Relativistic light-matter interaction
Tor Kjellsson
Abstract:
In this licentiate thesis light-matter interaction between hydrogen and superintense attosecond
pulses is studied. The specific aim here is to identify for what intensities the non-relativistic
calculations, given by solving the time dependent Schrödinger equation, no longer are valid.
In order to do this the time dependent Dirac equation has been numerically solved for
interaction beyond the so called dipole approximation, where spatial dependence of the pulse
is neglected.
The spatial part of the pulse is taken into account by a power series expansion truncated at a
certain order. It is shown that the relativistic description demands more terms in this
expansion compared to the non-relativistic description. As spatial dependence is
computationally heavy to take into account, several optimizations have been made.
As relativistic effects are expected when the classical quiver velocity of the electron reaches a
substantial fraction of the speed of light, this thesis considers cases up to v≈0.23 c
Akademisk avhandling för avläggande av licentiatexamen vid Stockholms universitet, Fysikum
Licentiatseminariet äger rum 28 september kl.10 i sal FB55, Fysikum, Albanova universitetscentrum,
Roslagstullsbacken 21, Stockholm.
1
Relativistic light-matter interaction
Tor Kjellsson
Akademisk avhandling for avläggande av licentiatexamen i teoretisk fysik
vid Stockholms Universitet, september 2015
Abstract
In this licentiate thesis light-matter interaction between hydrogen and superintense attosecond
pulses is studied. The specific aim here is to identify for what intensities the non-relativistic
calculations, given by solving the time dependent Schrödinger equation, no longer are valid. In
order to do this the time dependent Dirac equation has been numerically solved for interaction
beyond the so called dipole approximation, where spatial dependence of the pulse is neglected.
The spatial part of the pulse is taken into account by a power series expansion truncated at a
certain order. It is shown that the relativistic description demands more terms in this expansion
compared to the non-relativistic description. As spatial dependence is computationally heavy to
take into account, several optimizations have been made.
As relativistic effects are expected when the classical quiver velocity of the electron, vquiv ,
reaches a substantial fraction of the speed of light, this thesis considers cases up to vquiv ≈ 0.23c.
©
Tor Kjellsson, Stockholm 2015
List of Papers
PAPER I: Relativistic ionization dynamics for a hydrogen atom exposed to super-intense
XUV laser pulses
Tor Kjellsson, Sølve Selstø, Eva Lindroth
In manuscript
Contents
Abstract
ii
List of Papers
iii
Abbreviations
vii
List of Figures
ix
List of Tables
xi
1
Introduction
13
2
The hydrogen atom
2.1 Non-relativistic spectrum . . . . . . . . . . . . .
2.1.1 Time independent Schrödinger equation
2.1.2 Numerical implementation . . . . . . . .
2.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Relativistic spectrum . . . . . . . . . . . . . . . .
2.3.1 Numerical spectrum and spurious states
3
4
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15
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29
29
30
31
33
34
36
37
38
40
40
Dynamics
4.1 Time evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Expansion in field dressed eigenstates . . . . . . . . . . . . . . . . . . . . . . . .
43
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Interaction with an external field
3.1 Maxwell’s equations . . . . . . . . . . . . . . . . . . .
3.2 Interaction in general . . . . . . . . . . . . . . . . . . .
3.3 Choosing a pulse . . . . . . . . . . . . . . . . . . . . .
3.4 Spatial dependence of the field . . . . . . . . . . . . .
3.5 Spherical tensor operators and Wigner-Eckart theorem
3.6 Non-relativistic matrix elements . . . . . . . . . . . .
3.6.1 Dipole approximation . . . . . . . . . . . . . .
3.6.2 Beyond the dipole approximation . . . . . . .
3.7 Relativistic matrix elements . . . . . . . . . . . . . . .
3.7.1 Coupling of tensor operators . . . . . . . . . .
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4.3
4.4
5
6
4.2.1 Negative energy states . . . . . . .
Limiting the box size with complex scaling
Krylov subspace method . . . . . . . . . .
4.4.1 Optimization . . . . . . . . . . . . .
Results
5.1 Model space and pulse parameters . .
5.2 Dipole interaction . . . . . . . . . . . .
5.3 Beyond dipole interaction . . . . . . .
5.4 TDDE beyond dipole approximation .
5.4.1 Order of interaction . . . . . .
5.4.2 Negative energy states to HD0
5.4.3 Towards relativistic effects . .
Summary and Outlook
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44
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48
48
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53
53
54
57
58
58
60
61
63
Appendix
lxv
References
lxix
Abbreviations
TDSE
Time dependent Schrödinger equation
TDDE
Time dependent Dirac equation
BYD
Beyond dipole approximation
List of Figures
Energy spectrum for κ = −1. In this case, there are 183 values in total, one
for each basis element in the model space. The model spectrum splits into two
subsets and towards the extreme values the discrete nature of the model spectrum
is seen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2
The first few eigenvalues of the positive energy spectrum for κ = −1. . . . . . . .
25
2.3
Probability distribution for the two radial components of the ground state. The
Q-component has been multiplied by a factor of 100. . . . . . . . . . . . . . . .
26
Comparison of mathematical model (blue) with experimentally produced pulse
(red). The bottom figure is adapted from [1]. . . . . . . . . . . . . . . . . . . . .
32
Radial probability of an electron initially in the ground state. The pulse it interacts with is a sine-square pulse with ω = 3.5 a.u and E0 = 10 a.u. The times are
given in percentage of the pulse length with zero field strength after 100%. . . .
45
Radial probability of an electron initially in the ground state. Here E0 = 60 a.u.
while the other pulse parameters are the same as in Fig. 4.1. . . . . . . . . . . . .
46
Radial probability of an electron initially in the ground state. The field strength
here is E0 = 10 a.u. In the left figure uniform scaling has been employed while the
right figure presents dynamics with exterior complex scaling, both with θ = 10
degrees. The black dashed line indicates where the rotation starts. . . . . . . . .
47
Relativistiv couplings matrices. Each coloured block denotes a coupling of the
corrsponding operator between two states. In general, the operator contains a lof
of zero entries and as such is said to be sparse. . . . . . . . . . . . . . . . . . . .
49
A comparison between the full BYD3 angular couplings with the reduced matrix
elements not duplicated. The left figure can be reproduced from the right by just
multiplication of the correct angular factor. . . . . . . . . . . . . . . . . . . . . .
50
Probability to find the electron in the ground state as a function of time. The
pulse here has a total time of T = 30π a.u, E0 = 40.0 a.u and ω = 3.5 a.u. The
dots are the results from dynamics with full coupling matrices while the red solid
line is an optimized Krylov implementation. . . . . . . . . . . . . . . . . . . . . .
51
2.1
3.1
4.1
4.2
4.3
4.4
4.5
4.6
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
6.1
Ionization probability for hydrogen as a function of the peak electric field strength.
The vertical line denotes the value of E0 for which vq ≈ 0.1c. Note the convergence with increasing lmax towards the black solid line which is globally converged to 10−5 (see figure below). . . . . . . . . . . . . . . . . . . . . . . . . . . .
Global convergence test of the dipole approximation. The difference between
lmax = 40 and lmax = 15,20,30 are shown. The residual of lmax = 30 is of the order
10−5 which is in the current context will be accepted as converged. Note the
second vertical line here indicating another 10%-increment of speed of light. . .
Comparison of TDSE and TDDE calculation (lmax = 40) within the dipole approximation. At vq ≈ 0.15c, the non-relativistic Pion starts to increase compared
to the relativistic Pion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pion predicted by TDSE within (dashed line) and beyond (solid line) dipole approximation. The acronym BYD1 stands for first order beyond dipole approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison of TDDE and TDSE in the non-relativistic regime. The numbers
trailing the acronym "BYD" are the numerical value of N in eq. 5.2. For all
cases lmax = 15 has been used which in this region is enough for convergence. . .
The radial probability distributions of the electron initially in the ground state.
The percentage refers to elapsed interaction time. While BYD2 and BYD3 are
overlapping after the pulse the wavefunction following the BYD1 Hamiltonian
is visibly different. The peak electric field strength is here E0 = 40.0 a.u. . . . . .
A comparison of Pion computed with and without the time independent negative
energy states in the basis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pion for TDDE BYD3. For E0 > 80 a.u. lmax > 15 is requred while for E0 > 100
a.u. lmax > 20 is required for convergence. . . . . . . . . . . . . . . . . . . . . . .
55
55
56
57
58
59
60
61
B-splines of orders k = 2,3,4,7 on the sequence {0,1,2,3,4}. Note that order
k implies piecewise polynomials of order k − 1, clearly visible k = 1,2. At each
value r the B-splines sum up to unity, reflecting their completeness. . . . . . . . lxvi
List of Tables
2.1
2.2
4.1
5.1
6.1
The set of quantum numbers (omitting m j ) that are related to each other for the
relativistic eigenfunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The four lowest eigenvalues belonging to the positive energy spectrum for the
state lP = 1, j = 1/2. When using k1 = k2 the lowest eigenvalue in the positive set
is identical to the ground state for which lP = 0. . . . . . . . . . . . . . . . . . . .
27
Comparing figures for full and Krylov implementation. The model space here
consists of 2 × 20 B-splines per angular symmetry, with 128 such included.
(lmax = 7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Grid parameters for model box. The same equidistant knot sequence is used for
TDSE and TDDE computation. Note that the relativistic computations require
twice as many radial states due to having two components. Since the angular
basis is twice as large due to spin, the relativistic basis is four times larger than
the non-relativistic basis for the same box and lmax . . . . . . . . . . . . . . . . . .
54
Conversion table for physical quantities. α is the fine structure constant, as it is
dimensionless it has the same value in atomic units as in SI-units. . . . . . . . .
lxv
26
1. Introduction
Prediction is very difficult, especially about the future - Niels Bohr
There seem to be rules by which nature abides when the current state of «everything»
evolves. The scientific community is not only trying to find these rules but also comprehend
them, ultimately to a level where they perhaps can be applied to benefit society. The past century
presented a series of triumphs of this kind, with applications for instance of special and general
relativity to calibrate GPS-systems and the use of quantum mechanics for medical imaging.
One widely used method to obtain information about a system is to study its interaction
with light. It may be applied on the very largest of systems such as stars, galaxies and even the
universe itself, as well as on the tiny systems of microcosm: the world of molecular and atomic
systems.
On the microscopic scales the rules are given by quantum mechanics. In this theory a system
typically evolves on the time scale Tsc given by:
e−i
∆E⋅t
̵
h
→ Tsc ∼
h̵
∆E
(1.1)
where ∆E = E f in − Ein is the energy difference between an initial and final state. For an atomic
system ∆E ≈ eV giving a time scale of:
Tatom ∼ 10−16 s.
(1.2)
Recent advances in laser technology have now progressed the field to a point where subfemtosecond laser pulses, also called attosecond pulses can be generated. These have a time resolution
on the scale of 10−18 s, thus enabling the study of electronic motion in atoms on its natural time
scale [2]. With this experimental opportunity, theoretical predictions for dynamics on this scale
may be tested.
The development has not just made pulses short enough to monitor dynamics but also
strong enough to influence it. While the intensity of the Z = 1 Coulomb field at one bohr radius is Ic ∼ 1016W /cm2 , pulses with peak intensities as high as I ∼ 1022W /cm2 have been
reported [3]. Although pulses with this intensity have not yet been used in experiments, theoretical efforts have been made to study light-matter interaction in the relativistic regime where
Irel > 1018W /cm2 [4] and even beyond that [5]. However, due to the extremely challenging nature of phenomena occurying in these regimes most studies are carried out in simplified models.
In this thesis light-matter interaction is studied within the fully relativistic description given by
the time dependent Dirac equation (TDDE). Although there exist some work in this field [6–9]
13
there is to our knowledge no work that, in a full-dimensional calculation, incorporates more than
the lowest order relativistic interaction for a realistic pulse type and problem size. In order to do
this here, the focus is put on the interaction itself by studying one electron systems interacting
with external electromagnetic fields.
14
2. The hydrogen atom
The purpose of computing is insight, not numbers - Richard Hamming
One electron systems are of fundamental importance since they are the only atomic system for
which the electronic structure can be solved exactly. In quantum mechanics a system is characterized by its state ∣Ψ(t)⟩ and in coordinate space, information about the system is available
in the wave function Ψ(r,t) = ⟨r∣Ψ(t)⟩. The evolution of the wave function is governed by the
equation of motion:
ih̵
∂ Ψ(r,t)
= Ĥ(t)Ψ(r,t)
∂t
(2.1)
where the Hamiltonian Ĥ(t) contains all operators associated with energy within the chosen
framework. In subsequent chapters eq. 2.1 is solved for a non-relativistic and relativistic Hamiltonian to compare when the relativistic effects starts influencing the dynamics.
At any point in time the set of eigenfunctions {Φ(r,t)} of Ĥ(t) span the Hilbert space of Ψ(r,t)
and can thus be used as a basis to work in. In this work the time dependent wave function Ψ(r,t)
will be spectrally decomposed in the set of eigenfunctions {Φ(r)} of the Hamiltonian prior to
any interaction: Ĥ(t = 0) = Ĥ0 .
The next sections cover the basics of finding this basis and how to represent it in a practical
calculation. Once the basis is computed eq. 2.1 can be addressed for an external field.
2.1
Non-relativistic spectrum
For hydrogen, assuming a non-relativistic description with a point-like nucleus of infinite mass
the Hamiltonian is [10];
p⋅p
e2
Ĥ0 =
−
(2.2)
2m 4πε0 r
where p is the canonical momentum, e the electric charge, m the electron mass, ε0 the permittivity of free space and r the distance to the nucleus. To see that this Hamiltonian consists of terms
associated with energy, note that the last term in eq 2.2 is the Coulomb potential while the first
term represents the classical kinetic energy:
Ekin =
mv2 p2
=
2
2m
(2.3)
15
of a particle with mass m and velocity v.
Inserting eq. 2.2 into 2.1 gives:
ih̵
∂ Ψ(r,t)
p2
e2
=[
−
]Ψ(r,t).
∂t
2m 4πε0 r
(2.4)
which is the time dependent Schrödinger equation without an external field. The next section covers how to solve this equation to obtain the spectral basis of the field-free Schrödinger
Hamiltonian.
2.1.1
Time independent Schrödinger equation
The method to find the spectrum {En } and spectral basis {Φn } of the Schrödinger Hamiltonian
is covered in most elementary textbooks and hence only the key points from [11] are listed here:
First assume that a separation of variables with respect to spatial coordinates and time parameter
is possible:
Ψ(r,t) = Φ(r)T (t).
(2.5)
Inserting this into eq. 2.4 gives the time independent Schrödinger equation:
p2
e2
−
]Φ(r)
(2.6)
2m 4πε0 r
̵∇ and assumes a separable solution of the form:
To find Φ(r) and E one now substitutes p → −h∇
EΦ(r) = [
Φ(r) =
P(r)
Y (θ ,φ )
r
(2.7)
which after some basic manipulations yields the two differential equations
−
h̵2 d 2 P(r)
h̵2 l(l + 1)
e2
+
[
−
]P(r) = EP(r)
2m dr2
2m r2
4πε0 r
1
∂
∂Y (θ ,φ )
1
∂ 2Y (θ ,φ )
(sin(θ )
)+ 2
= −l(l + 1)Y (θ ,φ ).
sin(θ ) ∂ θ
∂θ
sin (θ ) ∂ φ 2
(2.8)
(2.9)
The angular equation 2.9 has the known analytical solutions:
2l + 1 (l − ∣m∣)!
Yml (θ ,φ ) = b[
]
4π (l + ∣m∣)!
1/2
eimφ Pml (cos(θ ))
(2.10)
called spherical harmonics. Here b = (−1)m if m ≥ 0 and b = 1 else, l is a non-negative integer
called orbital angular momentum and m is an integer ranging from −l to l.
16
Solving the radial equation 2.8 provides both the allowed eigenvalues and radial eigenfunctions.
The eigenvalues can be divided into two sets, one discrete set with only negative values E < 0
and one continuous set of positives values E ≥ 0.
For E < 0, corresponding to bound states, a discrete set of solutions exist:
⎡
2⎤
⎢ m
⎥ 1
e2
⎢
En = − ⎢ ̵ 2 (
) ⎥⎥ 2 , n = 1,2,3,...
⎢ 2h 4πε0 ⎥ n
⎣
⎦
2 3 (n − l − 1)!
]
Pnl (r) = [( )
na 2n((n + 1)!)3
1/2
e−r/na (
(2.11)
2r l 2l+1
) Ln−l−1 (2r/na)
na
where l is the previously encountered orbital angular momentum, a =
in eq. 2.12 is an associated Laguerre polynomial.
4πε0 h̵2
me2
(2.12)
and the last term
In the case of positive values E > 0, corresponding to unbound states, eq. 2.8 admits a continuous
set of values. For a derivation of the unbound eigenfunctions the reader is referred to section
1.3.2 in [12]
Although eq. 2.8 has analytical solutions the standard procedure in practice is to use numerical
methods to represent the spectral basis. Unlike the analytical basis, which is infinite, the numerical basis is finite and complete (within a model space). If the model space approximates reality
well enough for the scenario of interest this approach provides a good description of the relevant
physics. To ensure that this is the case, convergence with respect to the model space size must
be checked. Since any sophisticated problem quickly grows out of hands it is crucial that an
efficient numerical approach is employed to keep this size as small as possible.
2.1.2
Numerical implementation
The key point in minimizing the model space without compromising the physics is to correctly
identify what may be neglected. For instance, even though the formal boundary conditions on
the wave function are:
Ψ(∣r∣ = 0) = 0
and
Ψ(∣r∣ → ∞) → 0
(2.13)
the interaction of an initially localized electron with a finite pulse should not extend to infinity.
In other words, there should exist a value r = Rmax such that:
Ψ(∣r∣ → Rmax ) ≈ 0
(2.14)
to a desired accuracy. Employing this boundary condition is referred to as making calculations
inside a box.
17
A similar argument regarding the angular part of the model space can be made. If the electron is
initially prepared in a specified state, say the ground state, there should be a limit to how many
angular states the wave function may transit to with a non-negligible probability. That is, the
angular resolution of the wave function should be sufficiently described by a linear combination
of a limited number of spherical harmonics. Convergence checks with respect to the number of
angular states is discussed at a later point in the chapter on dynamics.
Recall that the time dependent wave function will be spectrally decomposed in eigenfunctions
of Ĥ0 :
Pnl (r) l
Ψ(r,θ ,φ ,t) = ∑ cn,l,m (t)
Ym (θ ,φ )
(2.15)
r
n,l,m
with a finite number of spherical harmonics, given by eq 2.10, covering the angular distribution. What remains to complete the model space is to find the numerical radial eigenfunctions
Pnl (r)
inside the box.
r
There exist different ways to numerically solve eq. 2.8. One efficient method that is popular in
the field of atomic and molecular physics [13] is to expand the eigenfunction in B-splines [14] :
Pnl (r) = ∑ ci Bki (r).
(2.16)
i
A major advantage with this method is the flexibility it provides in distributing the resources
freely to specifically tailor the basis to the problem at hand. The reader interested in details
about B-splines is referred to appendix.
To find Pnl (r), expansion 2.16 is inserted into. 2.8. Omitting constants and radial arguments for
brevity one obtains:
[−
d 2 l(l + 1) 1
+
− ] ∑ ci Bki (r) = E ∑ ci Bki (r)
2
2
dr
r
r i
i
(2.17)
Multiplication by Bkj and integration give:
k
∑ ci ∫ [−B j
i
d 2 Bki
l(l + 1) 1 k k
+(
− )B j Bi ]dr = E ∑ ci ∫ Bkj Bki dr
2
dr
r2
r
i
(2.18)
Denoting the integral on the left as H ji and right as B ji this equation is now on the form:
∑ ci H ji = E ∑ ci B ji
i
(2.19)
i
Note that each side of this equation can be expressed as the dot product of two vectors:
⃗ Tj ⋅ c⃗ = E B⃗Tj ⋅ c⃗
H
(2.20)
⃗ Tj , B⃗Tj are row vectors and c⃗ is a column vector. Since there is one equation of this sort
where H
for each index j the collection of these can be summarized in a matrix equation:
18
H ⋅ c⃗ = EB ⋅ c⃗
(2.21)
with matrix elements:
H ji = ∫ [−Bkj
d 2 Bki
l(l + 1) 1 k k
+(
− )B j Bi ]dr
2
dr
r2
r
(2.22)
B ji = ∫ Bkj Bki dr.
(2.23)
Since B-splines are polynomials the integrals can be computed exactly to machine accuracy
using Gauss-Legendre quadrature [15]. What remains now is to solve the (generalized) matrix eigenvalue problem 2.21 which is efficiently done with diagonlization routines from LAPACK [16]. Comparison of the numerical basis with analytical values is shown in subsequent
sections where the relativistic implementation is discussed.
2.2
Spin
Before discussing the relativistic Dirac Hamiltonian it is instructive to discuss spin. The famous
Stern-Gerlach experiment (covered in for instance [17]), showed that atoms with zero orbital
angular momentum (l = 0) carries a magnetic moment. This turns out to originate from the electronic spin which is an angular momentum not associated with spatial motion.
The spin magnetic moment is approximately [11]:
µ =−
eh̵
σ
2m
(2.24)
where σ = (σx ,σy ,σz ) is a vector containing the Pauli matrices. Taking this magnetic moment
into account the Hamiltonian for an electron in an external field A is1 :
ĤPauli =
(p + eA)2
e2
−µ ⋅B−
2m
4πε0 r
(2.25)
which is also known as the Pauli Hamiltonian. Since the Hamiltonian now contains spin matrices
the state must also have components expressed in the eigenvectors of these:
a
Ψ(r,s) = ψ(r) ⊗ ( ).
b
(2.26)
where the column vector has two components, as is known from introductory courses on quantum mechanics, because electron carries a spin of s = 12 with two projection values ms = ± 12 .
1
The next chapter recapitulates the basics of electromagnetic fields and their description through
potentials.
19
Apparently the non-relativistic Hamiltonian is missing operators that must be added "by hand"
in order to describe observations made in different scenarios. The next section gives a short
derivation of the Dirac Hamiltonian that inherently contains spin.
2.3
Relativistic spectrum
Recall that the Schrödringer equation (using p2 = −h̵2 ∇2 and without a potential for brevity):
ih̵
∂ Ψ(r,t)
h̵2
= − ∇2 Ψ(r,t)
∂t
2m
(2.27)
is a differential equation of first order in time but second order in the spatial coordinates. It is
thus not compatible with the theory of special relativity since it is not Lorentz covariant. (This
does of course also apply to the Pauli operator in the previous section). The following paragraphs, inspired by [18], are dedicated to the derivation of the Dirac equation.
In the theory of special relativity the energy of a free particle is given by:
Erel =
√
c2 p2 + m2 c4
(2.28)
which turned into operator form gives:
∂ Ψ(r,t) √ 2 ̵2 2
= −c h ∇ + m2 c4 Ψ(r,t).
∂t
(2.29)
Eq. 2.29 is called the square-root Klein Gordon equation. There are several reasons why this
equation is not an acceptable description of nature; it gives for instance purely scalar wave functions (which in the last section was shown to not be compatible with spin) and is non-local in
the sense that a local change immediately affects the wave function everywhere. However, even
though the equation itself carries serious flaws it is conceptually important since a slightly different route leads to the accepted Dirac equation.
For simplicity, consider first one spatial dimension. Dirac made the ansatz that the relativistic
energy eq. 2.28 could be written on a linear form:
β mc2
Ematrix = α cp +β
(2.30)
β are assumed to be square matrices with constant matrix elements. If so:
if α ,β
2
2
Ematrix
= Erel
I2
2
20
(2.31)
α cp +β
β mc2 ) = (c2 p2 + m2 c4 ) I 2
(α
(2.32)
α β mc3 p +β
β α mc3 p +β
β 2 m2 c4 = (c2 p2 + m2 c4 ) I 2 .
α 2 c2 p2 +α
(2.33)
where I 2 denotes the unit matrix for 2 × 2-matrices. Eq. 2.33 gives the conditions:
α 2 = β 2 = I2
(2.34)
βα.
α β = −β
(2.35)
Finally, to create a Hamiltonian based on eq. 2.30 the condition of hermiticity must be put on
β.
α ,β
It does not take long to realize that the Pauli matrices themselves satisfy all the above conditions,
meaning that the Dirac Hamiltonian in one dimension, say the x-dimension, could be written as:
mc2
cp
ĤD (x) = σx cp + σz mc2 = (
).
cp −mc2
(2.36)
in the basis where σz is diagonal.
Expanding the derivation to higher dimension requires some attention since one α -matrix per
spatial dimension will be needed. Thus, for three dimensional space one needs four matrices
fulfilling the conditions above but it is impossible to find four (2 × 2)-matrices of this kind. It
turns out that (4 × 4)-matrices are needed1 and one (of many) possibility is:
αi = (
02 σi
), i = x,y,z
σi 02
I
02
β =( 2
)
02 −I2
(2.37)
The three dimensional Dirac Hamiltonian can thus be cast on the form:
α ⋅p−
ĤD = cα
e2
+ mc2β
4πε0 r
(2.38)
with the matrix definitions in eq. 2.37. Note that the matrix β has the eigenvalues ±1, leading
to eigenvalues of ĤD that are unbounded from both above (as the non-relativistic case) and below. The states that are unbounded from below are closely connected to what is perceived as
positrons. The role of these negative energy states is a key feature in this thesis and will be
discussed in subsequent sections.
Since ĤD consists of (4 × 4)-matrices the eigenfunctions must have four components. These
have the form (section 3.8 of ref [20]):
ψn,κ, j,m (r) =
1 Pnκ (r)Xκ, j,m
(
)
r iQnκ (r)X−κ, j,m
where Pnκ (r),Qnκ (r) are purely radial functions while Xκ, j,m contains the spin-angular dependency.
1
The dimension must be even, see for instance p.120 of [19].
21
There are now three new quantum numbers;
1
j = l+s → j = l ± ,
2
m = − j,− j + 1,..., j
⎧
⎪
l,
j = l − 12
⎪
κ ≡⎨
1
⎪
⎪
⎩−(l + 1), j = l + 2
The spin-angular states are given by:
(2.39)
(2.40)
Xκ, j,m = ∑ ⟨l,ml ;s,ms ∣ j,m⟩Yml l (θ ,φ )χms
(2.41)
ms ,ml
where Yml l (θ ,φ ) is a spherical harmonic (encountered in section 2.3.1) and ⟨l,ml ;s,ms ∣ j,m⟩ is a
Clebsch-Gordan coefficient. The spinor is given in the basis where σz is diagonal and thus:
1
χ1/2 = ( )
0
0
χ-1/2 = ( )
1
(2.42)
The radial functions are coupled through the two differential equations [20] (section 3.8):
̵
− hc(
d κ
e2
− )Q = (E −
− mc2 )P
dr r
4πε0 r
2
̵ d + κ )P = (E − e + mc2 )Q.
hc(
dr r
4πε0 r
(2.43)
(2.44)
which must be solved to find the allowed eigenvalues E and eigenfunctions of ĤD . Just as in
section 2.3.2, a numerical method is for practical purposes desirable but this time caution is required. Unlike eq. 2.8, the two equations above carry a potential "trap" when numerically solved.
Already explained in 1981 by Drake and Goldman [21] incorrect states may appear when solving a discretized version of the Dirac equation. The reason for their appearance, explained in a
nice and concise way in ref [22], is attributed to variational collapse and is due to the discretization process itself. Fortunately there exist ways to avoid these incorrect states, also known as
spurious states, if proper measures are taken. In the discussion on the adopted method below an
example of a contaminated spectrum is shown.
Froese-Fischer and Zatsarinny have developed a stable B-spline approach for the Dirac equation [23]. It builds on two separate B-spline expansions:
P(r) = ∑ ai Bki 1 (r)
i
Q(r) = ∑ b j Bkj2 (r).
(2.45)
j
where k1 and k2 denote the order of the B-splines. Whereas the combination k1 = k2 gives spurious states they found that the combination k1 = k2 ± 1 is free of them. Due to the flexibility that
B-splines provide this method is employed in this work to produce the relativistic model space
22
basis.
Just as in section 2.3.2 each B-spline expansion 2.45 is inserted into eq. 2.43 and eq.2.44. Following similar steps as in the derivation of eq. 2.18 the following matrix equation is obtained:
V
K 12 ⃗
B
0 ⃗
( 1
)d = E ( 1
)d
K 21 V 2
0 B2
(2.46)
d⃗= (a1 ,a2 ,...,aN1 ,b1 ,b2 ,...,bNb )T
(2.47)
with
k
k
(B p )i j = ∫ Bi p B j p dr,
p = 1,2
(2.48)
(V1 )i j = a j ∫ (−
e2
+ mc2 )Bki 1 Bkj1 dr
4πε0 r
(2.49)
(V2 )i j = b j ∫ (−
e2
− mc2 )Bki 2 Bkj2 dr
4πε0 r
(2.50)
(K12 )i j = −c ⋅ b j ∫
Bki 2
(K21 )i j = c ⋅ a j ∫ Bki 1
dBkj1
dr
dBkj2
dr
κ
− Bki 2 Bkj1 dr
r
(2.51)
κ
+ Bki 1 Bkj2 dr
r
(2.52)
All matrix elements are evaluated with Gauss-Legendre quadrature. Eq. 2.46 is then solved using diagonlization routines from LAPACK [16] to obtain the coefficients in d⃗and eigenvalues E.
2.3.1
Numerical spectrum and spurious states
The analytical spectrum of the Dirac Hamiltonian is divided into two sets:
E ∈ (−∞,−mc2 + Eg ] ∪ [mc2 + Eg ,∞)
(2.53)
where Eg is the ground state energy. In order to compare it with non-relativistic spectrum:
Enrel ∈ [Egnrel ,∞)
(2.54)
it is useful to shift the relativistic energies by mc2 . In practice this is done by subtracting mc2 on
the main diagonal of left hand side of eq. 2.46 which in no way perturbs the solutions themselves.
The eigenvalues obtained by solving eq. 2.46 for a specific value of κ, say κ = −1 are show in
fig. 2.1:
23
Figure 2.1: Energy spectrum for κ = −1. In this case, there are 183 values in total, one for each basis
element in the model space. The model spectrum splits into two subsets and towards the extreme
values the discrete nature of the model spectrum is seen.
In Fig. 2.1 the blue lines represent the obtained eigenvalues from solving eq. 2.46 with κ = −1
shifted downwards by mc2 ≈ 18769 a.u. This value is given in atomic units, listed in appendix,
and for the rest of this thesis these units will be used if nothing else is stated. The subset containing positive elements will be referred to as the positive energy spectrum while the other subset
will be called negative energy spectrum.
In Fig. 2.2 the first few eigenvalues of the positive energy spectrum κ = −1 are shown.
24
Figure 2.2: The first few eigenvalues of the positive energy spectrum for κ = −1.
Turning now to the eigenfunctions, the relativistic ones:
ψn,κ, j,m j (r) =
1 Pnκ (r)Xκ, j,m j
(
)
r iQnκ (r)X−κ, j,m j
and non-relativistic ones:
1
φn,l,ml (r) = Pnl (r)Yml l (θ ,φ )
r
are seen to be described by slightly different sets of quantum numbers. The quantum numbers
of the ground states are:
∣nrel⟩ = ∣n = 1,l = 0,ml = 0⟩
∣rel⟩ = ∣n = 1,κ = −1, j = 12 ,m j = - 12 ⟩
(2.55)
(2.56)
where only the principal quantum number n has the same meaning. Using eq. 2.40 one can
transform the value of κ to a combination of l and j, so that at least the orbital angular momentum
can be compared. The problem is however that l is not the same for both radial components:
25
κ
-1
1
-2
2
-3
3
j
1/2
1/2
3/2
3/2
5/2
5/2
lP
0
1
1
2
2
3
lQ
1
0
2
1
3
2
Table 2.1: The set of quantum numbers (omitting m j ) that are related to each other for the relativistic
eigenfunctions.
The dilemma can be cleared by observing the relative size of the two components. In Fig 2.3 the
probability distributions of the two radial components of the ground state are shown.
Figure 2.3: Probability distribution for the two radial components of the ground state. The Qcomponent has been multiplied by a factor of 100.
For the positive energy spectrum ∣Pnκ ∣ >> ∣Qnκ ∣ and is for this reason often referred to as the
large component. It is therefore customary to speak about the orbital angular momentum lP in
table 2.1 when the positive energy states are concerned. The size relation is however reversed
for states in the other subset of the spectrum. This will be seen to play a major role in the chapter
covering relativistic dynamics.
By using the quantum numbers of the larger component for the positive energy spectrum it is
26
finally time to evaluate the numerical solutions. For comparison the computed eigenvalues are
compared to the analytically known formula (section 3.8 in [20]):
En j − mc2 =
1 + [c2
√
mc2
− mc2 .
−1
√
n − j − 1/2 + ( j + 1/2)2 − 1/c2 ]
(2.57)
The numerical eigenvalues are computed for the cases k1 = 7,k2 = 8 and k1 = 8,k2 = 8 in a box of
size Rmax = 200 a.u.
Exact
-0.1250020802
-0.0555562952
-0.0312503380
-0.0200001811
k1 = 7 & k2 = 8
-0.1250020802
-0.0555562952
-0.0312503380
-0.0200001811
k1 = 8 & k2 = 8
-0.5000066566
-0.1250020802
-0.0555562952
-0.0200001811
Table 2.2: The four lowest eigenvalues belonging to the positive energy spectrum for the state
lP = 1, j = 1/2. When using k1 = k2 the lowest eigenvalue in the positive set is identical to the ground
state for which lP = 0.
Marked in red is the incorrect energy. Even though lP = 1, this spurious energy is identical to the
ground state energy for lP = 0. This inclusion of an eigenvalue from another angular symmetry
appears for all values of κ > 0.
Before concluding the section on the relativistic model space the matter of boundary conditions
must be addressed. In the non-relativistic case it was assumed that Φ(0) = Φ(Rmax ) = 0. For
the relativistic eigenfunction containing two radial components there are several combinations
of boundary conditions one may apply for P(r) and Q(r). For a discussion on these, see for
instance [23; 24].
In this work the following boundary conditions are used:
P(0) = P(Rmax ) = 0
(2.58)
Q(0) = Q(Rmax ) = 0.
(2.59)
which have not been found to give problems with the relation of B-spline orders k1 = k2 − 1 that
has been exclusively used in the simulations to be presented.
With the method of computing the model space presented the next chapter addresses how to
incorporate interaction with an external electromagnetic field.
27
28
3. Interaction with an external field
I am no poet, but if you think for yourselves, as I proceed, the facts will form a poem in your
minds - Michael Faraday
The external electromagnetic field will be strong enough to be described by classical electromagnetism. Therefore, this chapter is initiated with a recapitulation of Maxwell’s equations and
then followed by the computation of light-matter interaction.
3.1
Maxwell’s equations
The Maxwell equations describe any phenoma in classical electromagnetism. In free space they
read [25]:
∇ ⋅E =
ρ
ε0
∇ × B = µ0 J +
1 ∂E
c2 ∂t
∇ ⋅B = 0
(3.1)
(3.2)
(3.3)
∂B
(3.4)
∂t
where B and E are the magnetic and electric field, ρ and J the total charge and current density,
ε0 and c the permittivity and speed of light in free space.
∇ ×E = −
The set of four equations may be summarized in two equations coupled by a vector potential A
and a scalar field Φ. From eq. 3.3 one obtains:
∇ ⋅B = 0 → B =∇ ×A
(3.5)
while eq. 3.5 inserted into eq. 3.4 gives:
∇ × [E +
∂A
∂A
] = 0 → E+
= −∇Φ
∂t
∂t
(3.6)
Similar substitutions and some more algebra give:
29
∇2Φ +
∂
ρ
∇ ⋅ A) = −
(∇
∂t
ε0
(3.7)
1 ∂ 2A
1 ∂Φ
∇ (∇
∇⋅A+ 2
−∇
) = −µ0 J.
(3.8)
2
2
c ∂t
c ∂t
There is a freedom in choosing the form of A and Φ as long as B and E, the objects of interest,
remain unchanged. From eq. 3.5 and eq. 3.6 the conditions are:
∇2 A −
∇χ
A → A +∇
(3.9)
∂
(3.10)
Φ → Φ− χ
∂t
for a scalar function χ. The freedom in scaling the fields A and Φ provides the means to transform them into a suitable form for a specific scenario. This type of transformation goes under
the name gauge transformation.
For the cases in this thesis there will be no sources present, i.e ρ = 0 and J = 0. In such a case
there exists a particularly useful gauge called the Coulomb gauge where ∇ ⋅ A = 0. With no
sources present the electric and magnetic fields in this gauge have the form:
∂A
∂t
∇
B = ×A
E=−
3.2
(3.11)
(3.12)
Interaction in general
The mathematical formulation of interaction is given by all operators1 in the Hamiltonian that
depend on the field A (omitting arguments (r,t) for brevity). Incorporating the external field is
done by substituting p → p + eA [17]:
Hnrel (t) =
(p + eA)2 1
p2 1
1
−
=
− + p ⋅ A + A2
2
r
2 r
2
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
(3.13)
HI−nrel (t)
1
1
α ⋅ (p + A) − +β
β c2 = cα
α ⋅ p − +β
β c2 + cα
α ⋅A .
Hrel (t) = cα
r
r
²
(3.14)
HI−rel (t)
note that eq. 3.13 assumes p ⋅ A = A ⋅ p which indeed is valid in Coulomb gauge. The interaction within the model space is then evaluated by computing the matrix with elements:
HI ab = ⟨Φa ∣ĤI (t)∣Φb ⟩
which may be done once the form of the pulse is known.
1
30
In what follows, if it is clear from the context that an object is an operator the hat is left out.
(3.15)
3.3
Choosing a pulse
A common mathematical model for a pulse is the sine-square pulse:
A(ωt − k ⋅ r) =
E0 2 π(ωt − k ⋅ r)
sin (
)sin(ωt − k ⋅ r + φ ) û
ω
ωT
(3.16)
where ω is the central frequency of the pulse, k the wave vector with k = ωc , φ a phase term, T
the total pulse length in time and û the direction of the vector field. In this work the following
choices are made:
- Electric field linearly polarized in one direction, say ẑ
- Propagation direction in one spatial dimension, say x̂.
These choices give the pulse:
A(ωt − kx) =
E0 2 π(ωt − kx)
sin (
)sin(ωt − kx + φ ) ẑ.
ω
ωT
(3.17)
Computing matrix elements with the complete form of eq. 3.17 is however too demanding. The
key, as pushed in the former chapter, is to identify what approximations may be applied to
capture its relevant features. For instance, the spatial term:
kx =
ω
x
c
(3.18)
can normally be regarded as small for a range of k in the vicinity of the atom. As a first approximation it may thus be the case that kx can be neglected, giving:
A(ωt) =
E0 2 πt
sin ( )sin(ωt + φ ) ẑ.
ω
T
(3.19)
Consider for instance an electron localized in the ground state, where 99% of the wave function
is localized in the first 5 a.u (c.f fig 2.3). A pulse with ω = 1 a.u (corresponding to an energy of
27.2 eV) has the wave number k ≈ 1/137, giving:
kx ≈ 0.036.
(3.20)
Interaction with a pulse that does not have any spatial dependence gives rise to the electric dipole
approximation which is a lowest order type of interaction that is particularly easy to work with.
Although kx ≈ 0.036 may seem too large to be neglected, results of simulations will show that
the dipole approximation holds for moderate field strengths E0 for even larger values of ω. In
order to deduce this one must of course have a way to take spatial dependence into account,
which is soon to be covered. First a small detour is made to discuss the reproducability of the
mathematical form assumed for the pulse.
31
Pulse in the lab
Before doing anything more a pressing question should be addressed - how well can the pulse
assumed be produced in the lab? In the figure below eq. 6 is plotted, for parameters given in the
figure, together with an experimentally produced pulse by ref [1]. The purpose of this figure is
to show that the mathematical model given in the previous section is not too much of an idealization.
Figure 3.1: Comparison of mathematical model (blue) with experimentally produced pulse (red).
The bottom figure is adapted from [1].
32
3.4
Spatial dependence of the field
Since only the z-component of the vector field is non-zero, from now on the ẑ will be implicit
for brevity.
A natural way to study the effect of the term kx is to perform an expansion of eq 3.17. Defining
η = ωt − kx, an expansion in (kx) is achived by expanding A(η) around the point η0 = ωt:
A(η) ≈ A(η0 ) + (η − η0 )
R
R
1
dA RRRR
d 2 A RRRR
RRR + (η − η0 )2
R + O((−kx)3 ).
dη RR
2
dη 2 RRRR
Rη0
Rη0
(3.21)
where each term with spatial dependence provides an interaction beyond the dipole approximation. Using:
η − η0 = −kx
(3.22)
R
dA RRRR
dt dA(t) 1 dA(t)
RRR =
=
dη RR
dη dt
ω dt
Rη0
(3.23)
where A(t) carries no spatial dependence and thus is differentiated with a total derivative, the
expansion can now be written in terms of (x,t) (omitting ω for brevity):
x dA(t) 1 x 2 d 2 A(t)
A(η) ≈ A(t) − ( )
+ ( )
+ O((−kx)3 ).
2
c
dt
2 c
dt
(3.24)
Another approach, that for instance ref [6] took, is to make a plane wave expansion of the field
(here on general form):
(3.25)
A(ωt − k ⋅ r) = ∑ an eibn (ωt−k⋅r)
n
followed by an expansion of the spatial part by each exponential in spherical harmonics (see
section 10.3 of ref [25]):
∞
λ
∗
eiak⋅r = 4π ∑ ∑ iλ jλ (an kr)(Yµλ (θ ′ ,φ ′ )) Yµλ (θ ,φ )
(3.26)
λ =0 µ=−λ
where (θ ′ ,φ ′ ) are the angular coordinates of the wave vector k and jλ (an kr) is a spherical
bessel function of the first kind. Even though the sum over λ is infinite, only a finite number
give non-zero contribution1 . Since the pulse in eq. 3.17 could be written with six exponentials
this approach admits full spatial dependence for a small number of terms.
Both approaches above give a way to evaluate effects of including spatial dependence in the
vector field. The second approach could in principle be seen as more correct in the sense that
all possible interactions between two states in the model space are computed. It is however not
1
To be explained in the next section.
33
practical1 and cannot be used to distinguish the importance of different types of higher order
interactions. The reason for this is that all possible interactions between each individual pair of
states are included.
This thesis is mainly based on the first method, in which a peculiar observation has been made:
TDDE does not seem to be stable for expansion 3.24 to first order in x.
Since there exist work published on first order corrections beyond the dipole approximation for
TDSE, i.e [26], this observation was at first believed to be an artifact of incorrect source code.
As it turns out, continuing the expansion leads to stability and agreement with the plane wave
method. It also agrees with predictions by TDSE in a non-relativistic regime. These observations give strong reasons to believe that this instability is not an artifact.
Now that the final forms of the vector field have been chosen the matrix elements can be computed. There are different ways to do this, for instance by using powerful tools from angular
momentum theory.
3.5
Spherical tensor operators and Wigner-Eckart theorem
When working in the spectral basis of an atomic system it is convenient to define a set of basis operators which have known properties when acting upon angular momentum states. By
expressing the interaction operator in this operator basis, complicated matrix elements may be
computed. The elements of this basis are called spherical tensor operators.
Let j = ( jx , jy , jz ) be a general angular momentum vector and define j± = jx ± i jy . Consider an
operator with components of the form tqk where q ∈ {−k,−k + 1,...,k}. If this operator satisfy:
[ jz ,tqk ] = qtqk
(3.27)
k
[ j± ,tqk ] = [k(k + 1) − q(q ± 1)]1/2 tq±1
(3.28)
it is called a spherical tensor operator of rank k [27]. For such an operator the following powerful
theorem holds for matrix elements involving angular momentum (of any kind):
l
k lb
⟨na la ma ∣tqk ∣nb lb mb ⟩ = (-1)la −ma ( a
)⟨na la ∣∣tk ∣∣nb lb ⟩
-ma q mb
(3.29)
which is known as the Wigner-Eckart theorem. Note that the states are here characterized by
their quantum numbers, which from now on is the default notation if not stated otherwise.
1
34
Ref [6] discusses an additional physical argument that enforces even more computational load.
The Wigner-Eckart theorem tells us that all information regarding the projection quantum numbers ma ,mb can be put into a factor called a 3j-symbol which is a scaled Clebsch-Gordan coefficient:
(
la k lb
) = (−1)la −k−mb (2lb + 1)−1/2 ⟨la ,-ma ;k,q∣lb ,-mb ⟩.
-ma q mb
(3.30)
The Clebsch-Gordan coefficients can be written on the form (see eq. 2.76 in ref [27]):
⟨la ,ma ;k,q∣lb ,mb ⟩ = δ (ma + mb + M) × N
(3.31)
where δ (ma +mb +M) = 0 unless ma +mb +M = 0 and N is a pure number that can be computed
by a series of factorials using all quantum numbers. Together with a last (not apparent from this
discussion) condition: ∣la − k∣ ≤ lb ≤ la + k called the triangular rule the following useful facts
can be deduced:
1) The conditions ma + mb + M = 0 and ∣la − k∣ ≤ lb ≤ la + k form part of the selection rules for
the operator.
2) If the matrix elements are computed for one pair (ma ,mb ) of states with fixed la ,lb then
all other matrix elements where only (ma ,mb ) changes are the same up to a factor.
3) Eq. 3.30 is trivially computed by basic arithmetic operations following a known formula.
To demonstrate the power of the properties listed the claim made in the previous section regarding the plane wave expansion 3.26 will now be proven.
First, the expansion is first written in a slightly easier way to read:
∞
λ
eik⋅r = ∑ ∑ fλ (kr) ⋅Yµλ (θ ,φ )
(3.32)
λ =0 µ=−λ
which is completely valid for a given direction k̂ . The spherical harmonic Yµλ (θ ,φ ) actually
fullfills the criteria to be a spherial tensor operator of rank λ , so the following matrix element
⟨na ,la ,ma ∣eik⋅r ∣nb ,lb ,mb ⟩
(3.33)
can be cast on the form:
∞ λ
l
λ lb
⟨na ,la ,ma ∣eik⋅r ∣nb ,lb ,mb ⟩ = ∑ ∑ (-1)la −ma ( a
)⟨na la ∣∣ fλ (kr)Yλ ∣∣nb lb ⟩
-m
µ
m
a
b
λ =0 µ=−λ
(3.34)
due to the Wigner-Eckart theorem. From the triangular condition imposed by the 3j-symbol it
follows that for a given pair of (la ,lb ) the only terms in the infinite sum over λ that give non-zero
35
contributions are ∣la − lb ∣ ≤ λ ≤ la + lb .
To evaluate the non-zero elements the other term must now be addressed. This is known as the
reduced matrix element and needs to be derived separately for every type of spherical tensor
operator (unlike the 3j-symbol that only depends on the quantum numbers). This however poses
no obstacle here since all operators considered in this thesis have known expressions for their
reduced matrix elements.
A final note: recall that the model space is factorized into a radial and (spin)-angular part. The
total number of states is thus given by:
Nstates = Nrad ⋅ Nang
(3.35)
where Nrad is the number of radial eigenfunctions obtained from the discretization process (as
described in the previous chapter) for every angular state. By understanding the angular properties of the operators, the minimally required Nang can be found which directly influences the
model space size.
3.6
Non-relativistic matrix elements
In this section the non-relativistic interaction matrix elements are derived for the dipole approximation and first order beyond that. Recall that A = (0,0,A(ωt − kx)), so the interaction
operator 3.13 is:
1
HI (t) = pz A(ωt − kx) + A(ωt − kx)2 .
2
In the dipole approximation this becomes (denoting ωt → t for brevity):
1
Hdip (t) = pz A(t) + A(t)2
2
(3.36)
while using the first two terms in eq. 3.24;
HBY D1 (t) = pz A(t) − pz
x dA(t) x
dA(t) 1
− A(t)
+ A(t)2
c dt
c
dt
2
(3.37)
where the x2 -term has been neglected. From now on the acronym BYDX, standing for beyond
dipole to order X, will be employed.
Before moving on to the evaluation, two statements must be made;
The A(t)2 -term appearing in both Hamiltonians is actually unnecessary.
The pz x-term appearing in HBY D1 should be omitted,
which will now be addressed.
36
The A(t)2 is unnecessary due to at least two reasons. One argument, perhaps not pointed out so
often, is that a purely time dependent operator is diagonal in a spatial basis. Hence it only shifts
eigenvalues - it does not change the eigensolutions.
Another (often employed) argument is that A(t)2 can be removed by performing a gauge
transformation on the wave function itself, see e.g ref [28]. Although not of more relevance in
the present work, gauge transformations in light-matter interaction is a very important subject
that deserves more than just a reference in passing. The interested reader is therefore referred to
appendix for a short discussion on this subject.
The second statement is only motivated but not proven in detail here. In ref [28] the argument for
removing pz x from HBY D1 is based on a discussion of its physical significance. The authors
present compelling reasons why the operator should be regarded as a second order correction
which therefore the current work also adopts.
The final forms of H I (t) to work with are:
Hdip (t) = pz A(t)
x
dA(t)
HBY D1 (t) = pz A(t) − A(t)
.
c
dt
Note that the computation of the matrix elements will omit all factors for brevity.
3.6.1
(3.38)
(3.39)
Dipole approximation
The matrix elements to compute are:
(Hdip )ab = ⟨na la ma ∣pz ∣nb lb mb ⟩.
(3.40)
Using eq. 3.28 and eq. 3.27 the cartesian components of p can be combined to form a spherical
tensor operator of rank 1 with components p1±1 = px ± ipy and p10 = pz . The Wigner-Eckart
theorem then gives:
l
1 lb
(Hdip )ab = (-1)la −ma ( a
)⟨na la ∣∣p1 ∣∣nb lb ⟩
-ma 0 mb
(3.41)
Note now that the 3j-symbol gives the well-known selection rules for the electric dipole approximation (with polarization in ẑ):
i) The triangular condition gives that lb = la ± 1, (but of course with the restriction l ≥ 0).
ii) −ma + 0 + mb = 0 gives that the projection number can not change.
The selection rules provide two things. The first obvious thing is a direct overview of which
angular symmetries have non-zero matrix elements, also known as couplings.
37
The second thing is the required Nang for a given maximum angular momentum lmax considered.
While there are 2l +1 different values of ml for every value of l, the pz -operator can only connect
states with the same ml . If the initial state prior to interaction has a fixed value of ml , i.e ml = 0,
the required angular model space has:
Nang = lmax + 1
(3.42)
number of states. It is of course not incorrect to include states with different values of ml as well
but it is in this case just dead weight.
What remains now is to complete the matrix element is to compute the reduced matrix element.
The explicit form of this for p1 can be found in ref [29]. As there is no aspect from its appearance that is needed to follow the development of this thesis, it is not addressed further here.
3.6.2
Beyond the dipole approximation
The matrix elements to compute here are:
⟨na la ma ∣pz ∣nb lb mb ⟩
(3.43)
⟨na la ma ∣x∣nb lb mb ⟩.
(3.44)
The x-operator can be rewritten in spherical harmonics of rank 1:
√
2π 1
(Y −Y 1 )
x = r⋅
3 -1 1
(3.45)
and applying the Wigner-Eckart theorem to this gives:
√
⟨na la ma ∣x∣nb lb mb ⟩ =
2π
l
1 lb
l
1 lb
⋅ (-1)la −ma (( a
)−( a
))⟨na la ∣∣rY1 ∣∣nb lb ⟩.
-m
-1
m
-m
1
m
3
a
a
b
b
(3.46)
The conditions from the 3j-symbol give:
i) The triangular condition gives lb = la ± 1.
ii) −ma ± 1 + mb = 0 gives that the projection number changes; mb = ma ± 1.
This operator connects states with different values of ml and thus forces the model space to grow.
Given a specific initial state, say the (angular) ground state ∣l = 0,ml = 0⟩ the following states (up
to l=2) can be reached:
∣l = 1,ml = ±1⟩, ∣l = 2,ml = ±2⟩, ∣l = 2,ml = 0⟩.
If the interaction was only given by the x-operator some states could be omitted; for instance
∣l = 1,ml = 0⟩, ∣l = 2,ml = ±1⟩. The interaction however also includes the pz -operator that connects
38
states with the same ml . Due to the change of ml by the x-operator, pz couplings must now be
computed for all ml , which in turn gives that x-couplings must also be computed for all ml .
In short, the two operators together leave no state entirely disconnected and thus enforce full
degeneracy in the model space.
The required number of angular symmetries now grows significantly from the dipole approximaton. For a given lmax :
lmax
Nang = ∑ (2l + 1) = (lmax + 1)2
(3.47)
l=0
the model space grows quadratically.
Since the reduced matrix element of the x-operator is relevant for the upcoming section it is now
given in some detail. Note that from the inner product:
∗
⟨na la ma ∣rY11 ∣nb lb mb ⟩ = ∫ Pn∗a la r Pnb lb dr ∫ (Ymlaa ) Y11Ymlbb dΩ
r
Ω
⟨na la ma ∣rY11 ∣nb lb mb ⟩ = F(r) ⋅ ⟨la ma ∣Y11 ∣lb mb ⟩
(3.48)
where the purely radial integral is denoted by F(r). The Wigner-Eckart theorem applied to the
angular part now gives:
l
1 lb
)⟨l ∣∣Y1 ∣∣lb ⟩
⟨la ma ∣Y11 ∣lb mb ⟩ = (-1)la −ma ( a
-ma 0 mb a
(3.49)
from which one may deduce that the total reduced matrix element is:
⟨na la ∣∣rY1 ∣∣nb lb ⟩ = F(r) ⋅ ⟨la ∣∣Y1 ∣∣lb ⟩.
From eq. 2.127 and eq. 2.100 in ref [27] one obtains:
√
√
4π
l 1 lb
1
⟨la ∣∣Y ∣∣lb ⟩ =
(-1)la (2la + 1)(2lb + 1) ( a
)
0 0 0
3
(3.50)
(3.51)
which inserted into eq. 3.50 gives the complete form of the reduced matrix element.
In the same spirit as this section, the next one covers the computation of relativistic couplings
within and beyond the dipole approximation.
39
3.7
Relativistic matrix elements
The relativistic interaction operator was given in eq. 3.14:
α ⋅ A.
HI (t) = cα
Recall the definition of α :
0 σi
αi = ( 2
),
σi 02
σx = (
0 1
),
1 0
0 -i
σy = (
),
i 0
σz = (
1 0
)
0 -1
It should come as no surprise that combinations of the Pauli matrices can form components of
a spherical tensor operator. Such an operator of rank 1 can be formed by the three components
1
= σx ± iσy1 and σ01 = σz , which will soon be used when computing the matrix elements.
σ±1
The dipole and beyond dipole version of the Hamiltonian can be written on the same form. With
the vector field pointing in ẑ this is:
N
α z ⋅ ∑ (−kx)n
Hrel (t) = cα
n=0
1 d n A(t)
.
n! dt n
(3.52)
Due to the presence of spin the evaluation of the couplings is slightly more advanced than the
previous cases encountered. On the bright side it is possible to generalize the computation to
any desired order n, which will now be shown.
3.7.1
Coupling of tensor operators
Note that any order of x can be expressed as a sum of spherical harmonics, with a few first
examples being:
√
x0 = 4π ⋅ r0 ⋅Y00
√
2π 1
1
⋅ r ⋅ (Y-11 −Y11 )
x =
(3.53)
3
√
1
1
1
x2 = 4π ⋅ r2 (− √ Y02 + √ [Y22 +Y-22 ] + Y00 ).
3
30
15
The interaction operator will consist of terms of the form:
02
( 1 γ λ
σ0 r Yµ
σ01 rγ Yµλ
)
02
which evaluated in the relativistic spectral basis:
ψn,κ, j,m j (r) =
1 Pnκ (r)Xκ, j,m j
(
)
r iQnκ (r)X−κ, j,m j
takes on the form:
α z rγ Yµλ ∣nb κb jb mb ⟩ =
⟨na κa ja ma ∣α
40
(3.54)
i ∫ [Pn∗a κa (r)rγ Qnb κb (r)⟨κa ja ma ∣σ01Yµλ ∣-κb jb mb ⟩−
(3.55)
r
Q∗na κa (r)rγ Pnb κb (r)⟨-κa ja ma ∣σ01Yµλ ∣κb jb mb ⟩dr].
While the radial integrals are of the same form as before, the spin-angular part is different.
This time the interaction operator is a tensor product with one tensor (σ01 ) acting in spin-space
while the other term acts in the orbital angular momentum space. While such a point of view is
perfectly valid another useful approach exists: to couple the operators together to form a single
operator acting in the combined spin-angular Hilbert space1 :
λ +1
K
σ 1 Yλ } ,
σ01Yµλ = ∑ ⟨10;λ µ∣KQ⟩{σ
Q
K=∣λ −1∣
Q = −µ
With this choice the spin-angular part may be computed as2 :
λ +1
K
σ 1 Yλ } ∣κ̃ j˜m̃⟩
⟨κ jm∣σ01Yµλ ∣κ̃ j˜m̃⟩ = ⟨κ jm∣ ∑ ⟨10;λ µ∣KQ⟩{σ
Q
K=∣λ −1∣
λ +1
K
σ 1 Yλ } ∣κ̃ j˜m̃⟩
⟨κ jm∣σ01Yµλ ∣κ̃ j˜m̃⟩ = ∑ ⟨10;λ µ∣KQ⟩⟨κ jm∣{σ
Q
K=∣λ −1∣
using eq. 3.30:
λ +1
⟨κ jm∣σ01Yµλ ∣κ̃ j˜m̃⟩ = ∑ (-1)λ −1−Q
K=∣λ −1∣
√
K
1 λ K
σ 1 Yλ } ∣κ̃ j˜m̃⟩
2K + 1(
)⟨κ jm∣{σ
0 µ −µ
Q
(3.56)
The Wigner-Eckart theorem can be applied to the matrix element of the combined operator
K
σ 1 Yλ }Q :
{σ
K
σ 1 Yλ } ∣κ̃ j˜m̃⟩ = (-1) j−m (
⟨κ jm∣{σ
Q
K
j K j˜
σ 1 Yλ } ∣∣ j˜⟩
)⟨ j∣∣{σ
−m Q m̃
(3.57)
Lastly, the reduced matrix element of the combined operator is (see eq. 4.21 in ref [27]):
K
σ Y } ∣∣ j˜⟩ =
⟨ j∣∣{σ
1
λ
√
⎧
l l˜ 1 ⎫
⎪
⎪
⎪
⎪
⎪
˜ ⎨ s s̃ λ ⎪
σ ∣∣s̃⟩⟨l∣∣Y ∣∣l⟩
⎬
(2 j + 1)(2K + 1)(2 j˜ + 1)⟨s∣∣σ
⎪
⎪ j J˜ K ⎪
⎪
⎪
⎪
⎩
⎭
1
λ
(3.58)
where the last expression is a 9 j-symbol, which can be written as the product of 3 j-symbols (see
eq. 3.71 in ref [27]). The term 3.55 can now be computed by starting with eq. 3.58 and trace
back through the numbered equations.
1
2
See for instance eq. 2.95 in ref [27] which is the inverted relation.
To tidy up, the indices are temporarily replaced.
41
It is perhaps not trivial to digest what just happened. However, the overwhelming power of the
approach should be clear: these steps are programmed just once. Incorportating couplings to
any spatial order, say n, only requires the extra work of expressing xn in spherical harmonics no additional source code need to be developed.
On a final note, due to the presence of spin, the number of angular states in the relativistic framework is twice the size of the non-relativistic angular basis. Together with the two radial components the relativistic model space is four times larger than the correspnding non-relativistic
space.
42
4. Dynamics
Premature optimization is the root of all evil - Donald Knuth
This chapter lays the foundation for computing the dynamics using the couplings from the previous chapter. One final numerical technique to alter the model space called complex scaling is
introduced. In order to prepare for higher order dynamics with model spaces containing up to
106 states, several successful (and some unsuccessful) steps of optimization have been made.
4.1
Time evolution operator
The equations of motion may be put on another form than have been seen thus far:
∣Ψ(t1 )⟩ = Û(t0 ,t1 )∣Ψ(t0 )⟩
(4.1)
where Û(t0 ,t1 ) is known as the time evolution operator. For a time dependent Hamiltonian that
commutes with itself at different times it is:
t1
Û(t0 ,t1 ) = e−i ∫t0
H(t ′ )dt ′
which can be found in for instance ref [17]. Given a small enough time separation ∆t = t1 −t0 → 0
this may be approximated as:
Û(t0 ,t1 ) ≈ e−iH(t0 +∆t/2)⋅∆t
(4.2)
For the attentive, this approximation is just the mean-value theorem. Propagation now amounts
to understanding how to compute the action of the matrix exponential on the state ∣Ψ(t0 )⟩.
4.2
Expansion in field dressed eigenstates
Recall that the spectral basis {Φ(t)} of the Hamiltonian, at any time, is complete. It is therefore
possible to express the unity operator as:
1̂ = ∑ ∣Φn (t)⟩⟨Φn (t)∣.
(4.3)
n
Inserting the last two equations into eq. 4.1 one obtains:
∣Ψ(t0 + ∆t)⟩ ≈ e−iH(t0 +∆t/2)⋅∆t 1̂∣Ψ(t0 )⟩
43
∣Ψ(t0 + ∆t)⟩ ≈ ∑ e−iEn (t0 +∆t/2)⋅∆t ∣Φn (t + ∆t/2)⟩⟨Φn (t)∣Ψ(t0 + ∆t/2)⟩
n
where the exponential operator acted on the eigenstates of H(t0 + ∆t/2). The way to achieve
this in a numerical basis is to compute the time dependent Hamiltonian:
H(t) = H0 + HI (t)
(4.4)
where the interaction matrix elements are computed by multiplying each operator with the
corresponding time factor, i.e:
(HI (t))ab = A(t) ⋅ ⟨Φa ∣σz ∣Φb ⟩ −
ω dA
⋅ ⟨Φa ∣σz x∣Φb ⟩
c dt
(4.5)
for the first order relativistic BYD Hamiltonian. Once the Hamiltonian is built the eigenvalues
and eigenvectors of this matrix are obtained from diagonalizing the matrix.
Now an extremely important question arises - what about the negative energy states?
4.2.1
Negative energy states
Recall that the relativistic spectrum contains negative energy states. Dirac’s interpretation of
these states were to regard them as naturally occupied. If they were reachable, then any positive
energy eigenstate would decay into the negative spectrum. Now the question arises: unless there
is real pair-production, can these states be left out of the basis set? That is; from the spectral
basis of the time independent Dirac Hamiltonian, is it valid to remove all eigenstates belonging
to the negative energy spectrum? In general, the answer is no (see ref [6; 7]), which at first seems
to contradict Dirac’s original hypothesis.
However, the conclusion reached in ref [6] is that the hypothesis applies to the time dependent basis, since what constitutes a negative energy state changes with the field. The omission
of the "true" states is then achieved by only projecting the state ∣Ψ(t0 )⟩ onto the time dependent
positive energy states P:
∣Ψ(t0 + ∆t)⟩ ≈ ∑ e−iEn (t0 +∆t/2)⋅∆t ∣Φn (t + ∆t/2)⟩⟨Φn (t)∣Ψ(t0 + ∆t/2)⟩
n∈P
.
The conclusion above works nicely with the technique employed to compute a small and efficient radial model space: complex scaling.
4.3
Limiting the box size with complex scaling
One condition to put on a model space is that it should not introduce artificial effects. If the
model space is not big enough unreliable results are to be expected:
44
Figure 4.1: Radial probability of an electron initially in the ground state. The pulse it interacts with
is a sine-square pulse with ω = 3.5 a.u and E0 = 10 a.u. The times are given in percentage of the
pulse length with zero field strength after 100%.
45
In fig. 4.1 it is seen that as the wave function moves outwards it hits the box end and reflects
back. At the field strengths to be considered (E0 ∼ 130 a.u) it is not feasible to just make the box
larger, which is seen in Fig. 4.2 below. Here the box is increased to Rmax = 300 a.u while the field
strength is E0 = 60 a.u. While the majority of the wave function is localized within 50 a.u from
the nucleus small ripples of probability continuously shoot out towards the box end. Eventually
that miniscule part hits the box end and propagates back to interfere with the rest of the wave
function.
Figure 4.2: Radial probability of an electron initially in the ground state. Here E0 = 60 a.u. while
the other pulse parameters are the same as in Fig. 4.1.
One way to overcome this obstacle is to use complex scaling. In this method the radial coordinate
is rotated with an angle θ into the complex plane; either from r = 0, called uniform complex
scaling:
r → eiθ r
46
(4.6)
or starting from some value r0 :
r → r0 + (r − r0 )eiθ r,
r > r0
(4.7)
called exterior complex scaling. The applicability of complex scaling is wide; it is for instance
useful for computing resonances, represent continuum spectrum efficiently (see e.g [30]) and
absorb outgoing flux [31]. In the present case, it is the latter two properties that are of interest
and in the figure below the last one is represented:
Figure 4.3: Radial probability of an electron initially in the ground state. The field strength here
is E0 = 10 a.u. In the left figure uniform scaling has been employed while the right figure presents
dynamics with exterior complex scaling, both with θ = 10 degrees. The black dashed line indicates
where the rotation starts.
While both implementations absorb outgoing flux they each have their particular strength. Uniform scaling requires fewer states but any information about the ionized part is very hard to
retrieve. Exterior complex scaling admits full analysis until the point of scaling but the price is
a larger number of required states for the unscaled part.
Due to the severe computational load the TDDE calculations were anticipated to cause the
results in this thesis have been computed with uniform complex scaling.
One final thing should be mentioned about complex scaling: the energies of the rotated continuum states are roughly given by
θ
Econ
≈ e−i2θ Econ
(4.8)
while the bound state energies are left more or less unaltered [30]. Thus, positive continuum
θ
energies get surpressed1 during time evolution (by the factor e−iE dt ) while for negative energies the opposite situation occurs. This is the reason why the findings of ref [6] regarding the
omission of time dependent negative energy states are of importance - if this was not the case
1
A complex scaled Hamiltonian is not Hermitian.
47
complex scaling would probably only cause problems when applied to TDDE.
4.4
Krylov subspace method
With uniform complex scaling, the radial basis size can be kept small. Despite this, a full diagonalization is for practical purposes not feasible as the number of angular symmetries requried
will be very large. In such a case a Krylov subspace technique might be useful.
The idea, made readily accessible in ref [32] together with ref [33], is to approximate the action
of the time evolution operator:
eH̃ x⃗ ≈ pm−1 (H̃)⃗
x
(4.9)
where H̃ = −iH(t + ∆t/2)∆t and pm−1 (H̃) is a polynomial in H̃ of order m − 1. This polynomial
belongs to the space spanned by:
Km = {⃗
x, H̃ x⃗,..., H̃ m−1 x⃗}
(4.10)
where Km is called a Krylov space of order m. Note that most of the symbols and indices are
kept similar to the notation in [32; 33] for the benefit of the reader.
Following the Arnoldi algorithm outlined in sec.2.1 of ref [32], an orthonormal basis Vm of
the Krylov space can be constructed. In [33] the following approximation is made:
eH̃ ≈ Vm eUm VmT
(4.11)
where Um = VmT H̃Vm . The geometrical interpretation of this is that the matrix exponential in the
Hilbert space, eH̃ , is projected onto the Krylov subspace spanned by the basis Vm . Given a short
enough time step ∆t, the dimension of this space m can be made very small. For the strongest
fields considered in this thesis some typical values are ∆t ≈ 0.02 ∼ 0.04 with Krylov subspaces
of order m ∼ 15. The convergence with respect to m can be monitored by computing one of the
error estimates given in ref [32].
Note that the matrix exponential in Krylov space can be evaluated in complete analogy as
the full diagonalization:
m
eH̃ ≈ Vm eUm 1̂mVmT = Vm [ ∑ eEn ∣φn ⟩⟨φn ∣]VmT
(4.12)
n=1
with the exception that this time the diagonalization is made in the 15 × 15 space which is
done in an instance. Now the time consuming part is dominated by matrix-vector products.
These can however be siginificantly optimized which is the subject up next.
4.4.1
Optimization
Consider the basis of the relativistic model space ordered in increasing (l, j,m j ) (for the large
component), i.e ∣0, 21 ,- 12 ⟩, ∣0, 21 , 21 ⟩, ∣1, 12 ,- 12 ⟩,∣1, 12 , 12 ⟩, ∣1, 23 ,- 32 ⟩, ∣1, 32 ,- 12 ⟩ and so on up to l = 6.
The four first spatial operators then take on the following appearance:
48
Figure 4.4: Relativistiv couplings matrices. Each coloured block denotes a coupling of the
corrsponding operator between two states. In general, the operator contains a lof of zero entries
and as such is said to be sparse.
Each block is an (Nrad × Nrad ) submatrix, typically with Nrad ∼ 100. While the coloured blocks
represent interaction the rest are just zeros that fill up system memory - but the tragedy does not
necessarily stop there. In case these matrices are multiplied with vectors using normal routines
(non-sparse) a lot of zero-multiplication take place.
By only using the non-zero elements significant memory and computational time is freed
up. It is also easily parallelized both on shared memory machines as well as distributed systems
which can give a new level of performance to the program. The optimization does however not
stop just quite yet.
Recall the essence of the Wigner-Eckart theorem: all information regarding projection numbers
can be factored out of the matrix element. This means that a lot of the blocks in the previous
figure should only differ by a scalar.
49
Figure 4.5: A comparison between the full BYD3 angular couplings with the reduced matrix elements not duplicated. The left figure can be reproduced from the right by just multiplication of the
correct angular factor.
The final form of the propagation builds a Krylov subspace for many small time steps. At each
step, products of the form H(t)∣Ψ⟩ are computed by applying the angular and radial factors
separately to ∣Ψ⟩ to minimize the total number of operations. As a comparison, consider the
Fig. 4.6
50
Figure 4.6: Probability to find the electron in the ground state as a function of time. The pulse here
has a total time of T = 30π a.u, E0 = 40.0 a.u and ω = 3.5 a.u. The dots are the results from dynamics
with full coupling matrices while the red solid line is an optimized Krylov implementation.
To demonstrate the increased performance a table showing the memory and time consumption
of the two approaches is given below.
Full
Optimized Krylov
Time [s]
3 ⋅ 106
3 ⋅ 102
Time step [a.u]
0.25
0.02
Memory
2.0 GB
40 kB
Table 4.1: Comparing figures for full and Krylov implementation. The model space here consists
of 2 × 20 B-splines per angular symmetry, with 128 such included. (lmax = 7).
The same Krylov implementation is adopted for the non-relativistic simulations. In the next
chapter results of simulations are shown.
51
52
5. Results
Everything uptil now has been in preparation to solve the equations of motion. By doing so one
should in principle be able to obtain any desired information about the system. However, due to
the use of uniform complex scaling some observables are for practical purposes out of reach for instance the angular distribution of the electron.
One observable that still is computable is the total photoionization yield, Pion , which is the
probability that the electron due to the interaction leaves the system.
Relativistic effects are expected when the classical quiver velocity of the electron vq ∼ Eω0 , reaches
a substantial fraction of the speed of light [28] [2]1 . In this chapter results are presented for dynamics up to vq ≈ 0.23c (E0 = 110 a.u) to explicitly compare the predictions by TDSE and TDDE.
5.1
Model space and pulse parameters
The size of the required model space depends on the properties of the pulse that the electron
interacts with. The type of pulse, as stated in previous chapters, is:
A(ωt − kx) =
E0 2 π(ωt − kx)
sin (
)sin(ωt − kx + φ ) ẑ.
ω
ωT
The following parameters are kept fixed:
ω = 3.5
φ =0
T = 2πNc
(5.1)
Nc = 15
where the extra parameter Nc quantifies the number of cycles the carrier makes during the pulse
length.
The ionization yield will be presented as a function of increasing peak electric field strength E0 .
In ref [34] a non-relativistic study with the same fixed parameters was carried out for a maximum value of E0 = 60 a.u, corresponding to vq ≈ 0.13c, with Nc = 15,30,...,100. The authors
report that simulations carried out in the following boxes:
1
See section VI.B for an equivalent argument.
53
i) A very large box of Rmax = 800 a.u
ii) A smaller box of Rmax = 40 a.u with an absorbing potential for outgoing flux
give the same values for the ionization yield. Given the much shorter propagation time used
here (15 cycles, compared to their 100 cycles) the smaller box should be sufficient for the field
strengths considered here (E0 ≤ 110 a.u). Thus, computations are carried out in the following
box:
Rmax
dr
θ
B-spline order
Number of B-splines
40.0 a.u
0.34 a.u
2.5○
7 (Pnonrel ,Prel ), 8 (Qrel )
121 (Pnonrel ,Prel ), 122 (Qrel )
Table 5.1: Grid parameters for model box. The same equidistant knot sequence is used for TDSE
and TDDE computation. Note that the relativistic computations require twice as many radial states
due to having two components. Since the angular basis is twice as large due to spin, the relativistic
basis is four times larger than the non-relativistic basis for the same box and lmax .
A sufficient size of the angular model space is monitored by increasing the maximum orbital angular momentum lmax (including all states with lower l and full degeneracy) until the dynamics
have converged. Once the final model space is acquired Pion can be computed by solving TDSE
and TDDE in the various spatial approximations of the pulse that were discussed in chapter 3.
Pion is in this work computed as the sum of population in the bound numerical box states.
Before moving on to discuss the results of the dynamics a reminder of the time dependent Hamiltonians is in order:
x
dA(t)
Hnrel (t) = H0 + pz A(t) − A(t)
.
c
dt
N
Hrel (t) = HD0 + cαz ⋅ A(t) + cαz ⋅ ∑ (−kx)n
n=1
1 d n A(t)
.
n! dt n
(5.2)
where the value of N will be discussed in subsequent sections. Before that, the next section
presents Pion predicted by the dipole approximation, where any dependence on x is neglected.
5.2
Dipole interaction
Consider first the interaction without a spatial dependence of the pulse. Fig. 5.1 shows Pion for a
the box described in tab. 5.1 with increasing values of lmax .
54
Figure 5.1: Ionization probability for hydrogen as a function of the peak electric field strength. The
vertical line denotes the value of E0 for which vq ≈ 0.1c. Note the convergence with increasing lmax
towards the black solid line which is globally converged to 10−5 (see figure below).
Figure 5.2: Global convergence test of the dipole approximation. The difference between lmax = 40
and lmax = 15,20,30 are shown. The residual of lmax = 30 is of the order 10−5 which is in the current
context will be accepted as converged. Note the second vertical line here indicating another 10%55
increment of speed of light.
The discontinuous jumps of Pion (E0 ) for lower values of lmax in fig. 5.1 are typical for an incomplete basis, that is, a basis that does not provide enough channels for the dynamics. This is also
the reason why the jumps disappear when more channels (larger lmax ) are included.
Before leaving the section on dipole approximation, converged calculations for TDSE and
TDDE are compared:
Figure 5.3: Comparison of TDSE and TDDE calculation (lmax = 40) within the dipole approximation. At vq ≈ 0.15c, the non-relativistic Pion starts to increase compared to the relativistic Pion .
Note the somewhat unintuitive behaviour of the ionization yield: at first it increases with increasdP
> 0, which seems natural - stronger field gives more ionization.
ing electric field strength, dE
0
At E0 ≈ 12 a.u this however changes - suddenly the ionization becomes less probable with increasing field strength. This phenomenon is known as stabilization and has been studied quite
extensively, see e.g ref [35].
Lastly, note that from fig. 5.3 one could be tempted to interpret the relativistic effects on Pion
as surpressive for E0 > 70 a.u. However, this only bears meaning if the dipole approximation is
valid in this region.
56
5.3
Beyond dipole interaction
Consider first the non-relativistic interaction beyond the dipole approximation:
Figure 5.4: Pion predicted by TDSE within (dashed line) and beyond (solid line) dipole approximation. The acronym BYD1 stands for first order beyond dipole approximation.
In fig. 5.4 it is seen that the dipole approximation breaks down for E0 ≥ 30 a.u. In light of the
discussion in section 3.3 the dipole approximation should be regarded as valid for quite a large
E0 .
In the next section the ionization yield predicted by the relativistic Hamiltonian beyond dipole
approximation is presented.
57
5.4
5.4.1
TDDE beyond dipole approximation
Order of interaction
In fig. 5.5 the relativistic beyond dipole interactions for N = 1 ∼ 3 in eq. 5.2 are shown in a region
where non-relativistic dynamics are believed to be valid:
Figure 5.5: Comparison of TDDE and TDSE in the non-relativistic regime. The numbers trailing
the acronym "BYD" are the numerical value of N in eq. 5.2. For all cases lmax = 15 has been used
which in this region is enough for convergence.
Some features in fig. 5.5 to highlight:
1) BYD1 completely overestimates Pion .
2) BYD2 agrees with TDSE until E0 = 50 a.u, where it starts to underestimate Pion .
3) BYD3 gives good agreement with TDSE in the whole range.
The inadequacy of truncating the series in eq. 5.2 after one term is also visible in the radial
probability density plotted in fig. 5.6 below.
58
Figure 5.6: The radial probability distributions of the electron initially in the ground state. The
percentage refers to elapsed interaction time. While BYD2 and BYD3 are overlapping after the
pulse the wavefunction following the BYD1 Hamiltonian is visibly different. The peak electric field
strength is here E0 = 40.0 a.u.
Recall that complex scaling was employed to absorb outgoing flux, which in fig. 5.6 can be
hinted by the lack of probability towards the box end. In regards of this, the BYD1 wavefunction
is seen to be less easily killed off than BYD2 and BYD3. This is a reflection of its instability for E0 ≥ 50 a.u. the ionization yield predicted by BYD1 even diverges, which is a consequence
of the non-Hermitian Hamiltonian that results from complex scaling.
Although not shown here, for E0 ≥ 70 a.u. BYD2 gives diverging ionization yields as well.
For all values of E0 to be considered BYD3 remains non-diverging and thus showing good signs
of stability for the current box.1
Before increasing the field strength further one small stop should be made.
1
Note that also BYD3 would diverge at modest electric field strengths if too large values of the scaling
angle or box size would be used.
59
5.4.2
Negative energy states to HD0
Throughout this thesis the importance of the negative energy states have been mentioned. In
fig. 5.7 the same observation as ref [6] made can be seen:
Figure 5.7: A comparison of Pion computed with and without the time independent negative energy
states in the basis.
while Pion remains (almost) the same within the dipole approximation, the BYD effects completely vanish. This is not so hard to understand since the matrix elements are computed using
eq. 3.55 which connects the P(r)-component of one state with the Q(r)-component of another.
Recall now the discussion following fig. 2.3: for positive energy states ∣P(r)∣ >> ∣Q(r)∣ but for
negative energy states the relation is reversed. It thus follows that the largest matrix elements
are computed between positive and negative energy states.
It should also be mentioned that for relativistic calculations in the so called length gauge, predictions made within the dipole approximation are altered, see e.g ref [7].
60
5.4.3
Towards relativistic effects
As E0 is increased convergence with respect to lmax must once again be checked. Below are
results obtained for electric field strengths up to E0 = 110 a.u:
Figure 5.8: Pion for TDDE BYD3. For E0 > 80 a.u. lmax > 15 is requred while for E0 > 100 a.u.
lmax > 20 is required for convergence.
In fig. 5.8 deviations between the TDDE and TDSE calculations start to appear where the
relativistic effects seem to increase the ionization yield. For E0 = 100 a.u, corresponding to
vquiv ≈ 0.21c, the relativistic Pion is about 2.3% higher than the non-relativistic prediction.
Regarding convergence, what can be said with confidence is that lmax = 30 is converged for
E0 = 100 a.u, which secures the results obtained up to this value. For higher electric field
strengths yet larger values of lmax must be used. Such simulations are currently running but
did not complete in time for the printing of this thesis.
61
62
6. Summary and Outlook
To search for relativisic effects in the dynamics induced by an attosecond pulse TDDE has been
solved for light-matter interaction beyond the dipole approximation. It has been demonstrated
from an expansion of the vector field in the spatial coordinate
N
x n 1 d (n)
A(η) ≈ A(t) + ∑ (- )
A(t)
n! dt n
n=1 c
that at least N = 3 must be used for full stability if spatial dependence of the field is to be incorporated at all. This is in contrast to the non-relativistic TDSE for which N = 1 suffices to give
the lowest order spatial effects.
The following pulse:
E0 2 πη
sin (
)sin(η + φ ) ẑ
ω
ωT
with ω = 3.5, T = 30π and φ = 0 was used to find at which E0 the ionization yield (Pion ) given by
TDDE would differ from the prediction by TDSE which then would define the threshold of the
relativistic regime.
For the pulse considered in this thesis the relativistic effects start to increase Pion around
E0 = 80 a.u, corresponding to a classical quiver velocity of the electron vquiv ≈ 0.17c. At E0 = 100
a.u., corresponding to vquiv ≈ 0.21c, a difference of 2.3% is observed. For yet higher electric field
strengths further simulations with larger basis sets must be conducted to ensure convergence.
A(η) =
Once full saturation of the ionization yield is observed for the case tested in this thesis, the next
step is naturally to repeat the process for other values of the fixed parameters. However, there
are many further applications of the program that have not yet been presented. Dynamics for
highly charged hydrogenlike systems, spin-flip studies and angular distribution of the electron
can already be computed as it is. An exciting application that requires very little extra development is the dynamics induced by multiple pulses. For instance, ref [36] considered two keV
pulses with a small energy separation interacting with hydrogen. Although the electron energy
distribution was dominated by two single peaks centered about each pulse energy, a small peak
centered at the energy difference of the two pulses was observed and identified to be caused
by stimulated Compton scattering. The pulse parameters in their specific study were chosen to
justify non-relativistic calculations, but for yet higher energies a solution of the time dependent
Dirac equation is called for.
63
64
Appendix
Atomic units
Physical quantity
Charge
Mass
Length
Intensity
Action
Velocity
Constant’s name
Electron charge
Electron mass
Bohr radius, a0
1/r intensity at a0
Reduced Planck’s const.
Speed of light
SI-units
1.60 ⋅ 10−19 C
9.11 ⋅ 1031 Kg
5.29 ⋅ 10−11 m
3.51 ⋅ 1012 W/m2
1.05 ⋅ 10−34 Js
3.00 ⋅ 108 m/s
Atomic units
1
1
1
1
1
1/α ≈ 137.06
Table 6.1: Conversion table for physical quantities. α is the fine structure constant, as it is dimensionless it has the same value in atomic units as in SI-units.
B-splines
B-splines are piecewise polynomials of order k − 1, locally confined between points defined on
the radial axis. For an non-decreasing sequence of points {r0 ≤ r1 ≤ r2 ≤ ...,rN } from r = 0 up to
r = Rmax , the B-splines are generated between each point according to the recursion formula:
B1j = 1 if r j ≤ r ≤ r j+1 , else B1j = 0
Bkj =
r j+k − r k−1
r −rj
Bk−1
B (r)
j (r) +
r j+k−1 − r j
r j+k − r j+1 j+1
(6.1)
(6.2)
for the j:th interval. Some properties that are worth highlighting are:
- The points can be distributed in any fashion as long as r j+1 ≥ r j .
- A B-spline of order k is confined between k + 1 points. Hence the total number of B-splines
NB is of the form NB = Nint + k where Nint is the number of intervals.
- They form a complete set on the sequence.
- When evaluating integral products, such as inner products between wave functions, their
polynomial behaviour allows for exact integration to machine accuracy using Gauss-Legendre
quadrature.
The properties listed above give quite some power to this technique. Since the points can be
distributed freely it is possible to specifically tailor the sequence to the problem at hand. By
putting densely spaced points in regions where the functions are changing rapidly and sparsely
spaced points where their behaviour is more calm the dimension of the complete set {NB } can
be minimized without compromizing the quality of the projected function.
Figure 6.1: B-splines of orders k = 2,3,4,7 on the sequence {0,1,2,3,4}. Note that order k implies
piecewise polynomials of order k − 1, clearly visible k = 1,2. At each value r the B-splines sum up
to unity, reflecting their completeness.
Gauge transformations in light-matter interaction
A gauge transformation in light-matter interaction is similar in spirit to the gauge transformation
of the vector field encountered in chapter 3 (i.e Coulomb gauge) but in this case it is a wave
function that is transformed. By using a suitable unitary transformation the equations of motion
can be cast on a form that is suitable for the computations to be made.
As an example, consider the following transformation that removes the A(t)2 -term in ch. 3.6:
t 1
A(ωt ′ )2 dt ′
2
Ψ̃(r,t) = ei ∫0
Ψ(r,t).
(6.3)
The equation of motion 2.1 in dipole approximation is:
i
∂Ψ
p2
1
= [ + p ⋅ A(t) + A(t)2 ]Ψ
∂t
2
2
in which the wave function is replaced through 6.3:
i
t 1
′ 2 ′
p2
1
∂ −i ∫0t 1 A(t ′ )2 dt ′
2
[e
Ψ̃(r,t)] = [ + p ⋅ A(t) + A(t)2 ]e−i ∫0 2 A(t ) dt Ψ̃(r,t)
∂t
2
2
(6.4)
Carrying out the differentiation on the left hand side;
t 1
′ 2 ′ ∂ Ψ̃(r,t)
1 ′ 2 −i ∫0t 1 A(t ′ )2 dt ′
2
Ψ̃(r,t) + e−i ∫0 2 A(t ) dt
A(t ) e
2
∂t
(6.5)
and reinserting this gives:
t 1
A(t ′ )2 dt ′
2
e−i ∫0
t 1
′ 2 ′
∂ Ψ̃(r,t)
p2
= [ + p ⋅ A(t)]e−i ∫0 2 A(t ) dt Ψ̃(r,t).
∂t
2
(6.6)
Finding the Hermitian conjugate of U;
t 1
A(t ′ )2 dt ′
2
U = e−i ∫0
t 1
A(t ′ )2 dt ′
2
→ U † = ei ∫0
(6.7)
and applying it on both sides from the left gives:
∂ Ψ̃(r,t)
p2
= [ + p ⋅ A(t)] Ψ̃(r,t).
∂t
2
(6.8)
The last step is valid because the terms in the square brackets commute with this particular U † .
This equation is equivalent to the original one and the physics described by Ψ̃ is the same as Ψ,
at least in theory. The Hamiltonian in eq. 6.8 is said to be in velocity gauge. Other gauges that
are popular are length gauge and acceleration gauge, also known as the Kramers-Hennerberger
frame [10].
If a model space is sufficiently large the numerical predictions give the correct results. Therefore,
converged results computed in different gauges should agree, giving the means to test the numerical implementations adopted. Note also that this implies that convergence may be achieved
faster in a specific gauge, which of course is of major interest. Velocity gauge is adapted throughout this work but see [7] for a discussion on convergence behaviour in different gauges within
the dipole approximation.
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1
Relativistic ionization dynamics for a hydrogen atom exposed to super-intense XUV laser pulses
Tor Kjellsson,1 Sølve Selstø,2 and Eva Lindroth1
1
Department of Physics, Stockholm University, AlbaNova University Center, SE-106 91 Stockholm, Sweden
2
Centre of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway
We study the ionization dynamics of a hydrogen atom exposed to attosecond laser pulses in the extreme
ultra violet region at very high intensities. The pulses are such that the electron is expected to reach relativistic
velocities, thus necessitating a full relativistic treatment. We solve the time dependent Dirac equation and
compare its predictions with those of the corresponding non-relativistic Schrödinger equation. Our findings
indicate surprisingly small relativistic corrections for cases in which free electrons are expected to be accelerated
towards the speed of light.
PACS numbers:
I.
INTRODUCTION
For lasers operating in the high intensity regime, ionization processes in which magnetic interactions play a crucial
role are seen, see, e.g., Refs. [1, 2]. At laser facilities such
as FLASH in Hamburg, such effects should also be seen for
photoionization in the ultra violet regime and beyond. With
even stronger free electron lasers, now operating or on their
way, such at the XFEL in Germany, the SACLA in Japan and
the LCLS, USA, or the coming pan-European extreme light
infrastructure (ELI), we expect so see experiments in which
electrons are accelerated towards relativistic velocities in the
near future. In order to describe such processes, it is not sufficient for theoretical studies to just go beyond the much applied
dipole approximation; a relativistic treatment based on the full
solution of the Dirac equation is called for [3, 4].
In Ref. [5] the Dirac equation for hydrogen-like systems
exposed to strong attosecond laser pulses was solved numerically. Unfortunately, for the case of hydrogen, computational
constraints did not allow for calculations penetrating into the
relativistic regime It was found, however, that even below
relativistic velocities, the negative energy part of the energy
spectrum is crucial in order to account for dynamics beyond
the dipole approximation, in which the spatial dependence of
the external laser field is neglected all together. Moreover,
in Ref. [6] it was shown that even within the dipole approximation, negative energy states are crucial for ionization processes involving more than one photon. While the solution
to technical challenges regarding the Dirac equation is a goal
of its own, eventually we would like to explore relativistic effects. More specifically, here we investigate to what extent
the ionization probabilities predicted by the Dirac equation,
differ from those of its non-relativistic counterpart, i.e., the
Schrödinger equation, in a regime where relativistic corrections are to be expected.
Of course, relativistic effects arise when the electron is
accelerated to velocities comparable to the speed of light.
This may come about in two ways; for highly charged nuclei
high velocities may be induced by the the Coulomb potential
alone, or the electron may be driven to relativistic speeds by
some external field. In the former case, relativistic corrections to the energy structure do of course influence the ionization dynamics, e.g., by modifying the ionization potential
[6–8]. However, in the present work we will restrict ourselves
to hydrogen and investigate cases in which the external field
alone is strong enough to potentially induce relativistic dynamics. As a measure we may take the maximum quiver
velocity of a classical free electron exposed to a homogenous electric field of strength E0 oscillating with frequency
ω: vquiv = eE0 /(mω); as vquiv becomes comparable to the
speed of light c, relativistic effects are to be expected. In order
to actually see such effects in the ionization probability, the
laser specifications must, of course, be such that saturation is
avoided even in this limit.
Various techniques have been applied in order to solve the
Dirac equation numerically, e.g., split operator methods, combined with Fourier transforms [8, 9] or the method of characteristics [10, 11], the close coupling method [7, 12] and
Krylov methods [13]. In the present work the Dirac equation is solved by expanding the state in a set of eigenstates of
the light-field free Hamiltonian, as in Ref. [5, 6], and propagated by a low order Magnus expansion [14]. The actual
matrix exponentiation is approximated by a Krylov sub-space
approach as in Ref. [13]. We aim to identify relativistic effects by comparing the predicted ionization probabilities with
the corresponding non-relativistic ones, which are obtained in
an analogous manner.
Atomic units are used throughout unless stated otherwise.
II.
THEORY
Our starting point is the time-dependent Dirac equation
(TDDE):
i~
d
Ψ = H(t)Ψ,
dt
(1)
with the Hamiltonian
H(t) = cα · [p + eA(η)] + V (r)14 + mc2 β = H0 + ecα · A.
(2)
For the representation of α, the Pauli matrices are used,
α=
0 σ
σ 0
,
(3)
2
and
β=
12
0
0 −1 2
.
(4)
The potential V (r) is simply the Coulomb potential from
a point nucleus, i.e., we neglect retardation effects in the
electron-nucleus interaction and take the nuclear mass to be
infinite, thus allowing for separation between the electronic
and the nuclear degrees of freedom. We choose to work in
Coulomb gauge, ∇ · A = 0, with the external vector potential
A linearly polarized along the z-axis and propagating along
the x-axis;
A(η) = A0 f (η) sin(ωη + ϕ) ẑ ,
(5)
where η = t − x/c. The widely applied dipole approximation,
which is generally not applicable for the cases of interest here,
see, e.g., the discussion by Reiss [3], consists in substituting η
with the time t, i.e., neglecting the spatial dependence of the
laser pulse completely. The envelope function is chosen to be
sine-squared;
sin2 πη
, 0<η<T
T
f (η) =
.
(6)
0,
otherwise
To obtain the non-relativistic dynamics, the time-dependent
Schrödinger equation (TDSE) must be solved:
2
d
p
e
e 2 A2
i~ Ψ =
+ V (r) + p · A +
Ψ
(7)
dt
2m
m
2m
for the same vector potential.
A.
Propagation and projection
The state Ψ(t) is propagated by means of a second order
Magnus propagator,
Ψ(t + τ ) = exp[−iτ H(t + τ /2)]Ψ(t) + O(τ 3 ) .
(8)
One of the major advantages of a Magnus-type propagator
for the Schrödinger equation is stated clearly in Ref. [15]:
“In contrast to standard integrators, the error does not depend on higher time derivatives of the solution, which is in
general highly oscillatory”. Due to the stiffness inherent in
the mass energy term, i.e., the mc2 β term of the Hamiltonian in Eq. (2), this becomes even more advantageous for
the Dirac equation than for the Schrödinger equation. In
fact, the accuracy of many time-propagation schemes, such
as Crank-Nicholson and Runge-Kutta, suffer greatly from the
2mc2 energy splitting. In Ref. [16], e.g., it is stated that
“The major drawback of the Dirac treatment is the temporal
step size ∆t . ~/E required, which has to be significantly
smaller than for Schrödinger treatments, because of the large
rest mass energy mc2 that is contained in the particle’s total energy E.” Indeed, several works dealing numerically with
TDDE apply time-steps of the order 10−5 a.u. or smaller, see,
e.g., [7, 8, 11, 17]. However, for Magnus-type propagators,
lemma 4.1 of Ref. [15] provides an error bound for Eq. (8) in
which the time-independent mass term mc2 β does not enter at
all, thus circumventing the above problem. Extremely small
time steps are, of course, always required when resolving phenomena which really do take place at very short time scales,
such as Zitterbewegung [13].
In our implementation the propagator in Eq. (8) does not
tell the whole story. We have solved the Dirac equation on
the manifold consisting of only positive energy states – of the
time-dependent Hamiltonian, H(t), that is. Intuitively, this
choice may be motivated by Dirac’s original idea of the “filled
sea” of negative energy states, population of such states are
prohibited. We emphasize that in a time-dependent context it
is crucial to keep in mind that a time dependent Hamiltonian
leads to a time-dependent distinction between positive and
negative energy states; the Dirac sea is not calm, so to speak.
A more technical justification for excluding negative energy
states in the propagation would be that actual population of
such states would correspond to (real) pair production, thus
necessitating an approach based on field theory rather than a
conserved number of particles. Thus, in the present regime
it should not matter whether we include them or not. In our
implementation, however, exclusion is actually necessary due
to the use of complex scaling, see Sec. III below. Negative energy states have then positive imaginary parts, which causes
the norm of the state to blow up upon application of the propagator in Eq. (8). We note in passing that a correct field theory
treatment of pair-production in any case would require a distinction between positive and negative energy states. For the
latter to represent positrons they have to be propagated backwards in time [18], and thus the sign of the imaginary part
of the negative energy state continuum after complex rotation
would cause absorption of the positron states when they travel
outwards from the nucleus, just as is the case for the (positive
energy) electrons.
Following the above discussion, the propagator of Eq. (8)
is modified to
Ψ(t + τ ) = P(t + τ /2) exp(−iτ H(t + τ /2))Ψ(t) + O(t3 ) ,
(9)
where P(t) projects the state onto the time dependent subspace spanned by the positive spectrum of the Hamiltonian
H(t). This projection does not prevent the evolution from being unitary.
Before we continue to the implementation of the timedependent Dirac equation it is appropriate to discuss its nonrelativistic limit since from that some numerical complications that we have encountered can be understood.
B.
The non-relativitic limit of the light-matter interaction
While the time-dependent Dirac equation, Eq. (1) is linear
in the vector potential A, the Schrödinger equation, Eq. (7),
has both a linear and a quadratic term. We can study the nonrelativistic-limit of the time-dependent Dirac equation by using the form of the wave function given in Eq. (15) below and
3
write Eq. (1) as:
V F + cσ · (eA + p) G = i~
∂F
∂t
(10)
which can be rewritten as
∂G
,
(11)
cσ · (eA + p) F + V − 2mc2 G = i~
∂t
with F and G being the large and small component, respectively. For the sake of simplicity, we have omitted the indices. Here a Foldy-Wouthuysen [19] type transformation can
be performed: From the second line one gets (keeping only the
dominating term, i.e. assuming that the potential V 2mc2 ,
and that the time variation of the small component is modest)
1
σ · (eA + p) F.
2mc
and inserting this on the first line we get
G≈
(12)
∂F
1
2
(σ · (eA + p)) F = i~
2m
∂t
1
e~
∂F
2
V +
(eA + p) +
σ · B F = i~
,
2m
2m
∂t
VF +
(13)
j
i
(18)
Ref [22] has demonstrated that specific choices of B-spline
orders k1 and k2 controls the occurrence of the so-called spurious states, that are known to appear in the numerical spectrum after discretization of the Dirac Hamiltonian. While
the choice k1 = k2 contaminates the spectrum with spurious states the choices k1 = k2 ± 1 are reported to be stable
combinations that do not produce them.
The energy index n corresponds to both bound and pseudo
continuum states, with the latter referring to both the negative and the positive energy continuum. In general a rather
large number of continuum states are necessary in order to
achieve convergent results. This number is however drastically reduced by imposing complex scaling of the spatial coordinates,
r → r exp(iθ),
where again the A2 -term is reappearing. We see that the A2 term is implicitly present in the Dirac equation and enters the
equation for the large component of the wave function through
the small component. Whenever we try to formulate the interaction with the electromagnetic field through operators that
are diagonal with respect to the small and large component
it reappears. A practical consequence of this is that when the
spatial dependence of A is included to a specific order through
a Taylor expansion, cf. Sec. III A, the implicit A2 -term will
always be included to the square of this order. As a result,
higher order contributions in this expansion are needed for the
solution of the Dirac equation than previously was found for
the Schrödinger equation [20].
III.
where Ymlκl (θ, φ) is a spherical harmonic and χms is an eigenspinor. The radial components Pn,κ (r) and Qn,κ (r) are expanded in B-splines [21];
X
X
Qn,κ (r) =
bj Bjk2 (r).
Pn,κ (r) =
ai Bik1 (r)
IMPLEMENTATION
We expand our wave function in eigenstates of the unperturbed Hamiltonian H0 ,
X
Ψ(t) =
cn,j,m,κ ψn,j,m,κ ,
(14)
Fn,j,m,κ (r)
Gn,j,m,κ (r)
,
where
1
Pn,κ (r)Xκ,j,m (Ω)
Fn,j,m,κ (r)
=
Gn,j,m,κ (r)
r iQn,κ (r)X−κ,j,m (Ω)
(15)
. (16)
Here κ = l for j = l − 1/2 and κ = −(l + 1) for j =
l + 1/2, and Xκ,j,m represents the spin-angular part which
has the analytical form:
X
Xκ,j,m =
hlκ , ml ; s, ms |j, miYmlκl (θ, φ)χms
(17)
ms ,ml
(19)
A.
Interaction
With the pulse linearly polarized in ẑ the time dependent
part of the Hamiltonian becomes HI (t) = cαz A(η). As this
term depends on both time and space, t and x, a direct calculation of such couplings would have to be performed at each
and every time step in order to represent H(t) numerically.
This cumbersome feature may be removed by factorizing the
vector potential in a time-dependent and a spatially dependent
part,
A(η) ≈
nX
trunc
an T (t)X(x) .
(20)
n=0
with
ψn,j,m,κ (r) =
.
This renders the energies of bound states unaffected while
continuum energies acquire an imaginary contribution which,
effectively, broadens the energy of these states. Thus, the entire energy range is well represented by very few states [23].
The price to be paid for this advantage is that, numerically, we
lose information about the photo electron after ionization.
n,j,m,κ
with 0 ≤ θ ≤ π/4
By determining the couplings αz X(x) the numerical representation at any time is easily constructed by the above expansion. Such separations may be achieved by, e.g., a Fourier
expansion in η or a Taylor expansion in x. In Ref. [5] both
methods were used. In the Fourier implementation, the number of terms was minimized in two ways: First, by taking A
to have the pulse length T as period and, second, by neglecting the spatial dependence of the envelope f (η), c.f., Eq. (6).
Both of these approaches have severe shortcomings.
Thus we resort to a Taylor expansion in the present work:
A(η) ≈
nX
trunc
n=0
1 (n) x n
A (t) .
n!
c
(21)
4
Expressing xn as a sum of spherical harmonics Yµλ :
xn =
X
ci rn Yµλ
(22)
i
the interaction operator is given by terms of the form:
hnκjm|αz rγ Yµλ |ñκ̃j̃ m̃i =
i
Z h
∗
Pnκ
(r)rγ Qñκ̃ (r)hκjm|σ01 Yµλ |-κ̃j̃ m̃i−
r
i
Q∗nκ (r)rγ Pñκ̃ (r)h-κjm|σ01 Yµλ |κ̃j̃ m̃i dr.
HNR ≈
(23)
With the radial components expressed in B-splines, the integrals are computed to machine accuracy using GaussLegendre quadrature. To obtain the spin-angular part, the operator product can be expressed in a coupled tensor operator
basis:
σ01 Yµλ
=
λ+1
X
where just like the relativistic case, the radial component is
expanded in B-splines while the angular part is analytically
known.
The time-dependent Schrödinger equation is given in
Eq. (7), and also here the form of the vector field in Eq. (21)
will be used. Typically, the A2 term provides practically all
corrections to the dipole approximation [20, 24] so to first order (which Ref. [20] found sufficient) in x we have:
p2
e
e2 x
+ V (r) + pz A(t) −
A(t)A(1) (t) (28)
2m
m
mc
where the purely time dependent A(t)2 has been removed by
the gauge transformation:
i
Ψ̃(r, t) = e ~
Ψ(r, t).
(29)
The pz -couplings are given by:
l −ma
K
h10; λµ|KQi σ 1 Yλ Q ,
e2
A(ωt0 )2 dt0
0 2m
Rt
(-1) a
Q = −µ
la
-ma
hna la ma |pz |nb lb mb i =
1 lb
hna la ||p1 ||nb lb i
0 mb
(30)
while the x-couplings are computed as:
K=|λ−1|
With this choice the spin-angular part may be computed as:
λ+1
X
hκjm|σ01 Yµλ |κ̃j̃ m̃i =
(-1)
√
λ−1−Q
r
2K + 1
K=|λ−1|
K
1 λ K
×
hκjm| σ 1 Yλ Q |κ̃j̃ m̃i
0 µ −µ
2π
l −m
· (-1) a a
3
with the reduced matrix element given by:
q
K
hj|| σ 1 Yλ
||j̃i = (2j + 1)(2K + 1)(2j̃ + 1)


 l ˜l 1 
σ 1 ||s̃ihl||Yλ ||˜li s s̃ λ
×hs||σ
.
 ˜ 
j J K
(25)
(26)
With this scheme the couplings induced by αz xn , for n =
0, 1, . . . , ntrunc are readily computed.
B.
Non-relativistic interaction
The non-relativistic spectral basis has eigenfunctions of the
form
Φ(r) =
Pnl (r) l
Ym (θ, φ)
r
la
-ma
(27)
hna la ma |x|nb lb mb i =
1 lb
la 1 lb
−
-1 mb
-ma 1 mb
×hna la ||rY1 ||nb lb i.
(31)
(24)
The Wigner-Eckart theorem can be applied to the matrix ele
K
ment of the combined operator σ 1 Yλ Q :
K
hκjm| σ 1 Yλ Q |κ̃j̃ m̃i =
K
j K j̃
j−m
(-1)
hj|| σ 1 Yλ
||j̃i
−m Q m̃
C.
Propagation
The main drawback of the propagator projected onto the
positive energy spectrum of the Hamiltonian H(t), given in
Eq. (9) is the need for a full diagonalization of the Hamiltonian at each and every time step. This is not only required in
order to exponentiate the Hamiltonian but also to distinguish
between positive and negative energy states so that the projection P can be performed. To this end, Krylov subspace
methods are quite useful [13, 25]. Such methods provide accurate approximations to the action of an exponential of an
operator on a specific vector. Moreover, they may typically be
implemented numerically very efficiently. At each time step
the Krylov subspace of dimension m, Km (t), is spanning the
set of states obtained by iteratively multiplying the state with
the Hamiltonian,
n
[H(t + τ /2)] Ψ(t),
n = 0, ..., (m − 1)
.
(32)
Instead of diagonalizing the full Hamiltonian, we diagonalize
its projection into K(t) – with the dimension m considerably
smaller than the full dimension of the numerical problem. In
fact, high accuracy is achieved with surprisingly low m. By
exponentiating and projecting according to Eq. (9), propagation is performed within K(t), and the approximate state vector at time t + τ is constructed by transforming back to the
original Hilbert space.
5
To decrease memory consumption H(t + τ /2) is actually
never explicitly built even in sparse form but its action is
accounted for by directly operating with couplings and time
factors on Ψ(t). The bottleneck is then no longer the diagonalization but the operations representing the matrix-vector
product in Eq. 32. To reduce the computational time the
operations are distributed to several processing units.
D.
The representation of the vector field beyond the dipole
approximation
As discussed above we have chosen to represent the vector field through a Taylor expansion, see Eq. 21. Ref. [20],
which considered the time-dependent Schrödinger equation,
provides strong support that below the relativistic region, the
first order term alone (i. e. ntrunc = 1) provides practically
all corrections to the dipole prediction. At first sight there is
no a priori reason why this conclusion should not apply to the
relativistic situation as well. Yet, we typically have to resort
to a third order expansion in our calculations. The underlying problem is the implicit inclusion of the A2 term in the
Dirac-equation, discussed in Sec.II B above. When only first
order corrections to A are included some of the second order corrections in the implicit A2 -term will still be included,
namely the ∼ (A(1) )2 x2 contributions, while others, namely
the ∼ A(2) A(0) x2 contributions, require the inclusion also of
second order corrections to A. We note that Førre and Simonsen [20] has shown that in the non-relativistic limit there
are important cancellations between the x2 -terms. It is thus
natural to assume that these cancellations are a key-issue and
that higher-order correction terms to A are needed to achieve
them when using the Dirac equation [26]. The fact that the
instability problem is drastically reduced when higher orders
of x is included in Eq. (21) supports this conclusion.
The solution of the TDSE has been done along the same
lines as that of the TDDE.
IV.
RESULTS AND DISCUSSION
For the relativistic basis we use B-splines of orders k1 = 7,
k2 = 8 and apply the boundary conditions:
Pn,κ (0) = Pn,κ (rmax ) = 0
uniform complex scaling angle θ = 2.5◦ to avoid artificial reflection at the box end. This choice is motivated by the recent
non-relativistic study in Ref. [27] where a similar box was
used for pulses with the same central frequency, a maximum
electric field strengths up to 60 a.u. and number of optical cycles up to Nc = 100. Although the maximum electric field
strength considered in this work is E0 = 110 a.u., the much
shorter pulses, and thus also propagation times, should ensure
that the box is adequate to describe the ionization dynamics.
We have studied the predictions of the ionization probability, Pion , and compared the results obtained by the TDSE and
TDDE. In all computations convergence with respect to the
maximum orbital angular momentum included, lmax , have
been checked.
In Fig. 1 the relativistic and non-relativistic ionization probability, Pion , computed within the dipole approximation are
shown. For low field strengths the ionization probability is increasing linearly with increasing field strength, but for around
E0 ≈ 10 a.u. the so-called stabilization against ionization sets
in [28]. For even higher field strengths the ionization starts
to increase again. Although, as will be seen in Fig. 2, the
FIG. 1: Comparison of TDSE and TDDE calculations within the
dipole approximation. Each vertical line corresponds to an increment
of vquiv = 0.1c. A difference is emerging at about E0 = 70 a.u.,
corresponding to vq ≈ 0.15c. Both simulations have been carried
out with lmax = 40.
Qn,κ (0) = Qn,κ (rmax ) = 0 .
dipole approximation in itself is not valid for regions where
(33)
vquiv > 0.15c, it is interesting to note that the relativistic efFor the non-relativistic basis we choose k = 7 and
fect seem to result in a tiny suppression of the ionization yield.
In Fig. 2 the results from solutions of the TDDE using the
Pn,l (0) = Pn,l (rmax ) = 0 .
(34)
Taylor expansion of the vector field, Eq. 21, with ntrunc =
1 − 3 are shown on top of the non-relativistic calculations for
The ionization probability, Pion , of the hydrogen atom ground
E0 ≤ 60 a.u. Note that the dipole approximation is breaking
state electron after interaction with a pulse having the fixed
down for E0 ≥ 30 a.u. The effects beyond the dipole apparameters:
proximation causes the ionization yield to increase further for
higher strengths, the physics behind this behaviour has been
ω = 3.5, φ = 0, T = 2πNc , Nc = 15
(35)
discussed in Ref. [27].
has been studied as a function of E0 . The calculations are carThe vertical line denotes the electric field strength correried out in a box of rmax = 40 a.u. using 120 B-splines and a
sponding to a quiver velocity of the electron vquiv = 0.1c.
6
FIG. 2: Comparison of TDDE and TDSE in the non-relativistic
regime. The numbers trailing the acronym ”BYD” are the numerical value of ntrunc in eq. 21. For all cases lmax = 15 have been
used. The parameters are the same as in Fig. 1
For electric field strengths below and slightly above this value
the TDSE and TDDE should agree. However, the relativistic
first order approximation of the spatial dependence severely
overestimates the ionization yield. Addition of the second order corrections gives initial agreement but is then seen to underestimate the Pion predicted by TDSE. Finally, inclusion of
also the third order term in eq. 21 gives excellent agreement
with TDSE up to E0 = 60 a.u. Thus, relativistic effects must
come in play at yet higher field strengths.
We note that the smoother behaviour of the TDDE results
with increasing spatial order in the vector field supports the
argument made in section II B - there is clearly a complicated
implicit inclusion of higher order terms in the TDDE when
a certain order of the Taylor expansion of the vector field,
Eq. (21) is employed.
A.
FIG. 3: A comparison of the ionization probability, Pion , computed
with and without the time-independent negative-energy states in the
basis.
E0 = 80 a.u the three TDDE curves with lmax = 15, 20, 30
converge slightly above the TDSE curve. This indicates that
there is a small relativistic shift emerging here where vquiv ≈
0.17c. When we increase the field strength even more the
ionization probabilities, Pion , predicted by TDDE and TDSE
separate further, suggesting that the relativistic effect is increasing the ionization yield, rather than decreasing it as predicted within the dipole approximation.
Removing time independent negative energy states
In Ref. [5] it was shown that exclusion of the timeindependent negative-energy states from the propagation basis removed all effects beyond the dipole approximation but
had no effect within the dipole approximation. As a further
demonstration of this the TDDE calculations within the dipole
approximation, as well as calculations including the first and
third order corrections to this approximation, was repeated
without the time independent negative energy states. In Fig. 3
it is seen that these three different Hamiltonians now give the
same ionization probability, Pion .
B.
Relativistic effects
For relativistic effects to become visible E0 must now be
further increased. As this is done convergence with respect to
lmax must once again be checked. Fig. 4 shows results obtained for electric field strengths up to E0 = 110 a.u. For
FIG. 4: Comparison of TDSE and TDDE for higher peak-values of
the electromagnetic field, E0 . The vertical line indicates vquiv =
0.2c.
V.
CONCLUSION
We have solved the time-dependent Dirac equation for a hydrogen atom exposed to extreme laser pulses. Upon comparison with the non-relativistic counterpart, i.e., the Schrödinger
equation, it was found that effects beyond the dipole approximation are more complicated to incorporate correctly in the
relativistic framework. Whereas first order space-dependent
corrections to the vector field are sufficient for TDSE, TDDE
demands an expansion to at least third order for convergence.
Regarding the ionization yield, tiny relativistic effects for the
test case start to appear for a quiver velocity of vquiv ≈ 0.17c.
7
The relativistic corrections to Pion at E0 = 110 a.u, corresponding to vquiv ≈ 0.23c, is about 2%.
tations were performed at the cloud computing facility “Alto”
at Oslo and Akershus University College of Applied Sciences.
Financial support by the Swedish Research Council (VR) is
gratefully acknowledged.
Acknowledgments
The authors would like to thank M. Førre for fruitful discussions and for providing us with data for comparison. Compu-
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