonic
pulse 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 construct attosecond pulse trains by selecting a few phase-locked harmonics. Attosecond
pulses that are generated via harmonic generation have been used to probe the ionizing
laser field itself as well as fast atomic processes such as Auger decay. Figure 3.23 shows
an example for experimental high-order harmonic spectra.
For calculating the rate of harmonic emission one may follow the same route as in
the SFA treatment of ionization. Harmonic generation and above threshold ionization
(ATI) are complementary to each other: while in harmonic generation the electron
comes back to the ion and eventually recombines, upon emitting radiation, in ATI
it rescatters upon which it may gain additional energy. In harmonic generation one
observes a plateau reaching up to photon energies 3.17 Up + |Ef |. In ATI a plateau is
observed 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 given
by Larmor’s formula
2q 2
P = 3 |r̈ |2 .
(3.374)
3c
Thus, in a semiclassical approach, it appears reasonable to replace the acceleration by
its quantum mechanical expectation value and making use of Ehrenfest’s theorem. One
obtains
*
2
+ 2
2q 2 1
∂ Ĥ 2q 2 d2
(3.375)
−
P = 3 2 ⟨r̂ ⟩ = 3 .
∂ r̂ 3c dt
3c m
The last expression on the right-hand side is particularly suited for the numerical evalu-
98
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
Figure 3.23: Harmonic production efficiency in Ar and Ne at 2 × 1015 and 5 × 1014 Wcm−2 ,
respectively, using a 7-fs Ti:sapphire laser pulse. [From M. SCHNÜRER et al., Absorption-Limited
Generation 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 as
a Fourier-transform,
Z∞
q
dω exp(iωt )d(ω).
(3.376)
d(t ) = q ⟨r̂ ⟩ = p
2π −∞
The total emitted energy then is
Z∞
Z∞
2
Erad =
dt P = 3
dω ω 4 |d(ω)|2 ,
3c −∞
−∞
(3.377)
and we infer that the yield radiated into a spectral interval [ω, ω + dω] is
εrad, ω dω ∼ ω 4 |d(ω)|2 dω.
(3.378)
A full quantum mechanical treatment reveals that calculating the harmonic spectrum emitted by a single atom from the square of the dipole expectation value is actually
incorrect. Since the expectation value of the number of photons in a mode ω, k (with
creation and annihilation operators â † , â, respectively) at time t is
Zt
†
†
⟨â (t )â(t )⟩ = ⟨â (ti )â(ti )⟩ + 2C ℜ dt 0 ⟨r̂ (t 0 )â(ti )⟩ exp(−iωt 0 )
ti
3.6. STRONG FIELD APPROXIMATION
Z
+C
Z
t
2
dt
t
0
ti
99
dt 00 ⟨r̂ (t 00 )r̂ (t 0 )⟩ exp[iω(t 0 − t 00 )],
(3.379)
ti
where C is a coupling constant, one sees that if the mode under consideration is not
excited at the initial time t = ti , as it is the case here, only the third term survives. This
term accounts for spontaneous emission and scattering. Hence, the harmonic spectrum
of a single atom should be calculated from the two-time dipole-dipole correlation function ⟨r̂ (t 00 )r̂ (t 0 )⟩ instead of the Fourier-transformed one-time dipole. However, if one
considers a sample of N atoms
Zt
N X
N Z t
X
0
†
2
dt dt 00 ⟨r̂k (t 00 )r̂ j (t 0 )⟩ exp[iω(t 0 − t 00 )]
(3.380)
⟨â (t )â(t )⟩ = C
k=1 j =1
ti
ti
results and, by assuming that all these atoms are uncorrelated, that is ⟨r̂k (t 00 )r̂ j (t 0 )⟩ '
⟨r̂k (t 00 )⟩⟨r̂ j (t 0 )⟩, one arrives at
2
N Z t
X
⟨â † (t )â(t )⟩ ≈ C 2 dt 0 ⟨r̂k (t 0 )⟩ exp(iωt 0 )
(3.381)
k=1 t
i
if N 1 is assumed so that the self-interaction terms ∼ ⟨r̂k (t 00 )⟩⟨r̂k (t 0 )⟩ contribute negligibly. Moreover, if all atoms “see” the same field one obtains simply the absolute square
of N times the single dipole expectation value. Therefore, calculating the harmonic
spectra from the Fourier-transformed dipole, although not correct in the single atom
response case, is a reasonable method when comparison with high-order harmonic generation experiments in dilute gas targets is made. Hence, for the study of macroscopic
propagation effects the dipole expectation value may be inserted as a source in Maxwell’s
equations.
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 )|r̂ |Ψ(t )⟩
(3.382)
= −⟨Ψ0 (ti )|Û (ti , t ) r̂ Û (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 )|r̂ |Ψ0 (t )⟩
Zt
−i dt 0 ⟨Ψ0 (t 0 )|Ŵ (t 0 )Û (t 0 , t )r̂ |Ψ0 (t )⟩
t
Z it
+i dt 0 ⟨Ψ0 (t )|r̂ Û (t , t 0 )Ŵ (t 0 )|Ψ0 (t 0 )⟩
Z
ti
t
−
dt
ti
Z
t
0
dt 00 ⟨Ψ0 (t 0 )|Ŵ (t 0 )Û (t 0 , t )r̂ Û (t , t 00 )Ŵ (t 00 )|Ψ0 (t 00 )⟩.
ti
(3.383)
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
100
The first term vanishes for a spherically symmetric binding potential. The second and
third term are complex conjugates of each other and describe ionization (Ŵ ), propagation (Û ), and recombination (r̂ ) (i.e., the emission of harmonic radiation) in different
time-ordering. The last term involves one additional interaction with the laser field. We
will neglect it here without, however, omitting to point out that it has been discussed in
the literature (see “Further reading” below). Replacing—as in the SFA for ionization—
the full time evolution operator Û by Û (V) ,
Z
t
(L)
dt 0 ⟨Ψ0 (t 0 )|Ŵ (t 0 )Û (V) (t 0 , t )r̂ |Ψ0 (t )⟩ + c.c.,
d (t ) = −i
(3.384)
ti
we shall recover the Lewenstein-result. In length gauge [Ŵ (t ) = E (t ) · r̂ ] we obtain
(suppressing the +c.c.)
Z
Z
t
(L)
0
d3 p ⟨Ψ0 (t 0 )|E (t 0 ) · r̂ |p + A(t 0 )⟩⟨p + A(t )|r̂ |Ψ0 (t )⟩e−iS p (t ,t ) (3.385)
0
d (t ) = −i dt
ti
0
where we used (3.305) and Û (V) (t 0 , t )|p + A(t )⟩ = e−iS p (t ,t ) |p + A(t 0 )⟩. With
Z
S p,E0 (t , t ) =
t
0
dt
t0
00
1
2
00
2
[p + A(t )] − E0
(3.386)
we can write (3.385) as
Z
Z
t
(L)
d (t ) = −i dt
0
d3 p ⟨Ψ0 |E (t 0 ) · r̂ |p + A(t 0 )⟩⟨p + A(t )|r̂ |Ψ0 ⟩e−iS p,E0 (t ,t ) .
0
(3.387)
ti
Introducing the dipole matrix element
µ(p) = ⟨p|r̂ |Ψ0 ⟩ =
Z
1
d3 r e−ip·r r Ψ0 (r )
3/2
(2π)
(3.388)
we have
Z
t
(L)
d (t ) = −i dt
Z
ti
t
= i dt
Z
0
0
d3 p e−iS p,E0 (t ,t ) µ[p + A(t )]E (t 0 ) · µ∗ [p + A(t 0 )] + c.c. (3.389)
Z
0
0
d3 p e−iS p,E0 (t ,t ) µ∗ [p + A(t )]E ∗ (t 0 ) · µ[p + A(t 0 )] + c.c.
(3.390)
ti
The integration over momentum can be performed using the saddle-point approximation again:
!
∇ p S p,E0 (t , t 0 ) = 0.
(3.391)
3.6. STRONG FIELD APPROXIMATION
101
For a linearly polarized laser pulse with a slowly varying envelope we have
E (t ) = Êe z cos ωt ,
so that
Z
t
α(t ) − α(t ) =
t
∂ S p,E0 (t , t 0 )
∂ pz
Ê
dt 00 A(t 00 ) =
0
and
A(t ) = −
ω2
0
= p z (t − t 0 ) +
Ê
ω
Ê
ω
e z sin ωt ,
(3.392)
e z (cos ωt − cos ωt 0 )
!
2
(cos ωt − cos ωt 0 ) = 0
(3.393)
(3.394)
Ê[cos ωt − cos ω(t − τ)]
,
τ = t − t 0.
(3.395)
ω2 τ
The transverse stationary momentum vanishes, p x,s = py,s = 0. Plugging ps into (3.386)
and integrating yield the stationary action
⇒
p z,s (t , τ) = −
Ss (t , τ) = (Up − E0 )τ − 2Up
1 − cos ωτ
ω2τ
with
C (τ) = sin ωτ −
− Up
4
C (τ)
ω
sin2
cos[(2t − τ)ω]
(3.396)
ωτ
.
(3.397)
ωτ
2
In the original Lewenstein-paper it is shown that the cut-off law 3.17Up + |E0 | can be
derived from the function C (τ). Setting ti = 0, the final result for the SFA dipole after
saddle-point integration reads
Z
t
(L)
d (t ) = i
dτ
0
2π
iτ
3/2
µ∗z [ p z,s (t , τ) + A(t )]
×µ z [ p z,s (t , τ) + A(t − τ)]Ê cos ω(t − τ)e
(3.398)
−iSs (t ,τ)
+ c.c.
from which, via Fourier-transformation, the harmonic spectra εrad, ω [Eq. (3.378)] can
be calculated. Figure 3.24 shows an example for a harmonic spectrum calculated using
the Lewenstein model.
Harmonic generation selection rules
As the name “harmonics” suggests, the emission of laser-driven targets mainly occurs at
multiples of the fundamental, incoming laser frequency ω1 ,
ω = nω1
(3.399)
102
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
Figure 3.24: Harmonic spectrum obtained using the SFA for H(1s), 2×1014 Wcm−2 , and photon
energy 1.17 eV. The time integral in (3.398) was calculated either directly (labelled “exact”) or
applying the saddle-point approximation. The expected cut-off at harmonic order n = 72.3 is
confirmed. [From D.B. MILOŠEVI Ć and W. BECKER, Role of long quantum orbits in high-order
harmonic generation, Phys. Rev. A 66, 063417 (2002).]
3.6. STRONG FIELD APPROXIMATION
103
with n the harmonic order. In the case of atoms in a linearly polarized laser field, for
instance, 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 classical
simulations also show harmonic generation and not just continuous spectra. The selection rules governing which harmonic orders n are allowed and which are forbidden are
determined by the symmetry of the combined system target + laser field, i.e., the symmetry of the field dressed target, also called dynamical symmetry. We shall now employ
the Floquet theory introduced in Sec. 3.5.1 to derive the selection rules for harmonic
emission for a few exemplary systems.
Let Ĥ (t ) be the Hamiltonian of an electron in a linearly polarized monochromatic
laser field Ê cos(ω1 t )e z of amplitude Ê and an ionic potential V̂ (r̂ ),
Ĥ (t ) =
p̂ 2
2
+ V̂ (r̂ ) + Ê ẑ cos ω1 t .
(3.400)
The Schrödinger equation reads i∂ t |Ψ(t )⟩ = Ĥ (t )|Ψ(t )⟩, and since the Hamiltonian is
periodic in time,
Ĥ (t + 2π/ω1 ) = Ĥ (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
Hˆ(t )|Φ(t )⟩ = ε|Φ(t )⟩,
Hˆ(t ) = Ĥ (t ) − i∂ t
(3.403)
which looks like a stationary Schrödinger equation in an extended Hilbert space with
the time as an additional dimension and with the scalar product
0
⟨⟨Φ|Φ ⟩⟩ :=
ω1
2π/ω1
Z
dt ⟨Φ(t )|Φ0 (t )⟩.
2π
(3.404)
0
Hereafter, we refer to Hˆ 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 generation
selection rules we assume an infinitely long laser pulse. It is then reasonable to assume
that 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 )|ẑ|Ψ(t )⟩ = −⟨Φ(t )|ẑ|Φ(t )⟩.
(3.405)
104
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
We define the dipole strength of the harmonic n as
2
2π/ω
1
2
Z
2
|d (n)| = dt exp(−inω1 t ) d (t ) = ⟨⟨Φ| exp(−inω1 t ) ẑ|Φ⟩⟩
0
(3.406)
which is proportional to the absolute square of the Fourier-transformed dipole. The
squared extended Hilbert space matrix element on the right-hand side of (3.406) may
be also interpreted as the probability for a transition from a Floquet state to itself, generated by the operator exp(−inω1 t ) ẑ, accompanied by the emission of radiation of
frequency nω1 .
The Floquet-Hamiltonian Hˆ(t ) is invariant under space inversion plus a translation
in time by π/ω1 ,
P̂inv = (r → −r , t → t + π/ω1 )
(3.407)
−1
so that P̂inv |Φ⟩ = σ|Φ⟩ and σ is a phase, i.e, |σ|2 = 1. Inserting the unity P̂inv
P̂inv in the
matrix element in (3.406) twice yields
−1
−1
⟨⟨Φ| exp(−inω1 t ) ẑ|Φ⟩⟩ = ⟨⟨Φ|P̂inv
P̂inv exp(−inω1 t ) ẑ P̂inv
P̂ |Φ⟩⟩
| inv{z }
| {z }
⟨⟨Φ|σ ∗
(3.408)
σ|Φ⟩⟩
−1
= ⟨⟨Φ|P̂inv exp(−inω1 t ) ẑ P̂inv
|Φ⟩⟩ = − exp(−inπ) ⟨⟨Φ| exp(−inω1 t ) ẑ|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 pulses
impinging on spherically symmetric systems such as atoms.
In the case of monochromatic circularly polarized laser light (with the electric field
vector lying in the xy-plane) the Floquet-Hamiltonian may be written (using the cylindrical coordinates ρ and ϕ) as
Ê
Hˆ(t ) = Ĥkin + V̂ (r̂ ) + p ρ cos(ϕ − ω1 t ) − i∂ t .
2
(3.411)
This expression is invariant under the continuous symmetry operation
P̂rot = (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(−inω1 t ) ρ exp(∓iϕ)|Φ⟩⟩ = exp(inθ ∓ iθ) ⟨⟨Φ| exp(−inω1 t ) ρ exp(∓iϕ)|Φ⟩⟩
(3.413)
3.6. STRONG FIELD APPROXIMATION
105
where ρ exp(∓iϕ) is the dipole operator for circularly polarized light (with the same
helicity (−) and the opposite helicity (+) as the incident pulse, respectively). Equation (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 frequencies ω1 and mω1 , respectively, the interaction Hamiltonian reads
Ê1
Ê2
Ŵ (t ) = p ρ cos(ϕ − ω1 t ) + p ρ cos(ϕ + mω1 t ).
2
2
(3.415)
Ê1 and Ê2 are the electric field amplitudes of the first and second laser, respectively. The
symmetry operation under which the Floquet-Hamiltonian is invariant now reads
‚
Œ
2π
2π
(m+1)
P̂rot
= ϕ→ϕ+
, t→t−
.
(3.416)
m +1
ω1 (m + 1)
In the same manner as in the two previous examples one arrives at the condition
n ∓1
1 = exp i2π
.
(3.417)
m +1
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 the
incident laser, whereas those with n = k(m + 1) − 1 are oppositely polarized. With
increasing m more and more low order harmonics are suppressed. This might be a
promising 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 rotational symmetry axis parallel to the laser propagation direction. An example for such a
target is the benzene molecule with M = 6. The Floquet-Hamiltonian in this case may
be written as
Ê
Hˆ(t ) = Ĥkin + V̂ (ρ, ϕ, z) + p ρ cos(ϕ − ω1 t ) − i∂ t .
2
(3.419)
Owing to the discrete rotational symmetry CM of V̂ (ρ, ϕ, z) the symmetry operation
of interest now is
‚
Œ
2π
2π
(M )
P̂rot = ϕ → ϕ +
, t→t−
,
(3.420)
M
ω1 M
CHAPTER 3. ATOMS IN EXTERNAL FIELDS
106
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 the
purposes here because the electron-electron interaction term is invariant under the operation (3.420) anyway.
For deducing these selection rules one assumes that the incident laser pulse is infinitely long. This is required for (3.401) to be true. In finite laser pulses the simple
selection rules above may be violated and one has to consider not only a single Floquet
states 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, Theory
of high-harmonic generation by low frequency laser fields, Phys. Rev. A 49, 2117
(1994). Perhaps the clearest account of HOHG [including a discussion of the
fourth 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 polarization properties of the harmonics, Phys. Rev. A. 56, 645 (1997). A review is given
P. SALIÈRES, A. L’HUILLIER, P. ANTOINE, and M. LEWENSTEIN, Adv. At. Mol.
Opt. Phys. 4, 83 (1999). Concerning the proper quantum mechanical calculation
of the harmonic emission of an isolated atom see, for instance, B. SUNDARAM
and P.W. MILONNI, Phys. Rev. Lett. 41, 6571 (1990); J.H. EBERLY and M.V. FEDOROV , Phys. Rev. A 45, 4706 (1992); P.L. K NIGHT and P.W. M ILONNI , 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 incoming laser pulse itself is presented in M. DRESCHER et al., Time-resolved atomic
inner-shell spectroscopy, Nature 419, 803 (2002) and R. KIENBERGER et al., Steering attosecond electron wave packets with light, Science 297, 1144 (2002). For the realtion 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).