TIME-DEPENDENT STUDIES OF FUNDAMENTAL
ATOMIC PROCESSES IN RYDBERG ATOMS
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee. This
dissertation does not include proprietary or classified information.
T?urker Topc?u
Certificate of Approval:
Michael S. Pindzola
Professor
Physics
Francis Robicheaux, Chair
Professor
Physics
Eugene Oks
Professor
Physics
Allen Landers
Assoc. Professor
Physics
Joe F. Pittman
Interim Dean
Graduate School
TIME-DEPENDENT STUDIES OF FUNDAMENTAL
ATOMIC PROCESSES IN RYDBERG ATOMS
T?urker Topc?u
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
4 August 2007
TIME-DEPENDENT STUDIES OF FUNDAMENTAL
ATOMIC PROCESSES IN RYDBERG ATOMS
T?urker Topc?u
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
DISSERTATION ABSTRACT
TIME-DEPENDENT STUDIES OF FUNDAMENTAL
ATOMIC PROCESSES IN RYDBERG ATOMS
T?urker Topc?u
Doctor of Philosophy, 4 August 2007
(M.Sc., Auburn University, 2005)
185 Typed Pages
Directed by Francis Robicheaux
This dissertation consists of four time-dependent studies of various fundamental atomic
processes involving highly excited atoms and ions. We use time-dependent close-coupling
theory besides other perturbative and and non-perturbative techniques, which has become
standard in studying time-dependent atomic dynamics, to investigate these processes.
The atomic processes studied in the following chapters include photo-double ioniza-
tion of He from ground and the first excites states, electron-impact ionization of highly
excited hydrogen and hydrogen-like ions, radiative cascade from highly excited states of
hydrogen in the presence of strong external magnetic fields, and chaotic ionization of a
hydrogenic Rydberg wave packet in parallel electric and magnetic fields. Besides their
obvious fundamental importance for advancing our understanding of the physics of highly
excited atomic systems, all these problems have applications which are of practical and
fundamental importance.
iv
Throughout this work, we make an effort to carry out calculations that are inherently
different from our quantum mechanical non-perturbative time-dependent calculations to
benchmark and contrast our results with those that are either perturbative or classical.
We make special effort to carry out classical calculations, where possible, to compare our
quantum mechanical treatments with their classical counterparts for studying the classical-
quantum correspondence for various different highly excited systems.
v
ACKNOWLEDGMENTS
I feel very much indebted and privileged to have worked with my Ph.D. advisor Prof.
Francis Robicheaux, who has carefully guided my research. I would to thank my advisor
for his sincere attention and help in guiding and supporting me through my research as a
graduate student, while greatly enhancing my understanding of atomic physics, as well as
physics in general. I must also acknowledge my gratitude to Prof. Francis Robicheaux for
providing the data for the classical calculations presented in Chap. 4 and for providing the
CTMC code that was used to generate classical results of Chap. 6.
I also would like to thank to rest of the members in my Ph.D. committee, namely, Prof.
Michael S. Pindzola, Prof. Eugene Oks, and Assoc. Prof. Allen Landers for their time and
suggestions during the preparation of this work. I must further thank Prof. Michael S.
Pindzola for many elucidating and stimulating comments and discussions that helped me
extend my understanding of atomic physics.
I am grateful to Assoc. Prof. Stuart D. Loch for helping me with various data visual-
ization software and for conversations that have constituted the core of my understanding
of the role of atomic processes in fusion plasmas.
I would like thank my family, who despite being abroad, kept their emotional support
constant during my years away from home.
At last, I must thank the Office of Basic Energy Sciences and the Office of Fusion
Energy Sciences, U.S. Department of Energy for their continued support for the research
presented in this dissertation.
vi
Style manual or journal used Transactions of the American Mathematical Society
(together with the style known as ?auphd?). Bibliograpy follows the style used by the
American Physical Society.
Computer software used The document preparation package TEX (specifically LATEX)
together with the departmental style-file auphd.sty.
vii
TABLE OF CONTENTS
LIST OF FIGURES xi
1 INTRODUCTION 1
1.1 Photo-double ionization of two electron atoms . . . . . . . . . . . . . . . . 2
1.2 Electron-impact ionization from excited states . . . . . . . . . . . . . . . . 3
1.3 Radiative cascade in strong magnetic fields . . . . . . . . . . . . . . . . . 3
1.4 Chaotic ionization of a highly excited hydrogen atom in external electric
and magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 NUMERICAL GRID METHODS FOR SOLVING THE TIME-DEPENDENT AND THE
TIME-INDEPENDENT SCHR ?ODINGER EQUATIONS FOR ATOMIC SYSTEMS 6
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Grid techniques for the study of time-dependent atomic dynamics . . . . . 7
2.2.1 Square-root Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Methods for time-propagation of the time-dependent Schr?odinger Equation 12
2.3.1 Explicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Implicit Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.3 Split-Operator Technique . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . 23
2.5 Green?s Function Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.6 Time-independent methods for generating the one- and two-electron orbitals 27
2.6.1 The Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6.2 The Relaxation Methods . . . . . . . . . . . . . . . . . . . . . . . 28
3 THE PHOTO-DOUBLE IONIZATION CROSS SECTION OF HELIUM NEAR THRESH-
OLD IN A COLLINEAR s-WAVE MODEL 30
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Calculations and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 ELECTRON IMPACT IONIZATION OF HIGHLY EXCITED HYDROGEN-LIKE IONS IN
A COLLINEAR s-WAVE MODEL 47
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
viii
4.2.1 The time-dependent wave packet method using collinear s-wave
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Classical trajectory Monte Carlo method . . . . . . . . . . . . . . 54
4.2.3 Collinear distorted-wave method . . . . . . . . . . . . . . . . . . . 57
4.2.4 Collinear R-Matrix method . . . . . . . . . . . . . . . . . . . . . 59
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3.1 Effect of using a model potential: 1r1+r2 versus 1r> . . . . . . . . . . 60
4.3.2 Electron-impact ionization of hydrogen . . . . . . . . . . . . . . . 61
4.3.3 Electron-impact ionization of B4+ . . . . . . . . . . . . . . . . . . 64
4.3.4 Effect of the ion stage . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3.5 Node structure near the scattering center . . . . . . . . . . . . . . . 68
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5 RADIATIVE CASCADE OF HIGHLY EXCITED HYDROGEN ATOMS IN STRONG
MAGNETIC FIELDS 76
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.2.1 Calculation of the Energy Spectrum in the Magnetic Field . . . . . 79
5.2.2 Calculation of the Dipole Matrix Elements . . . . . . . . . . . . . 82
5.2.3 Solution of the Rate Equation . . . . . . . . . . . . . . . . . . . . 84
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.3.1 Rough Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.3.2 Energy Spectrum in the Magnetic Field . . . . . . . . . . . . . . . 89
5.3.3 Radiative Cascade Starting from Completely Random (?;m) Dis-
tribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
5.3.4 Radiative Cascade Starting from High-jmj Distribution . . . . . . . 96
5.3.5 Effect of Black Body Radiation . . . . . . . . . . . . . . . . . . . 99
5.3.6 Semiclassical Treatment . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
6 CHAOTIC IONIZATION OF A HIGHLY EXCITED HYDROGEN ATOM IN PARALLEL
ELECTRIC AND MAGNETIC FIELDS 110
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
6.2.1 Quantum mechanical time-dependent wave packet method . . . . . 114
6.2.2 Classical trajectory Monte Carlo method . . . . . . . . . . . . . . 121
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3.1 Flux of ionizing electrons . . . . . . . . . . . . . . . . . . . . . . 123
6.3.2 l-distributions of ionizing electrons . . . . . . . . . . . . . . . . . 132
ix
6.3.3 Angular distribution of the ionizing pulse trains on the detector . . . 134
6.3.4 Effect of the angular distribution of the source term . . . . . . . . . 137
6.3.5 Effect of core scattering . . . . . . . . . . . . . . . . . . . . . . . 144
6.3.6 Transition to regularity - the case of high magnetic fields . . . . . . 147
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
7 SUMMARY 159
BIBLIOGRAPHY 163
x
LIST OF FIGURES
3.1 PDPI=tpeak versus excess energy for photo-double ionization of He(1s2)
for different boxes and pulse durations. . . . . . . . . . . . . . . . . . . . . 37
3.2 PDPI=tpeak versus excess energy for photo-double ionization of He(1s2s)
for different boxes and pulse durations. . . . . . . . . . . . . . . . . . . . . 38
3.3 PDPI=tpeak as a function of time for double ionization out of He(1s2). . . . 40
3.4 PDPI=tpeak as a function of time for double ionization out of He(1s2s). . . 41
3.5 Calculated PDPI=tpeak (?) and the fitted convolved Wannier power law (?)
versus excess energy for photo-double ionization of He(1s2). The upper
panel is showing the difference between the two. . . . . . . . . . . . . . . 43
3.6 Calculated PDPI=tpeak (?) and the convolved Wannier power law (?) ver-
sus excess energy for photo-double ionization of He(1s2s). The upper
panel is showing the difference between the two. . . . . . . . . . . . . . . 44
4.1 Scaled final energy of the classical bound (solid curve) and incoming
(dashed curve) electrons versus the initial phase, ?, of the bound electron.
The initial position and velocity of the electron, ? and ?, are found by
solving the classical equations of motion for a time given by ? times the
Rydberg period with ?(0) = 1 and ?(0) = 0. This graph shows the result
for Z = 1 and Esc = 9:5. Note that the width of the region where ? is
positive for both electrons (shown in the insert) is the ionization probability. 58
4.2 Ionization probability out of n = 1 for H versus the scaled energy of the in-
coming electron from collinear s-wave DW (solid curve), Temkin-Poet DW
(dot-dash curve), collinear s-wave RMPS (dash curve) and Temkin-Poet
RMPS (dot-dot-dash curve) calculations. Note that the agreement between
the DW and RMPS is better for the weaker collinear s-wave model poten-
tial. The data are fitted using least squares method to obtain the smooth
curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
xi
4.3 Ionization Cross sections for e?-H(1S) scattering from n up to 4 calculated
in collinear DW, CTMC, RMPS, and s-wave TDCC methods as a function
of the scaled energy Esc of the incoming electron. . . . . . . . . . . . . . . 63
4.4 Ionization Cross sections for e?-B4+(1S) scattering from n = 1;2;4 and 8
calculated in collinear DW, CTMC, and s-wave TDCC methods as a func-
tion of the scaled energy of the incoming electron. . . . . . . . . . . . . . . 65
4.5 CTMC versus s-wave TDCC method for the electron impact ionization of
B4+ for up to n = 25 for the 1S symmetry of the initial wave packet. . . . . 66
4.6 CTMC versus s-wave TDCC method for the electron-impact ionization of
B4+ for up to n = 25 for the 3S symmetry of the initial wave packet. . . . . 67
4.7 Ratio of the quantal and classical ionization probabilities,
PQM(n;Z)=PCM(Z), as a function of n for Z = 1?6. . . . . . . . . . . 69
4.8 ln[PQM(n;Z)] versus ln(Z) for various n at Esc = 9:5. Note that the
straight lines have a slope of 2 which implies ? 1=Z2 scaling of the ioniza-
tion probability for n = 1 and 4 and for higher Z of n = 16 and 25. . . . . 70
4.9 Contour plots forj?n=16(r1;r2)j2 for H (top panel) and B4+ (bottom panel)
at the time of scattering from the nucleus at Esc = 9:5. The lines in each
figure represent classical trajectories that lead to ionization. . . . . . . . . . 73
5.1 Flow of probability in the completely ?;m mixed distribution for n = 35
in a 4:0 T field as a function of time. Each curve represents the total proba-
bility of finding the atom in a state whose energy is in one of the particular
energy ranges indicated on the legend. . . . . . . . . . . . . . . . . . . . . 91
5.2 Time required to populate the ground state of the hydrogen atom by (a) 10%
and (b) 50% as a function of the effective hydrogenic principal quantum
number n for 1:0 T, 2:0 T, 3:0 T and 4:0 T fields. Initial distribution of
states is completely ?;m mixed. . . . . . . . . . . . . . . . . . . . . . . . 93
5.3 Scaled time required to populate the ground state of the hydrogen atom by
(a) 10%, (b) 50% and (c) 90% as a function of the effective hydrogenic
principal quantum number n for 1:0 T, 2:0 T, 3:0 T and 4:0 T fields. Initial
distribution of states is completely ?;m mixed. . . . . . . . . . . . . . . . 95
xii
5.4 Time required to populate the ground state of the hydrogen atom by 10%
as a function of the effective hydrogenic principal quantum number n for
1:0 T and 4:0 T fields. Initial distribution of states only involves states with
high jmj. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.5 Scaled time required to populate the ground state of the hydrogen atom by
(a) 10%, (b) 50% and (c) 90% as a function of the effective hydrogenic
principal quantum number n for 1:0 T and 4:0 T fields. Initial distribution
of states only involves states with jmj? n. . . . . . . . . . . . . . . . . . 100
6.1 Current-time plots for ionization with excitation to (a) n = 40, (b) n = 60,
and (c) n = 80 states of hydrogen via a short laser pulse in parallel electric
and magnetic fields. Results from quantum mechanical wave packet (solid
curve) and CTMC (dotted curve) calculations are plotted on top of each
other along with the pure electric field case (dash-dotted curve). (d) shows
just the quantum mechanical current-time plots in (a), (b), and (c) plotted
together for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Classical calculations of Ref. [77] for the ionization of hydrogen in parallel
electric and magnetic fields of F = 19 V/cm and B = 0:49 T for n = 80.
(a) shows the ionization rate as function of scaled time, and (b) shows the
scaled time for a trajectory to escape as a function of its launch angle. The
solid curve in (a) is calculated using an ensemble of trajectories with a pre-
cise launch time and energy whereas the smooth solid curve is calculated
using a Gaussian distribution for the energy and the launch time. Each of
the escape segments seen in (b) correspond to a particular pulse in the ion-
ization rate and are matched with their pulse via the dotted lines. The scape
segment that gives rise to the direct pulse peaks at = ? and most of it falls
outside the range plotted. (We thank K. A. Mitchell for kindly supplying
the data for this partial replot of Fig. 2 of Ref. [77]. We also thank K. A.
Mitchell and J. B. Delos for their permission to include this replot in this
paper.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Snapshots of the absolute value squared wave function of hydrogen in par-
allel electric and magnetic fields for n = 80 case at (a) 58 ps, (b) 117 ps,
(c) 175 ps, (d) 233 ps, (e) 292 ps, and (f) 350 ps. In (b), marks the part
of the wave function that shows Stark oscillation whereas fl shows the one
that gives rise to the direct ionization pulse in Fig. 1(c). In (c), ? shows
the part of the wave function that is responsible for the second peak in the
pulse train. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
xiii
6.4 Time evolution of the angular momentum distribution of the electrons
within the space bounded by the spherical detector at (a) 44 ps, (b) 58
ps, (c) 85 ps, (d) 129 ps, (e) 234 ps, and (f) 338 ps for n = 80. The inserts
show the angular momentum distributions of the electrons at corresponding
times for the pure electric field case for which the time dependent dynamics
is regular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.5 Time evolution of the classical (dotted curve) and quantum (solid curve)
angular distributions of the electrons hitting the detector at (a) 85 ps, (b)
107 ps, (c) 151 ps, (d) 173 ps, (e) 289 ps, and (f) 347 ps for n = 80. . . . . 135
6.6 (a) Current-time plots from quantum mechanical wave packet calculations
for various angular distributions of the source term. All the angular distri-
butions we tried have at most a single node in the angular range [0;?] and
are plotted as a function of in (c). The solid curve has no nodes whereas
the dash-dotted curve has node at = ? rad, dashed curve at = 1:28 rad,
and the dotted curve at = 1:9 rad. Figure (b) shows ionization current
from the CTMC calculations for various exclusions from the angular range
of the launch angle shown in (d). The solid curve allows this entire range
whereas the dotted curve completely excludes it from its allowed launch
angles. The dash-dotted and dashed curves are obtained by excluding the
angular ranges [1:8;2:123] rad and [1:998;2:035] rad from the distribution
shown in (d). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.7 Quantum mechanical (a) and classical (b) current-time plots with the inclu-
sion of core scattering. In (a) the solid curve is for C = ?1 and a = 0:05
and the dotted curve is for C = ?1 and a = 0:5. The dashed curve shows
the pure electric field case in the absence of chaos. In (b) again the solid
curve is for C = ?1 and a = 0:05 and the dotted curve is for C = ?1
and a = 0:5. The difference is less prominent compared to the quantum
mechanical case in (a). (c) and (d) show the model core potentials curves
(solid lines) and the field free atomic potential curves for l = 0, 1, and 2
for a = 0:05 and a = 0:5 respectively. All the calculations are for n = 80. . 146
6.8 Quantum mechanical (solid curves) and classical (dashed curves) current-
time plots for four different scaled energies quantified by fi; (a) fi = 0:5,
(b) fi = 1:0, (c) fi = 2:0, and (d) fi = 4:0. All the calculations are for n = 80.149
xiv
6.9 Time-dependent autocorrelation functions for (a) fi = 1:0 and (b) fi = 4:0.
(c) and (d) are the spectral autocorrelation functions that correspond to (a)
and (b) respectively. The time-dependent autocorrelation functions clearly
show two types of oscillations whereas the spectral autocorrelation func-
tions show equally spaced resonances which indicates Landau quantization. 153
xv
CHAPTER 1
INTRODUCTION
In the last decade, the efforts made in studying highly excited atomic systems from
both experimental and theoretical point of view has substantially escalated. Advancements
made in cooling atoms down to only a few Kelvins made it possible to study atoms almost
individually without having to suffer from the effects of collisions and thermalization. This
has opened the possibility of studying highly excited atomic systems very efficiently from
the experimental point of view, which otherwise has proven impossible due to the large
dipole moments of these atomic systems.
Besides many immediate practical applications of Rydberg systems which we discuss
below, study of highly excited systems is also very important from the standpoint of ad-
vancing our understanding of classical-quantum correspondence. Due to their large quan-
tum numbers and high energies, Rydberg electrons in many cases can be treated classically
in accordance with the Bohr correspondence principle. According to the most commonly
adopted definition of the Bohr correspondence principle, any given quantum mechanical
system in a bound state should converge to its classical counterpart in the case of large
quantum numbers. Even though the Bohr correspondence principle is one of the corner-
stones of the early quantum physics, systems have been found that violates it [1]. There-
fore, study of fundamental atomic processes in Rydberg systems provides a very suitable
arena for investigating and understanding the classical-quantum correspondence.
1
Apart from its obvious fundamental importance, atomic processes involving highly
excited states also have important applications from modeling the plasma flow inside the
fusion tokamaks to finding efficient ways to get the highly excited anti-hydrogen atoms to
their ground states to study CPT violation in the universe. Below, we briefly discuss the
importance and applications of the problems that are included in this dissertation.
1.1 Photo-double ionization of two electron atoms
One of the most fundamental problems in atomic physics is the three-body Coulomb
breakup, which has long attracted the attention of both theorists and experimentalists. The
importance of this problem arises mainly because of our inability in describing analyti-
cally the two escaping electrons resulting from absorption of a photon in the presence of a
Coulombic core. The presence of the electron-electron interaction is vital for this process
to take place, which also makes this problem fundamentally important in understanding
the role of the electron correlations in the escape process. In this study, we particularly
pay attention to the double ionization near threshold where both electrons move slowly
and therefore can be approximately treated classically. Classical treatment of this problem
resulted in threshold laws describing the energy dependence of the ionization cross section
near threshold. We perform non-perturbative time-dependent calculations to investigate the
threshold behaviors predicted by these classical threshold laws.
2
1.2 Electron-impact ionization from excited states
Electron-impact scattering from hydrogen-like ions is an important atomic process
for modeling of tokamak plasmas and performing plasma diagnostics. Therefore, it is
important to have ionization data accurate enough to satisfactorily model and diagnose
laboratory plasmas. Usually, the cross sections for electron-impact ionization are obtained
from fully quantal calculations for ionization from the lowest states and from classical
calculations for ionization from highly excited states [2]. It has been shown [3] that
for denser plasmas the effective ionization rate changes substantially when contributions
through higher n levels are included in addition to the ground and metastable states. The
sensitivity of the plasma models to the atomic data illustrates the need for the investigation
of the extent to which the classical cross sections are accurate.
1.3 Radiative cascade in strong magnetic fields
Formation of highly excited anti-hydrogen atoms have been reported by two experi-
mental groups where cold anti-protons are merged with a cold trapped positron plasma at
roughly [4] 16 K and [5, 6] 4 K in magnetic fields of about 3 T and 5:4 T respectively.
The goal is to perform Lorentz and CPT violation checks in the hyperfine spectrum of the
2s ! 1s transition [7] for which the transition frequency is accurately known to about 1
part in 1014 for atomic beam of hydrogen [8] and to about 1 part in 1012 for trapped hy-
drogen [9]. The dominant process in the formation of the anti-hydrogen atoms is believed
to be three body recombination [10] which yields a small fraction of anti-hydrogen atoms
3
that are suitable for being laser stimulated down to the low-n states [11]. It has also been
noted that the anti-hydrogen atoms formed through three body recombination are likely to
be in highly excited m states [12]. Since Lorentz and CPT violation experiments require
ground state anti-hydrogen atoms, the highly excited anti-hydrogen atoms need to decay
down to the ground state in order to serve this purpose. The knowledge of the decay rates
and the time it takes to cascade down to the ground state from initial distributions of Ryd-
berg states with jmj? n and from completely random ?;m distributions of Rydberg states
in strong magnetic fields are therefore necessary.
1.4 Chaotic ionization of a highly excited hydrogen atom in external electric and
magnetic fields
Rydberg atoms and wave packets have been particularly useful in studying how clas-
sically chaotic dynamics is manifested in their quantum mechanical counterparts. This is
because Rydberg atoms provide a real non-separable physical system with just a few de-
grees of freedom whose classical counterpart exhibits chaos in external magnetic and com-
binations of magnetic and electric fields. Rydberg atoms in various combinations of static
[13, 14, 15] fields have also been studied and several useful theoretical and computational
tools have been developed during this process [16, 17]. In Chap. 6, we study ionization of a
hydrogen atom excited to a Rydberg wave packet in external parallel electric and magnetic
fields, whose classical counterpart have been shown to display chaos in the time domain.
We investigate various time dependent properties of this system and successfully reproduce
the classical results within a fully quantum mechanical framework.
4
The outline of this works is as follows. In Chap.2 we develop the general fundamen-
tal properties of the time-dependent methods we use throughout this work. We then turn
our attention to the photo-double ionization of He from ground and singlet first excited
state in Chap.3 to investigate the threshold law for double ionization. In Chap.4, we study
electron-impact ionization of highly excited hydrogen and hydrogen-like ions as function
of their principle quantum number and ion stage, and compare our non-perturbative results
with those from classical trajectory Monte Carlo calculations to investigate the extent of
which the Bohr correspondence principle holds. In Chap.5, we study radiative cascade of a
highly excited hydrogen atom in a strong external uniform magnetic field and calculate the
decay rates and lifetimes of its states in a fully quantum mechanical framework. Finally,
in Chap.6, we delve into the chaotic dynamics of a hydrogenic Rydberg electron whose
classical counterpart in time domain displays classical chaos. We calculate the ionization
current from a hydrogenic Rydberg wave packet excited by a short laser pulse as a function
of time in external uniform parallel electric and magnetic fields. We study the angular dis-
tributions of the electrons on a spherical detector through which we calculate the ionization
current and investigate the angular momentum distribution of the ionizing electrons as a
function of time. We further, study the onset of chaos as function of the scaled energy for
different magnetic field strengths.
5
CHAPTER 2
NUMERICAL GRID METHODS FOR SOLVING THE TIME-DEPENDENT AND THE
TIME-INDEPENDENT SCHR ?ODINGER EQUATIONS FOR ATOMIC SYSTEMS
2.1 Introduction
In this chapter, I describe the general characteristics of the numerical methods that will
be used in the following chapters. The numerical methods that are used in this dissertation
mainly employs time-dependent grid techniques which require discretizing the space and
time coordinates relevant to a given physical problem. Therefore, I will begin by describing
the methodology used in discretizing space which enables us to study atomic systems on
spatial lattices. Then, I will describe general characteristics of the time-dependent methods
that are used in the following chapters for time propagating the time-dependent Schr?odinger
equation in one and two spatial dimensions. Finally, I will conclude by discussing the
numerical methods used to generate the one- and two-electron orbitals by solving the time-
independent Schr?odinger equation. The discussions of this chapter are general in nature
and the specifics of the theoretical and numerical methods that are employed for each of
the problems in the following chapters are discussed in detail within each chapter. I will use
atomic units in all of the following discussions where the mass and charge of an electron is
taken to be unity and the unit of distance is a Bohr radius.
6
2.2 Grid techniques for the study of time-dependent atomic dynamics
Due to our inability to carry out exact analytical solutions to most of the problems
we encounter in physics, many numerical methods have been invented for the study of
complicated physical systems. With the advent of powerful computers that can carry out
numerical simulations, this vast amount of numerical techniques gave rise to the birth of
computational physics. Today?s powerful computers open up the road to studying and un-
derstanding many physical systems which would otherwise be impossible to cope with. For
our purposes, simulating an atomic system on a computer means storing and manipulating
actual wave functions and orbitals that are defined in the continuous space-time. Since
we cannot represent a continuous function on a computer, representing these functions re-
quires discretization of space and time. Discretization of space leads to the concept of grid,
or lattice, which is a set of points in space on which the wave functions and orbitals are
calculated, stored and manipulated.
Since the orbitals and wave functions we represent on a lattice are actually continuous
functions, it is important to define the lattice such that all the lattice spacings are smaller
than the smallest distance scales involved in the problem in order to have a good represen-
tation of the physical system at hand. As an example, consider the momentum of a plane
wave calculated on a uniform grid of points. Whereas the momentum of a one-dimensional
7
plane wave eikr is k, representing this plane wave on a grid gives,
peikr = ?ie
ik(r+?r) ?eik(r??r)
2?r
= ? i2?r(eik?r ?e?ik?r)eikr
= sin(k?r)?r eikr: (2.1)
Note that as ?r goes to zero, all the terms in the Taylor expansion of sin(k?r)=?r vanishes
except the first, and sin(k?r)=?r converges to k as expected. As a result, the kinetic energy
of the outgoing plane wave eikr on a uniform grid is
1
2p
2eikr = ?1
2
eik(r+?r) ?2eikr +eik(r??r)
(?r)2
= ? 12(?r)2(eik?r ?2+e?ik?r)eikr (2.2)
= 1?cos(k?r)(?r)2 eikr:
This means the plane wave eikr with momentum k becomes a plane wave with kinetic
energy (1 ? cos(k?r))=(?r)2 on a uniform grid of points. Again, note that as ?r ! 0,
eigenvalue of the kinetic energy operator in Eq. 2.2 converges to k2=2 as expected.
To make sure that we have accurate lattice representations of the atomic systems we
are studying in the following chapters, we take special care for performing convergence
checks to make sure that the lattices we use are fine enough such that our results have
numerical errors of only a few percent due to the discrete nature of our spatial lattices.
8
When using grid techniques, another important issue arises due to the finite nature
of the space enclosed within the boundaries of the grid, which we refer to as a box. The
problem of defining a large enough box with minimum number of points is important since
fewer points means less computational burden, which in turn translates into the ability to
study a wider array of problems. Although the uniform meshes are the most natural choices,
most of the time they are not the most optimum ones, meaning that they do not provide the
largest box for a given number of grid points. Specifically, in problems involving Rydberg
atoms, large boxes are required since the size of the atom in a high-n state scales as ? 2n2.
For such problems, we have found that using a non-linear mesh, particularly a square-root
mesh, is more suitable for the lattice representations of Rydberg systems.
2.2.1 Square-root Mesh
To be able to study atomic problems involving excited states, many types of non-linear
lattices have been employed in time-dependent studies of atomic systems [46]. Particu-
larly, square-root lattices have proved especially fruitful for the study of Rydberg systems
due to the fact that at large distances, a square-root mesh puts roughly the same number of
points in nodes of the highly excited atomic orbitals. Here, I will first describe the general
idea behind the institution of a one-dimensional non-linear lattice for the description of a
wave packet ?(r), which then I will specify for a square-root mesh.
9
Expectation value of the kinetic energy operator T on a one-dimensional non-uniform
mesh can be given by,
hTi = 12
Z
??
?i ddr
?
?i ddr
?
? dr
= 12
Z d??
dr
? d?
dr
?
dr
? 12
NX
j=0
(?0j+1=2)?(?0j+1=2)(rj+1 ?rj)
? 12
NX
j=0
(?j+1 ??j)?(?j+1 ??j)
(rj+1 ?rj)
? 12
NX
j=0
j?j+1 ??jj2
(rj+1 ?rj) (2.3)
where prime denotes derivative with respect to r, and we have employed the midpoint rule
for the derivative. The time-dependent Schr?odinger equation in terms of the kinetic energy
and the potential energy operators can be written as,
i _? = T? +V? (2.4)
where dot denotes the derivative with respect to time, and T and V are the kinetic and the
potential energy operators, respectively. Multiplying this equation with ?? from left and
integrating over r on the lattice yields,
i
2
NX
j
??j _?j(rj+1 ?rj?1) = hTi+ 12
NX
j
??jVj?j(rj+1 ?rj?1): (2.5)
10
Taking the derivative of both sides of Eq. 2.5 with respect to ??j and multiplying through
by two, the Schr?odinger equation on the lattice becomes,
i _?j(rj+1 ?rj?1) = 2@hTi@??
j
+Vj?j(rj+1 ?rj?1) (2.6)
where we have dropped the summation over the grid points to reduce the equation to a
particular lattice site j. From the kinetic energy expression in Eq. 2.3, we can evaluate the
derivative of hTi with respect to ?? as
@hTi
@??j = ?
1
2
??
j+1 ??j
rj+1 ?rj ?
?j ??j?1
rj ?rj?1
?
(2.7)
and with the use of this expression, the time-dependent Schr?odinger equation in 2.6 at
lattice site j becomes,
i(rj+1 ?rj?1) _?j = ?
??
j+1 ??j
rj+1 ?rj ?
?j ??j?1
rj ?rj?1
?
+Vj(rj+1 ?rj?1)?j: (2.8)
With the change of variables ?j = zj=p(rj+1 ?rj?1)=2, this equation can be further
written as,
i_zj = ? zj?1p(r
j+1 ?rj?1)(rj ?rj?2)(rj ?rj?1)
+
? 1
(rj+1 ?rj)(rj ?rj?1) +Vj
?
zj
? zj+1p(r
j+2 ?rj)(rj+1 ?rj?1)(rj+1 ?rj)
: (2.9)
11
In terms of the new variable z, the normalization integral reduces to a simple sum,
X
j
j?jj2(rj+1 ?rj?1) =
X
j
jzjj2 = 1 (2.10)
Eq. 2.9 can now be written in the matrix form,
i_~z =T~z +V~z (2.11)
where the kinetic energy matrix Tis tridiagonal and the potential energy matrix V is diag-
onal in r.
Note that this derivation holds for any non-linear spatial lattice and gives a squre-root
mesh when rj = rf j2=N2.
2.3 Methods for time-propagation of the time-dependent Schr?odinger Equation
In this section, we describe the methods used to simulate the time evolution of the ra-
dial part of a wave function according to the time-dependent Schr?odinger equation. First we
discuss the simpler explicit time-propagation scheme and then describe the implicit propa-
gator for one-dimensional atomic systems. For higher dimensional systems, we discuss the
split-operator scheme which in combination with one of the one-dimensional propagation
schemes yields a propagator for radial part of the wave function involving two or more
electrons.
12
2.3.1 Explicit Scheme
Since the time-dependent Schr?odinger equation is a partial differential equation in-
volving both time and space variables, obtaining a particular solution requires an initial
condition, i.e. the initial wave function, which we take to be ?(??t) where ?t is the time
step taken during the propagation and we assume a uniform mesh in time. Using the Taylor
expansion of the exact quantum mechanical time evolution operator e?iH?t, propagating
?(0) one time step backward in time gives,
?(??t) =
1+ iH?t? 12H2?t2 +O[?t3]
?
?(0) (2.12)
where we have expanded the exact quantum mechanical time evolution operator e?iH?t up
to terms of the order ?t2. Applying the time evolution operator a second time to ?(0) gives
the wave function at t = ?t,
?(?t) =
1? iH?t? 12H2?t2 +O[?t3]
?
?(0): (2.13)
Subtracting Eq. 2.12 from Eq. 2.13, and keeping terms up to order ?t2, yields the lowest
order leap-frog scheme,
?(?t) = ?(??t)?2i?tH?(0) (2.14)
13
The terms of the order ?t3 and up being ignored, the error in this method scales as ?t3.
Due to omitted terms in the Taylor expansion of the time evolution operator and the dis-
cretization of the space, two main types of errors emerge: the phase and the amplitude
errors.
To estimate the phase error, it is useful to decompose ?(t) as
?(t) =
X
j
Aj exp(?i??jt)`j (2.15)
where `j are the eigenstates of the time-independent Hamiltonian H with eigenvalues ??j on
the grid. Note that the eigenvalues ??j are just the actual eigenvalues ?j of the Hamiltonian
H with the phase accumulation. Substituting Eq. 2.15 into Eq. 2.14 and recalling that
e?ifi = cos(fi)? isin(fi), we get
X
j
Aj exp(?i??j?t)`j =
X
j
Aj exp(i??j?t)`j ?2i?tH
X
j
Aj`j (2.16)
?2i
X
j
Aj sin(??j?t)`j = ?2i?t
X
j
Aj??j`j: (2.17)
Multiplying both sides of this equation with `k and integrating over all coordinates, and
recalling the orthonormality of the eigenstates, i.e.,
h`jj`ki = ?j;k; (2.18)
one can realize that
?j?t = sin(??j?t): (2.19)
14
Provided that max(j?jj)?t ? 1 this relation between ?j and ??j can be inverted to give
??j = 1?t arcsin(?j?t): (2.20)
From Eq. 2.20 it is clear that the explicit scheme is not necessarily unitary unlike the
actual time evolution operator. If ?j?t > 1, then the right hand side of Eq. 2.20 would be
complex, which would give a time dependent norm. Using Eq. 2.15 and Eq. 2.20, we can
show that the lowest order leap-frog propagator is exactly unitary when max(j?jj)?t ? 1.
To see this, we need to evaluate j?(?t)j2 and show that it has the same norm as ?(??t).
j?(?t)j2 = j?(??t)j2 ?2iH?t??(0)?(??t)+2iH?t??(??t)?(0)
?(2iH?t)2j?(0)j2 (2.21)
Decomposing ? as in Eq. 2.15
j?(?t)j2 = j?(??t)j2 ?2iH?t??(0)?(??t)+2iH?t??(??t)?(0)
?(2iH?t)2??(0)?(0)
= j?(??t)j2 ?2i?t
X
j;k
A?jAk?j exp(i??k?t)`?j`k
+2i?t
X
k;j
A?kAj?k exp(?i??k?t)`?k`j ?(2i?t)2
X
j;k
?2jA?j`?jAk`k:(2.22)
15
To evaluate the norm of ?(t), we integrate both sides of Eq. 2.22 over all coordinates
Z
j?(?t)j2d3~r = 1?2i?t
X
j;k
A?jAk?j exp(i??k?t)
Z
`?j`kd3~r
+2i?t
X
k;j
A?kAj?k exp(?i??k?t)
Z
`?k`jd3~r
?(2i?t)2
X
j;k
?2jA?jAk
Z
`?j`kd3~r
= 1?2i?t
X
j
jAjj2?j exp(i??j?t)
+2i?t
X
j
jAjj2?j exp(?i??j?t)
?(2i?t)2
X
j
?2jjAjj2: (2.23)
where we have used the orthonormality condition 2.18 of the eigenstates `j, and taken
?(??t) to be normalized to unity. Combining second and third terms in Eq. 2.23
Z
j?(?t)j2d3~r = 1?2i?t
X
j
jAjj2?j[exp(i??j?t)?exp(?i??j?t)]
?(2i?t)2
X
j
?2jjAjj2
= 1?2i?t
X
j
jAjj2?j2isin(??j?t)?(2i?t)2
X
j
?2jjAjj2: (2.24)
Recalling Eq. 2.19, this results in
Z
j?(?t)j2d3~r = 1?(2i?t)2
X
j
jAjj2?2j ?(2i?t)2
X
j
jAjj2?2j
= 1: (2.25)
16
Note that the step taken in getting from Eq. 2.24 to Eq. 2.25 requires that the condition
max(j?jj)?t ? 1 is satisfied. This imposes the condition max(j?jj)?t ? 1 for unitarity of
the explicit scheme.
Keeping terms up to order ?t5 in the Taylor expansions of the exponential time evolu-
tion operator in Eqs. 2.12 and 2.13, we obtain the next order leap-frog scheme,
?(?t) = ?(??t)?2
iH?t? iH
3?t3
3!
?
?(0) (2.26)
where now the error scales like ?t5. Repeating the same procedure as we have carried out
above to estimate the phase errors, one obtains,
sin(??j?t) = ?j?t? ?
3
j?t
3
3! (2.27)
which can be inverted to give
??j = 1?t arcsin
?j?t? ?
3
j?t
3
3!
?
(2.28)
provided that fl
flfl
fl?j?t?
?3j?t3
3!
flfl
flfl? 1: (2.29)
This inequality can be solved for ?j?t to yield the condition for unitarity to be max(j?jj)?t .
2:8.
17
2.3.2 Implicit Scheme
Instead of using the Taylor expansion of the exponential time evolution operator as we
did in deriving the explicit scheme, we can simply integrate the time-dependent Schr?odinger
equation
i _? = H ? (2.30)
over one time step ?t,
i
Z ?t
0
_?dt =
Z ?t
0
H ?dt: (2.31)
Using the trapezoidal rule for the right hand side, this can be approximated as
i[?(?t)??(0)] ? H?t2 [?(?t)+?(0)]: (2.32)
Rearranging and solving for ?(?t) yields,
?(?t) ?
1? iH?t=2
1+ iH?t=2
?
?(0): (2.33)
The operator in front of ?(0) is the lowest order Pad?e approximation for the exact time
evolution operator to order ?t3.
e?iH?t =
1? iH?t=2
1+ iH?t=2
?
+O[?t3] (2.34)
We refer this approximation to the exact time evolution operator as the implicit propagator.
In contrast to the explicit operator, the Pad?e approximation to the quantum mechanical
18
time-evolution operator is exactly unitary regardless of the size of the time step ?t. This
can easily be seen by checking the usual unitarity condition as follows,
1? iH?t=2
1+ iH?t=2
?y 1? iH?t=2
1+ iH?t=2
?
=
1+ iHy?t=2
1? iHy?t=2
? 1? iH?t=2
1+ iH?t=2
?
=
1+ iH?t=2
1? iH?t=2
? 1? iH?t=2
1+ iH?t=2
?
= 1: (2.35)
Here we have used the fact that the Hamiltonian is hermitian, i.e. Hy = H.
Even though it is unconditionally unitary, due to the discretization of the space on the
lattice, the implicit propagator still suffers from the phase errors. To estimate the phase
errors over a time step, we will substitute the decomposed wave function 2.15 into the
implicit propagator,
?(t+?t) =
1? iH?t=2
1+ iH?t=2
?X
j
Aje?i??jt`j
?(t+?t) =
X
j
Aje?i??jt
1? i?
j?t=2
1+ i?j?t=2
?
`j
?(t+?t) =
X
j
Aje?i??jte?i(?jt+?)`j (2.36)
where we have noted that H?j = ?j?j and ? is the phase error, and we have used
1? i?j?t=2
1+ i?j?t=2 = e
?i(?j?t+?): (2.37)
19
This can be manipulated to give
2
1+(?j?t=2)2 ?1
?
? i ?j?t1+(?
j?t=2)2
= cos(?j?t+?)? isin(?j?t+?): (2.38)
which implies
cos(?j?t+?) = 21+(?
j?t=2)2
?1
sin(?j?t+?) = ?j?t1+(?
j?t=2)2
: (2.39)
From Eq. 2.39 the phase error ? can be deduced to be
? = arccos
2
1+(?j?t=2)2 ?1
?
??j?t (2.40)
or, equivalently,
? = arcsin
?
j?t
1+(?j?t=2)2
?
??j?t (2.41)
2.3.3 Split-Operator Technique
In our discussions of the explicit and implicit methods above, we have assumed a one-
dimensional Hamiltonian. In cases where there is more than one electron or dimension
involved, we can split the total Hamiltonian into pieces where at least one piece involves
only a single dimension. For example, in the case of the two-electron model Hamiltonian
20
of Chapter 4, we have divided the Hamiltonian as
H(r1;r2) = H1(r1)+H2(r2)+H3(r1;r2) (2.42)
where
H1(r1) = ?12 @
2
@r21 ?
Z
r1
H2(r2) = ?12 @
2
@r22 ?
Z
r2
H3(r1;r2) = 1r
1 +r2
(2.43)
for which the split operator looks like,
e?iH?t ? e?iH3?t=2e?iH1?te?iH2?te?iH3?t=2: (2.44)
For brevity, we will discuss the case where the total Hamiltonian is split into two
pieces, i.e. H = H1 + H2, for which the split operator technique is the lowest order split
operator technique,
?(t+?t) ? e?iH1?t=2e?iH2?te?iH1?t=2?(t): (2.45)
Note that the expression 2.45 is exactly unitary when H1 and H2 commute. When the com-
mutator [H1;H2] does not vanish, the error can be shown to be (i=24)[H1+2H2;[H1;H2]]?t3
21
by Taylor expanding both sides of
e?iH?t ? e?iH1?t=2e?iH2?te?iH1?t=2: (2.46)
It can be shown that the propagator 2.45 is exactly unitary for hermitian H1 and H2.
To show this, we need to calculate
??(t) = [e?iH1?t=2e?iH2?te?iH1?t=2]y[e?iH1?t=2e?iH2?te?iH1?t=2]?(t) (2.47)
and show that ??(t) identically yields ?(t), i.e. the operator in front of ?(t) is identity.
Noting that [e?iH?t]y = [eiHy?t] and (AB)y = ByAy, we can rewrite this expression as
??(t) = [eiHy1?t=2(e?iH1?t=2e?iH2?t)y][e?iH1?t=2e?iH2?te?iH1?t=2]?(t)
= [eiHy1?t=2eiHy2?teiHy1?t=2][e?iH1?t=2e?iH2?te?iH1?t=2]?(t): (2.48)
For hermitian H1 and H2, Hy1 = H1 and Hy2 = H2, and therefore we can regroup as follows,
??(t) = eiH1?t=2eiH2?t eiH1?t=2e?iH1?t=2| {z }
I
e?iH2?te?iH1?t=2?(t)
= eiH1?t=2 eiH2?te?iH2?t| {z }
I
e?iH1?t=2?(t)
= eiH1?t=2e?iH1?t=2| {z }
I
?(t)
= ?(t) (2.49)
which is what we wanted to show.
22
2.4 Time-dependent perturbation theory
When the interaction term in the total Hamiltonian is small such that the atom stays
mostly in the initial state it started out with, the time-dependent perturbation theory can be
used and directly incorporated into the time-dependent Schr?odinger equation:
i @@t ?H
?
?(~r;t) = 0: (2.50)
Splitting the total Hamiltonian into the perturbed and unperturbed parts, i.e. H = H0 +
H0(t), Eq. 2.50 can be rewritten as
i @@t ?H0
?
?(~r;t) = H0(t)?(~r;t): (2.51)
Since the Schr?odinger equation is linear, we seek the solution to Eq. 2.51 as,
?(~r;t) = ?(0)(~r)e?iE0t +?(1)(~r;t) (2.52)
where ?(0) is the eigenstate of the unperturbed Hamiltonian H0 with the energy E0. Note
that the entire time-dependence of the unperturbed solution in the first term on the right
hand side of Eq. 2.51 is in the complex exponential form and the effect of the perturbation
term in the equation is linearly reflected to the solution through ?(1). Substituting this
23
solution into Eq. 2.51,
i @@t ?H0
?
?(1)(~r;t) = H0(t)?(0)(~r)e?iE0t +H0(t)?(1)(~r;t): (2.53)
Ignoring the second term on the right hand side gives the time-dependent Schr?odinger
equation consistent with the first order time-dependent perturbation theory. Feeding the
solution of this equation back into itself gives higher order perturbation theory.
i @@t ?H0
?
?(1)(~r;t) = H0(t)?(0)(~r)e?iE0t 1st order
i @@t ?H0
?
?(2)(~r;t) = H0(t)?(1)(~r;t) 2nd order
i @@t ?H0
?
?(3)(~r;t) = H0(t)?(2)(~r;t) 3rd order
... (2.54)
The total solution ? would be
?(~r;t) = ?(0)(~r)e?iE0t +?(1)(~r;t)+?(2)(~r;t)+?(3)(~r;t)+:::: (2.55)
2.5 Green?s Function Method
Note that all of the equations in Eq. 2.54 are of the form
i @@t ?H0
?
?(1)(~r;t) = S(~r;t) (2.56)
24
which is an inhomogeniuous partial differential equation. There are mainly two reasons for
using this method to solve for ?(1) directly, instead of solving for the total wave function.
First, in the cases where perturbation theory is applicable, ?(1) is usually much smaller than
?(0), making it hard to extract the information carried in ?(1) from the total wave function.
The second advantage has to do with the time scales involved in the Schr?odinger equation.
Even though H0 and ?(0)e?iE0t may vary rapidly with time, their product usually varies
much slower with time, which enables us to use larger time steps for the time propagation
of the Schr?odinger equation.
Integrating Eq. 2.56 equation through one time step ?t, we get
i
Z t+?t
t
@?(1)(~r;t0)
?t0 = H0
Z t+?t
t
?(1)(~r;t0)?t0 +
Z t+?t
t
S(~r;t0)?t0: (2.57)
If we use the trapezoidal rule to evaluate this integral, we obtain
?(~r;t+?t) =
1? iH
0?t=2
1+ iH0?t=2
?
?(1)(~r;t)? i?t1+ iH
0?t=2
S
~r;t+ ?t2
?
+O[?t3] (2.58)
Note that at t = 0 ?(~r;t = 0) = ?(0)(~r;t = 0) therefore initially ?(1) = 0. The oper-
ator in the first term on the right hand side is the lowest order Pad?e approximation to the
exponential time evolution operator,
e?iH0?t =
1? iH
0?t=2
1+ iH0?t=2
?
+O[?t3]: (2.59)
25
Therefore we can rewrite Eq. 2.58 as,
?(1)(~r;t+?t) = e?iH0?t ?(1)(~r;t)? i?t1+ iH
0?t=2
S
~r;t+ ?t2
?
+O[?t3]: (2.60)
For example, if we take the total Hamiltonian to be the two-electron Hamiltonian of Chapter
3 with the electron-electron interaction being V(r1;r2) = 1=(r1 +r2), we can evaluate the
effect of the exponential time evolution operator onto ?(1)(~r;t) using lowest order split
operator technique with implicit propagator,
?(~r;t+?t) = e?iV?t=2
1? iH
1?t=2
1+ iH1?t=2
? 1? iH
2?t=2
1+ iH2?t=2
?
e?iV?t=2 ?(1)(~r;t)
?
i?t+ ?t
2
2 H0
?
S
~r;t+ ?t2
?
+O[?t3]: (2.61)
where H0 = H1 + H2 and Hj = p2j=2 ? Z=rj with (j = 1;2). Instead of evaluating the
integral 2.57 using the trapezoidal rule, if we change the integral limits to range from t??t
to t+?t and use the midpoint rule, we reach the lowest order leap frog scheme for solving
Eq. 2.56.
?(1)(~r;t+?t) = ?(1)(~r;t??t)?2iH0?t?(1)(~r;t+?t)
?2i?tS(~r;t)+O[?t3]: (2.62)
26
2.6 Time-independent methods for generating the one- and two-electron orbitals
As the wave function of an atomic system is propagated through time, the atomic
eigenstates are projected onto the total wave function to extract information about excita-
tion and ionization from the atomic system. For the problems that we will discuss, this
means generating and storing the one-electron orbitals beforehand to project them onto the
time-dependent wave function as it evolves through time. In the next two subsections, we
discuss two methods which we have found to be efficient in generating the atomic orbitals.
2.6.1 The Shooting Method
The Shooting method is simply integrating the Schr?odinger equation from both ends
of the box in which the orbitals need to be generated. To integrate the time-independent
Schr?odinger equation, we first discretize the Schr?odinger equation, H? = E?,
?12?j?1 ?2?j +?j+1(?r)2 +Vj?j = E?j: (2.63)
where we used a uniform grid for brevity. To start the forward integration, we need two
points on the grid, i.e. ?0 and ?1 which we take to be zero and ?r respectively. Solving for
?j+1 we get
?j+1 = 2(?r)2(E ?Vj)?j +2?j ??j?1: (2.64)
27
When the classical turning point is reached, the time-dependent Schr?odinger equation turns
from parabolic to elliptic in nature and the numerical integration picks up the diverging so-
lution. At this point, one stops the forward integration and start to integrate the Schr?odinger
equation backwards from the box edge. Taking ?N = 0 and ?N?1 = ?r, we can solve for
?j?1,
?j?1 = 2(?r)2(E ?Vj)?j +2?j ??j+1: (2.65)
One then stops at the point at which the forward integration was terminated and matches
the results of the forward and backward integrations at this point to obtain a smooth orbital.
2.6.2 The Relaxation Methods
Another effective method for generating the atomic orbitals involves the iterative ap-
plication of (Emax ?H) or 1=(Eg ?H) operator onto a chosen trial function. Here Emax is
the maximum energy that can be supported by the grid, i.e. 2=(?r)2, and Eg is a guess for
ground state energy of the Hamiltonian H. The choice of the trial function is not vital for
the success of the method, although it effects the number of iterations needed to converge
to the desired eigenstate. The closer the trial function to the actual eigenstate less iterations
it takes to converge.
The effect of the operators (Emax ?H) and 1=(Eg ?H) onto a trial function can be
understood if one considers the trial function a superposition of the actual eigenstates of
the Hamiltonian H. When either of the operators (Emax ?H) or 1=(Eg ?H) is applied to
the trial function, its action is to rescale the mixing coefficients of the eigenstates of H in
the expansion such that the coefficient of the lowest energy eigenstate gains a higher boost
28
relative to the coefficients of the higher energy eigenstates in the expansion. Repeating this
process with the energy guess E updated at every iteration, one checks the convergence of
the expectation values
?(fi)E = h`(fi)n;?j(H ?En;?)j`(fi)n;?i
?(fi)E2 = h`(fi)n;?j(H2 ?E2n;?)j`(fi)n;?i (2.66)
every time step, where `(fi)n;? is the initial trial function after fi iterations. The iteration
is stopped when ?E and ?E2 get below a chosen tolerance. Just as the trial function, the
closer the initial guess E(0) to the actual energy of the ground state less iterations it takes
to converge. Note that since neither of the operators (Emax ?H) or 1=(Eg ?H) is unitary,
the iterated function must be normalized after each time step.
Also note that the relaxation method cannot be used to generate an excited eigenstate
without first generating all the eigenstates that energetically lie below it. Once the ground
state is generated, the same trial function can then be used to generate the next eigenstate
by the same iterative process with subtracting the ground state generated previously. Since
the iteration always converges to the lowest energy component in the trial function, this
would force the iteration to converge to the eigenstate with the second lowest eigenenergy.
This process can be used to generate the spectrum of the Hamiltonian H one by one from
ground state up to a desired excited state. This makes this method most useful when one
needs all of the eigenstates and eigenenergies of the Hamiltonian from ground state up to
an excited eigenstate.
29
CHAPTER 3
THE PHOTO-DOUBLE IONIZATION CROSS SECTION OF HELIUM NEAR THRESHOLD IN
A COLLINEAR s-WAVE MODEL
3.1 Introduction
In this chapter, we investigate the threshold laws for double ionization by study-
ing the photo-double ionization of a helium atom from its ground and the first excited
state 1s2s(1S). The photo-double ionization problem near the threshold region presents
a formidable challenge from a theoretical point of view since both of the electrons move
slowly in the threshold region, and interact for a long time before they escape into the
continuum. Various theories developed for studying the two electron escape process near
threshold to explain the energy sharing of the escaping electrons, energy dependence of
the double ionization cross section, and the angular distributions of the outgoing electrons.
Due to the fact that both electrons move slowly near the threshold, this problem has been
treated on purely classical grounds and two conceptually different theories have been de-
veloped to describe the energy dependence of the photo-double ionization cross section
near the threshold.
First of these models, the Wannier theory [44], assumes that both of the electrons
travel along the Wannier ridge and have the same kinetic energy. They leave the atom along
the opposite sides of the atom and have approximately the same distance from the nucleus.
Wannier theory predicts the excess energy dependence of the photo-double ionization cross
30
section as 2e ? Efiexc where Eexc is the excess energy and fi is the Wannier exponent which
equals 1.056. Following this classical picture of the Wannier theory, both semiclassical
[20] and quantum mechanical [21] treatments of the threshold law have been carried out
and confirmed the Wannier law.
The second classical theory developed to describe the energy dependence of the dou-
ble ionization cross section near the threshold is the Coulomb-dipole theory developed by
Temkin and Hahn [53]. Contrary to the Wannier theory, Coulomb-dipole theory predicts
the double ionization cross section to be monotonically increasing while modulated by os-
cillations. This model does not assume that the electrons leave the atom with equal kinetic
energies, but rather one of the electrons leave the atom faster than the other. The faster
electron experiences a dipole potential formed by the residual ion and the slower leaving
electron.
To decide which of the models describe the actual double-ionization process near
threshold better, several experiments have been performed to test these threshold laws.
An experiment by Kossmann et al [23] has investigated the validity of the Wannier law for
the photo-double ionization of He along with its range of validity. They have found very
good agreement with the Wannier law and observed no oscillations modulating the thresh-
old law. On the other hand, more recent experiments by Wehlitz et al [24] and Luki`c et al
[25] displayed oscillations beyond experimental uncertainty in the photo-double ionization
cross sections of Li and Be, respectively.
In this chapter, we present results of our calculations for the photo-double ionization
of He from the ground and the first excited states within 1.0 eV excess energy above the
31
threshold. We perform our calculations in a collinear s-wave model which assumes that
both electrons are positioned on the opposite sides of the atom and that the electron-electron
interaction potential is 1=(r1+r2) where r1 and r2 are the radial coordinates of the electrons.
In the past, this model potential has been used to study two-electron dynamics and shown
to include the essential physics of the electron-electron interaction in the Coulomb field
[53, 41]. Using a model potential for the electron-electron interaction enabled us to extend
our calculations up to the first excited state which has proven challenging for the full three-
dimensional calculations. A full three-dimensional calculation investigating the Wannier
threshold law for the ground state He within 1.0 eV above threshold has been performed
by Kleiman et al [27] and agrees perfectly with our findings. Most of the work presented
in this chapter has been published as part of a letter in the Journal of Physics B [27].
3.2 Theory
Because the three-dimensional calculation of the photoionization of helium near thresh-
old is computationally difficult, we examined a simpler model with lower dimensionality.
We calculated the photo-double ionization probability in the collinear model, where the
interaction potential between the electrons is represented by 1=(r1 + r2). The choice of
this potential stems from the fact that the collinear potential yields the correct threshold
law for the two electron escape process, ? E1:056. On the other hand the the cross sec-
tion obtained from the exact s-wave potential exhibits an exponential supression due to
the cusp it has on the Wannier ridge r1 = r2 (Ref. [28]). On the other hand, this model
potential has been used successfully in the past to study two-electron dynamics and shown
32
to include the essential physics in the electron-electron interaction [53, 41]. These facts
combined with the computational burden of a three-dimensional calculation have incited
us to use the collinear s-wave model as a good test case for investigating the oscillations in
the photo-double ionization cross section.
With the choice of the collinear potential, the two electron Hamiltonian can be written
as
^H0 =
2X
fi=1
?
^p2fi
2 ?
2
rfi
!
+ 1r
1 +r2
: (3.1)
As described in Chap. 2.5, for the time dependent wave function of the two electron system
we made the ansatz ?(r1;r2;t) = ?(0)(r1;r2;t) + ?(1)(r1;r2;t), where ?(0)(r1;r2;t) =
?0(r1;r2)e?iE0t and E0 is the energy of the state ?0 according to ^H0?0 = E0?0. We inves-
tigated the threshold photo-double ionization of helium out of both 1s2(1S) and 1s2s(1S)
states which were generated on a radial box by relaxation of a trial function on the grid,
as described in Chap. 2.6. Generation of the one-electron orbitals ?1s20 and ?1s2s0 on the
box introduces continuum character into the eigenstates which is crucial for the double
ionization of the model atom.
The time-dependent Schr?odinger equation that is first order in photon interaction can
be written as
?
i @@t ? ^H0
!
?(1)(r1;r2;t) = F(t)e?i(!+E0)t(r1 +r2)?(0)(r1;r2)
= S(r1;r2;t) (3.2)
33
where ! is the energy of the incoming photon in atomic units, F(t) is the time dependence
of the laser pulse, and E0 is the energy of the initial state ?(0)(r1;r2). The right hand side
of Eq. 3.2 is the source term S(r1;r2;t) due to the laser field. Time dependence F(t) of the
electric field in the source term S(r1;r2;t) was chosen to be
F(t) =
8
>>>
<
>>>
:
1
2
h
1?cos
?
2?
tcutt
?i
for 0 < t < tcut;
0 for t > tcut
(3.3)
where tcut is the pulse duration, which we have taken to be half the total propagation time.
We have solved Eq. 3.2 as outlined in Chap. 2.5 using the lowest order leap-frog scheme:
?(1)(r1;r2;t+?t) = ?(1)(r1;r2;t??t)?2i ^H0?t?(1)(r1;r2;t)
?2i?tS(r1;r2;t)+O[?t3]: (3.4)
Since there is no analytical expression for the helium atom wave functions, ?(1)(r1;r2;t)
is placed in a radial box that extends to a finite rf. The finite size of the box imposes an
upper limit on the total propagation time because the fast ionizing components of the wave
function reflected at the end of the box would eventually rescatter from the remaining core.
On the other hand, longer propagation time is desired because it means that we can leave
the source term on for a longer period of time so that the photon is represented by a well
defined sharp peak in the energy space. Therefore, to increase the total propagation time
we put an absorption mask at the end of the box to absorb the fast, high energy components
of the wave function.
34
Note that the Wannier law E1:056 assumes an energy distribution with zero width and
goes to zero as E ! 0. Since we have have an energy distribution with finite width in
the energy space due to the fact that our pulse is on for a finite amount of time, we needed
to compare our results with the threshold law convolved with our energy distribution. We
convolved the Wannier law with the squared absolute value of the Fourier transformation
of F(t),
hE1:056i(Eexc) =
R jF(E ?E1:056
exc )j
2E1:056dE
R jF(E ?E1:056
exc )j2dE
(3.5)
where we have denoted the Fourier transform of F(t) by F, and fit this convolved threshold
law to our results for comparison. Note that hE1:056i(Eexc) is just the expectation value of
E1:056exc with respect to the energy distribution corresponding to our time-dependent pulse
F(t).
All the projections needed to obtain the photo-double ionization probabilities are done
after the source term is turned off. The probability for finding both of the electrons in the
continuum is calculated by
P2e(t) =
Z Z
j?(1)(r1;r2;t)j2dr1dr2 ?2
X
n
flfl
flfl
fl
Z
?(1)(r1;r2;t)`ns(r1)dr1
flfl
flfl
fl
2
+
X
n
X
m
flfl
flfl
fl
Z Z
?(1)(r1;r2;t)`ns(r1)`ms(r2)dr1dr2
flfl
flfl
fl
2
: (3.6)
Here `ns(r)s are the He+ negative energy eigenstates, i.e. (^p2=2?2=r)`ns(r) = ~?ns`ns(r).
These eigenstates are computed in a radial box that extends out twice as far as the box used
for the time propagation of ?(1)(r1;r2;t) for increased accuracy in the projections. Also,
35
generation of He+ states in a finite box yields a finite number of bound and continuum
eigenstates. Therefore, to minimize the error due to the finite box size we carefully chose
the box size so that the threshold is straddled between the highest energy bound state and
the lowest energy continuum state on the box.
3.3 Calculations and Results
To check the convergence of our results we have performed a series of calculations
where we have varied our box size, mesh spacing, time step and the total propagation time
independently for each energy value of the incoming photon. We have scaled our double
ionization probabilities with tpeak which is half of the pulse duration tcut. Figs. 3.1 and
3.2 show the Wannier convolved scaled probabilities for photo-double ionization out of
He(1s2(1S)) and He(1s2s(1S)) as a function of the excess energy for three different lattice
spacings and pulse durations. In all of these calculations, the box size rf was kept fixed at
200 (a.u.) and the time steps taken for the time propagation were ?t = 0:01 (a.u.). Note that
for the lattice spacings of 0.1 (a.u.) and 0.07 (a.u.), i.e. for the 2000 and 3000 point lattices,
the convolved scaled double ionization probabilities are in excellent agreement, whereas
the double ionization probability calculated for ?r = 0:2 (a.u.) is somewhat deviated from
the other two. We have also performed other convergence checks where only the lattice
spacing was varied with a much longer pulse duration than those presented in Figs. 3.1 and
3.2, and found that we could increase ?r up to 0.3 and still get results that are converged
within a few percent. Therefore, we conclude that the differences between the solid and the
other curves in Figs. 3.1 and 3.2 are mainly due to the differences in the pulse durations
36
0 0.2 0.4 0.6 0.8 1Excess Energy (eV)0
2.5
5.0
7.5
10.0
12.5
P DPI
/t peak
(10
-6 /(a.u.))
npts = 1000 and tcut = 300 (a.u.)npts = 2000 and t
cut = 600 (a.u.)npts = 3000 and t
cut = 900 (a.u.)
Figure 3.1: PDPI=tpeak versus excess energy for photo-double ionization of He(1s2) for
different boxes and pulse durations.
rather than the differences in ?r. All the results we present in this chapter are converged
within a few percent for all photon energies we consider.
Another problem we had to deal with was the slow convergence of the photo-double
ionization probability as a function of time. Since we are studying threshold ionization,
energy of the incoming photon in the energy space is centered close to the threshold with
a width controlled by how long the source term is on. The longer the source term is kept
on, sharper the energy distribution is and therefore smaller the contribution to the double
ionization probability from other energies in our energy distribution. But the propagation
37
0 0.2 0.4 0.6 0.8 1Excess Energy (eV)0
1.0
2.0
3.0
4.0
5.0
6.0
P DPI
/t peak
(10
-4 /(a.u.))
npts = 1000 and tcut = 300 (a.u.)npts = 2000 and t
cut = 600 (a.u.)npts = 3000 and t
cut = 900 (a.u.)
Figure 3.2: PDPI=tpeak versus excess energy for photo-double ionization of He(1s2s) for
different boxes and pulse durations.
38
time tf is also limited by the box size rf due to the reflections from the box edge. Further-
more, the source term must be turned off and the wave function should have time to evolve
before the the projections are done for the extracted double ionization probabilities to be
meaningful. We have found that for a given box size rf the most effective time we could
keep the source term on was half the propagation time which was twice the box size in
atomic units. Therefore tcut = tf=2 and tpeak = tcut=2 in Eq.(3.3). We then fit the double
ionization probability to the form A + B=(t ? tpeak) + C=(t ? tpeak)2 for the last quarter
of the propagation time. The coefficient A is then the t ! 1 extrapolated double ion-
ization probability. We examined the convergence of this extrapolated double ionization
probability with respect to the box size.
Figs. 3.3 and 3.4 show the scaled photo-double ionization probabilities at various
excess energies above the threshold for ionization out of the ground 1s2(1S) and the first
excited 1s2s(1S) states of He as a function of time. Notice the extremely slow convergence
of the scaled probability at each excess energy for t > 1000 (a.u.), which forced us to
extrapolate the scaled double ionization probabilities to t ! 1 to estimate the actual
ionization probability without having to propagate the wave function of the system too
much longer.
The extrapolated photo-double ionization probabilities scaled by tpeak for the ground
state 1s2(1S) Fig. 3.5 and the first excited state 1s2s(1S) Fig. 3.6 of helium are plotted
as a function of the excess energy within 1.0 eV above the double ionization threshold.
These scaled probabilities were obtained inside a box that is defined on a uniform two-
dimensional lattice of 3082?3082 points with a lattice spacing of ?r = 0:3 (a.u.) in both
39
0 500 1000 1500 2000Time (a.u.)0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
P DPI
/t peak
(10
-6 /(a.u.))
0.10 eV0.20 eV
0.30 eV0.40 eV
0.50 eV
Figure 3.3: PDPI=tpeak as a function of time for double ionization out of He(1s2).
40
0 500 1000 1500 2000Time (a.u.)0
1.0
2.0
3.0
4.0
P DPI
/t peak
(10
-4 /(a.u.))
0.10 eV0.20 eV
0.30 eV0.40 eV
0.50 eV
Figure 3.4: PDPI=tpeak as a function of time for double ionization out of He(1s2s).
41
directions. This extends the box out to 924.6 (a.u.) in both directions. On top of the scaled
probabilities, we also plotted the Wannier power law, i.e. aE1:056exc where a = 3:8387?10?4
for 1s2(1S) and a = 5:4223 ? 10?3 for 1s2s(1S), convolved to our spectral distribution
for tf = 1800 (au.). As one can see, our results are in good agreement with the convolved
Wannier law.
To see if there are any oscillations in the double ionization probability, we have also
plotted the difference ? between the calculated probabilities and the convolved Wannier
law on top of each graph. As is evident from the upper panels in Figs. 3.5 and 3.6, the
deviation of our results from the Wannier law is about two orders of magnitude smaller than
the double ionization probability itself, whereas the experimentally observed oscillations
reported in Refs. [24] and [25] are ? 10% effect for Be [25] and ? 20% effect for Li
[24].
This implies that the oscillations in our residual plots are not physical but merely due
to the fitting process and numerical uncertainties. Also note that the range of validity of
the Wannier law shrinks as one starts the ionization from the excited state. This is expected
since Wannier law assumes both electrons leave the atom back to back with equal energies
which is better realized if the atoms start out from same n-shell.
3.4 Conclusions
In this chapter, we have presented results of our non-perturbative time-dependent cal-
culations of photo-double ionization probabilities of He out of the ground and the first ex-
cited states 1s2s(1S) in a collinear s-wave model near double ionization threshold. Using
42
-2.0-1.0
01.0
? (10
?7 )
0 0.2 0.4 0.6 0.8 1Excess Energy (eV)0
0.5
1.0
P 2e/t
peak
(10
-5 /(a.u.))
Figure 3.5: Calculated PDPI=tpeak (?) and the fitted convolved Wannier power law (?)
versus excess energy for photo-double ionization of He(1s2). The upper panel is showing
the difference between the two.
43
-8.0-6.0
-4.0-2.0
02.0
4.06.0
? (10
?6 )
0 0.2 0.4 0.6 0.8 1Excess Energy (eV)0
0.5
1.0
1.5
P 2e/t
peak
(10
-4 /(a.u.))
Figure 3.6: Calculated PDPI=tpeak (?) and the convolved Wannier power law (?) versus
excess energy for photo-double ionization of He(1s2s). The upper panel is showing the
difference between the two.
44
the collinear model instead of the actual full three-dimensional electron-electron interaction
reduced the computational burden substantially, enabling us to perform many convergence
checks with respect to the box size, lattice spacing, time step, and the pulse width, which
let us reduce the size of the numerical uncertainties below a couple percent level. Due to
the slow convergence of the double ionization probability as function of time, we have ex-
trapolated the time-dependent ionization probabilities to t ! 1 for each incident photon
energy we considered, and taken the extrapolated values as the actual double ionization
probabilities. We have also convolved our excess energy dependent ionization probabil-
ities with the gaussian laser pulse we used to ionize the atom, to further account for the
contributions from the energy components in the gaussian laser pulse other than the central
energy component. This is important in getting the correct energy dependent behaviour
near threshold because the classical Wannier power law assumes an energy distribution of
zero width.
Our investigation of the ionization probability near threshold as a function of the ex-
cess energy confirmed the Wannier power law for the ionization out of both the ground and
the singlet excited state. We have observed deviations from the Wannier Power law which
are two orders of magnitude smaller than the ionization probabilities themselves in each
case and have concluded that these oscillations are well within the numerical uncertainties
of our calculations. We have found that the range of validity for the Wannier power law
is shorter for the singlet excited state 1s2s(1S) than it is for the ground state. This is ex-
pected since the assumption of the Wannier theory that the two electrons leaving the atom
have approximately the same distance from the nucleus holds best for the ground state than
45
for the excited in which the electrons start out at different distances from the nucleus. We
have observed that inclusion of the higher-order terms in the Wannier law in fitting our
convolved results to the Wannier law gives better agreement between the Wannier law and
our results at higher energies, extending the range of validity of the Wannier law.
46
CHAPTER 4
ELECTRON IMPACT IONIZATION OF HIGHLY EXCITED HYDROGEN-LIKE IONS IN A
COLLINEAR s-WAVE MODEL
4.1 Introduction
According to the Bohr correspondence principle, the quantum mechanical description
of a physical system should yield classical dynamics in the appropriate limit. Although
there is no common agreement on how this limit should be defined, the most widely ac-
cepted definition is in terms of large quantum numbers. The classical-quantum correspon-
dence in this context has been subject to many theoretical studies [29, 30, 31] as well as
to experiments with Rydberg atoms [32]. There are systems which were found to violate
the correspondence principle [30] as well as ones that fulfilled it exactly without taking
any limits [31]. In a recent experiment by Nagesha and MacAdam [1], highly excited
Na Rydberg atoms with principal quantum numbers of n = 35 ? 51 revealed quite large
electron-impact ionization cross sections compared to a formula designed to estimate the
ionization cross sections out of excited states within an n-manifold, and a theory [33]
based on low-n data. The discrepancy between the experiment and the theoretical models
suggests that electron-impact ionization from Rydberg states may exhibit peculiar dynam-
ics.
47
Ionization by electron-impact out of highly excited states would be expected to yield
cross sections that are consistent with classical models at sufficiently high quantum num-
bers. However, the convergence to classical dynamics may very well depend on the prop-
erties of the atom, such as its ionic charge or the strength that the electron interacts with
an external system or source. The question as to how well the atomic electron can be de-
scribed classically as the principal quantum number increases is the focus of the present
study. We investigate electron-impact ionization of H-like ions up to Z = 6 for various
electron-impact energies. We performed calculations up to n = 25 in initial principal quan-
tum number using a time-dependent close-coupling (TDCC) method in a collinear s-wave
model. We contrast results of these calculations with those from collinear s-wave classical
trajectory Monte Carlo (CTMC), distorted-wave (DW), and R-Matrix calculations.
Although electron-impact ionization has been covered extensively in the literature
(e.g. see Ref. [34, 35, 36] and references therein), there has only been a few non-
perturbative calculations for the electron-impact ionization from low excited states [37, 38,
39]. A recent paper by Griffin et al. [40] present results from a series of non-perturbative
quantal calculations for electron-impact ionization from H, Li2+, and B4+ up to n = 4
in comparison to fully three dimensional CTMC and perturbative distorted-wave calcula-
tions. Their non-perturbative calculations employed primarily the R-Matrix with pseudo
states (RMPS) method whose results compared very well with those from their TDCC cal-
culations for ionization from selected excited states. Using the non-perturbative results
as benchmarks, their study indicated that the CTMC results were reasonably accurate for
48
hydrogen, but for the ions the CTMC results were too large and did not improve with the
principal quantum number.
Since both the number of n;l terms and the size of the target increase rapidly with
principal quantum number, the computational resources required to study ionization out of
highly excited states become prohibitively large for a full three-dimensional model. There-
fore, in this study, we restrict ourselves to the collinear s-wave model where both electrons
move on a straight line. It has been shown that this model includes the essential physics
of the ionization process close to the threshold [44, 41, 42] and yields the correct Wannier
threshold law for single ionization, i.e. ? E1:128. The interaction potential between the
electrons is 1=(r1 +r2) which we refer to as the collinear s-wave potential.
In the next section, we discuss the theoretical methods employed; in Sec. III, we
present our results; and in Sec. IV, we discuss their signficance. We use atomic units
throughout this chapter, unless mentioned otherwise. The work presented in this chapter
has been published as a paper in Physical Review A [43].
4.2 Theory
We first describe the collinear s-wave TDCC model and discuss the various conver-
gence checks performed. Next we discuss the collinear s-wave CTMC, DW, and RMPS
methods that we employed. Since the accuracy of the results is important, we make special
effort to discuss the methods we used to insure convergence.
49
4.2.1 The time-dependent wave packet method using collinear s-wave model
In the collinear s-wave model where both electrons move on a straight line, the Hamil-
tonian reads [28]
H =
2X
fi=1
?
p2fi
2 ?
Z
rfi
!
+ 1r
1 +r2
: (4.1)
If the H-like ion is initially in the ns state, the symmetrized initial wave function at t = 0
is
?(r1;r2;t = 0) = 1p2
h
?ns(r1)Gks(r2)+?ns(r2)Gks(r1)
i
(4.2)
where ?ns is an eigenstate of the H-like ion and Gks is the Gaussian representing the
incoming electron of energy E = k2=2 and momentum k. It is given by:
Gks(r) = e?i`(r)e?
?
?
rf?r0
?2(
r?12(rf+r0))2; (4.3)
where r0 = 2n2 and the constant ? is chosen such that e?(?=2)2 ? 1 and Gks(r) can be
considered to be zero at r = r0 and r = rf. In our calculations, we have taken ? to be 12.
To reduce the energy spread of the packet due to the phase accumulations, the WKB phase
`(r) is evaluated as
`(r) =
Z r
0
p
2(E +(Z ?1)=r0)dr0: (4.4)
The wave packet is discretized on a 2 dimensional square root mesh where rj = j2?r
with ?r = rf=N2. The maximum grid spacing on this mesh can be evaluated by ?rmax =
(2N ?1)?r and is two times larger than that for a linear mesh.
50
Time propagation of the Schr?odinger equation is carried out by means of lowest order
split operator technique:
?(t+?t) = exp
?i ?t2(r
1 +r2)
?
exp(?iH2?t) (4.5)
?exp(?iH1?t)exp
?i ?t2(r
1 +r2)
?
?(t);
where Hfi = p2fi=2?Z=rfi and (fi = 1;2). Using the lowest order Pad`e approximation for
the exponential one particle operators exp(?iHfi?t=2), the time propagation scheme given
by Eq. (4.5) becomes
?(t+?t) = exp
?i ?t2(r
1 +r2)
??1?iH
2?t=2
1+iH2?t=2
?
(4.6)
?
?1?iH
1?t=2
1+iH1?t=2
?
exp
?i ?t2(r
1 +r2)
?
?(t):
The numerical error for this lowest order Pad`e approximation scales like ?t3 and the ap-
proximated operator is exactly unitary. The probability of finding both electrons in the
continuum at time t is calculated by making use of the bound states ?ns,
P2e?(t) = 1?2
X
n
Z flfl
fl
Z
?(r1;r2;t)?ns(r1)dr1
flfl
fl
2dr
2 (4.7)
+
X
n
X
m
flfl
fl
Z Z
?(r1;r2;t)?ns(r1)?ms(r2)dr1dr2
flfl
fl
2;
provided that the wave function is normalized to unity [45]. The ionization cross section
2e? is calculated from the ionization probability using 2e? = (?=k2)P2e?.
51
To increase the accuracy of the computed projections, we have calculated the eigen-
states in a box two times larger than the box in which ?j;k is discretized. This gives a finer
spacing of states in energy and decreases the effect of a finite box. In discretization of the
Hamiltonian on the square root mesh, we have used a three point differencing scheme for
the second derivatives. The action of p2=2 on the eigenstate ?j is calculated as [46]:
1
2p
2?j = ? ?j?1p
(rj ?rj?1)(rj?1 ?rj?2)(rj ?rj?1) (4.8)
+ ?j(r
j+1 ?rj)(rj ?rj?1)
? ?j+1p(r
j+1 ?rj)(rj ?rj?1)(rj+1 ?rj)
where ?j = Rj=p?rj and Rj is the actual radial orbital. The advantage of using a square
root mesh is that it enables us to employ much larger boxes than the uniform mesh does.
To check the effect of using a square root mesh on our results, we have also performed
a few trial calculations on a uniform mesh and compared the results with those from the
square root mesh calculations. We did not see any significant differences in the projected
probabilities.
We use scaled units to describe the electron-impact energy E in our calculations. En-
ergies of the incoming electron are chosen in multiples of the binding energy of the initial
state. The energy of the incoming electron is therefore defined by E = [Z2=(2n2)]Esc
where we call Esc the scaled electron-impact energy.
We have checked convergence of the TDCC results with respect to box size rf, number
of points N, and time step ?t for the time propagation. For example, for an atom in an initial
52
state with n = 12 and an incoming electron with scaled energy Esc = 9:5, the projected
ionization probability for all Z in a 2250 (a.u.) box with 2500 points remains well within
a percent if one increases the number points to 5000 while keeping rf fixed. When the
number of points is kept fixed at 2500 and the box size is doubled, the probability changes
by ? 2% for Z = 1. In cases of Z = 3 and Z = 5, the differences are within ? 1%.
When both rf and N are doubled to 4500 and 5000 respectively, the results were within
? 3% of the (rf;N) = (2250;5000) results for Z = 1. For Z = 5 the difference was
. 1%. We have found that for sufficiently high n, amplitudes for super-elastic scattering
down to low-n states are very small, enabling us to employ large ?rmax and rf. We carried
out similar analyses for all the initial states and several ion stages in our calculations and
the results we present are converged within a few percent.
To be able to use the same converged box parameters and time steps for all Z, we
scale the length and time according to rj = ?j=Z and t = ?=Z2. The full time-dependent
Shr?odinger equation that needs to be solved is
i @@t? =
" 2X
fi=1
?
1
2
@2
@r2fi ?
Z
rfi
!
+ 1r
1 +r2
#
?(r1;r2): (4.9)
After scaling and dividing through by Z, Eq. (4.9) becomes
i @@?? =
" 2X
fi=1
?
1
2
@2
@?2fi ?
1
?fi
!
+ 1Z(?
1 +?2)
#
?(?1;?2) (4.10)
which is equivalent to the hydrogen problem in scaled coordinates and time, except the
scaled electron-electron interaction potential is now 1=[Z(?1 + ?2)]. In the same manner,
53
scaling the time independent Schr?odinger equation for the evaluation of the eigenenergies
and the eigenstates, we solve the equation:
?12@
2`(?)
@?2 ?
`(?)
? = ~?`(?) (4.11)
with the Z-scaled energy ~? = ?=Z2. Note that with the use of the Z-scaled potential, the
electron-impact energy becomes E = [1=(2n2)]Esc.
For B4+, for an initial state with n = 4 and with a scaled impact energy of Esc = 9:5,
the difference in the ionization probability between the calculations using the collinear s-
wave potential and the Z-scaled collinear s-wave potential is found to be within a percent.
4.2.2 Classical trajectory Monte Carlo method
In the CTMC method, we solved the classical equations of motion to compute the
classical probability for ionization. The classical equations of motion scale. This means
the ionization probability in the collinear s-wave model only depends on Z and on the ratio
of the kinetic energy of the incident electron to the binding energy of the bound electron;
we define this to be the scaled incident energy, Esc. In addition to Z and Esc, the quantum
results depend on the binding energy of the target state.
The length and time scalings are
r = ? ZE
b
t = ? Z
E3=2b
(4.12)
54
where Eb is the binding energy of the attached electron. For this choice of scaling, the
scaled energy of the bound electron is ?1 and the incident energy is Esc. A bound energy
of ?1 corresponds to n = 1=p2 and a classical period of 2?n3 = ?=p2.
With this scaling the fully three dimensional equations of motion become
d~?1
d? = ?
~?1
?31 +
1
Z
~?12
?312
d~?1
d? = ~?1 (4.13)
d~?2
d? = ?
~?2
?32 ?
1
Z
~?12
?312
d~?2
d? = ~?2 (4.14)
where ~?12 = ~?1 ?~?2. For the collinear s-wave model, the scaled equations of motion are
d?1
d? = ?
1
?21 +
1
Z
1
(?1 +?2)2
d?1
d? = ?1 (4.15)
d?2
d? = ?
1
?22 +
1
Z
1
(?1 +?2)2
d?2
d? = ?2 (4.16)
with the additional condition of an infinitely hard wall at a small, positive value of ?, after
which the electron cannot move any closer to ? = 0. The infinitely hard wall reverses the
sign of the velocity and keeps both electrons at positive ?. The position of the wall affects
the ionization probability, but the effect decreases as the wall moves closer to ? = 0. We
chose a position, 5 ? 10?5, where the ionization probability was changed by much less
than a percent. Note that the nuclear charge Z manifests itself in the scaled equations of
motion by multiplying the electron-electron interaction by 1=Z. From this it is clear that the
ionization (which depends on the electron-electron interaction) decreases with increasing
Z.
55
If there were no scattering, the bound electron would be limited to a region 0 < ? ? 1.
The initial conditions of the bound electron were chosen to give a micro-canonical ensem-
ble. For the collinear case, this means the starting position and velocity can be specified by
a single random parameter, ?, in the range 0 ? ? ? 1; if ? is chosen correctly the distribu-
tion is flat in ?. The microcanonical ensemble is given where the parameter ? is the fraction
of a period of the bound motion and the position and velocity at ? = 0 is taken to be the
outer turning point, ? = 1,? = 0. The initial conditions for the fully three dimensional case
are somewhat more complicated but are well known (e.g. see Ref. [47]).
The incident electron is started at a distance ?2 = 100. In the collinear s-wave calcula-
tion, the initial velocity is ?2 = ?p2(Esc ?V0) where V0 = ?1=?2(0)+(1=Z)(1=[?1(0)+
?2(0)]) is the initial potential energy for electron 2. With this definition, the total energy
is exactly E = ?1 + Esc. We run the trajectories until at least one electron reaches the
distance ? = 120, at which point the energies of the electrons are computed. The ? range
for which the energies of both of the electrons are positive gives the ionization probability
P2e?, which is converted to cross section via 2e? = (?=k2)P2e?. The initial conditions
for the three dimensional calculation is similar but includes the impact parameter of the
incident electron.
For the collinear s-wave model, the initial conditions only depend on one parameter,
?. Therefore, it makes physical sense to examine the energies of the electrons versus ?. In
Fig. 4.1, we show the energies of the two electrons (solid and dashed curves) versus ? for
Z = 1 and Esc = 9:5. It is clear that relatively little energy is transferred to the bound elec-
tron (solid curve) except in a region near ? = 0:25 which is shown as an inset. The region
56
where both energies are positive corresponds to ionization. Because the microcanonical
distribution is flat in ? and the range of ? is one, the fraction ionized simply corresponds to
the range of ? where both energies are positive. The energies versus ? have similar types
of shape for all Z. There is a small region where substantial energy exchange can occur;
the width of the region decreases with increasing Z. The large energy transfer occurs when
the incoming electron and the bound electron are moving in the same direction near the nu-
cleus; for this type of motion, the incoming electron can do substantial work on the bound
electron, giving it enough energy to ionize.
4.2.3 Collinear distorted-wave method
In the collinear s-wave distorted-wave theory, the 1S cross section for single ionization
of a hydrogenic ion is given by:
= 32k3
i
Z E=2
0
d(k2e=2)
kekf
?R(k
es;kfs;ns;kis)+ R(kfs;kes;ns;kis)
?2 (4.17)
where the linear momenta (ki;ke;kf) correspond to the incoming, ejected, and outgoing
electron, respectively. The direct radial matrix element is:
R(kes;kfs;ns;kis) =
Z 1
0
dr1
Z 1
0
dr2 Pkes(r1)Pkfs(r2)Pns(r1)Pkis(r2)r
1 +r2
(4.18)
with a similar expression for the exchange term. The radial distorted waves needed to
evaluate the radial matrix elements are all Coulomb waves [48].
57
0 0.2 0.4 0.6 0.8 1?-40
-20
0
20
40
Scaled Electron Energy E
sc
0.2 0.22 0.24-10
0
10
20
Figure 4.1: Scaled final energy of the classical bound (solid curve) and incoming (dashed
curve) electrons versus the initial phase, ?, of the bound electron. The initial position and
velocity of the electron, ? and ?, are found by solving the classical equations of motion for
a time given by ? times the Rydberg period with ?(0) = 1 and ?(0) = 0. This graph shows
the result for Z = 1 and Esc = 9:5. Note that the width of the region where ? is positive
for both electrons (shown in the insert) is the ionization probability.
58
4.2.4 Collinear R-Matrix method
In the collinear s-wave R-Matrix method [49], symmetrized product states of single-
particle orbitals generated by the diagonalization of the one electron Hamiltonian
h(r) = p
2
2 ?
Z
r ; (4.19)
are used to span the basis for the diagonalization of the two electron Hamiltonian,
HR = H(r1;r2)+ 12?(r1 ?R) @@r
1
+ 12?(r2 ?R) @@r
2
(4.20)
where R is the box size and the Bloch operators (1=2)?(rfi ?R)(@=@rfi) ensure the Her-
miticity of HR. The ionization cross sections are calculated as a sum over all the excitation
cross sections to positive energy states of the basis in which HR is diagonalized to obtain
the R-Matrix. The R-Matrix is related to the K and J Matrices, which are then used to
determine the S Matrix. In this study, the excitation cross section from state i to state f is
given by
i!j(E) = ?k2
i
jSij ??ijj2 : (4.21)
For electron-impact scattering from hydrogen, we used a 2400-point lattice with a uniform
mesh spacing of ?r = 0:025 and a box size of R = 60:0 (a.u.).
59
4.3 Results
4.3.1 Effect of using a model potential: 1r1+r2 versus 1r>
Besides fully three dimensional treatments, the problem of electron-impact scatter-
ing from hydrogen has been studied within simple s-wave models primarily to reduce the
computational burden posed by solving the time-dependent Schr?odinger equation in three
dimensions. In one of these models, which was developed by Temkin [50] and Poet [51],
the interaction potential between the electrons is described by the s-wave term of the par-
tial wave expansion of the true electron-electron interaction potential, i.e. 1=j~r1 ?~r2j, and
is 1=r>. The Temkin-Poet model has been used extensively in the literature for studying
electron-impact scattering from hydrogen atoms (see e.g. Refs. [34, 52, 53, 54, 49] and
references therein). Our choice of the collinear s-wave model for studying electron-impact
ionization stems from the fact that although the Temkin-Poet potential is exact for s-waves,
it was shown not to yield the correct threshold law in one dimension [28] due to its cusp
along the r1 = r2 ridge. This cusp weakly pushes the probability density away from the
r1 = r2 ridge resulting in distortion of the depicted threshold law. On the other hand the
collinear s-wave potential yields the correct threshold law.
One of the differences between the Temkin-Poet and the collinear models is that the
collinear model potential is weaker than the exact s-wave potential of the Temkin-Poet
model. As a consequence, the collinear DW method agrees very well with both TDCC and
RMPS methods as demonstrated in the next two sections and Figs. 4.3 and 4.4. Due to the
60
perturbative nature of the DW method, a weaker potential means better agreement with the
non-perturbative methods.
To demonstrate this point, we have carried out DW and RMPS calculations for ioniza-
tion from ground state of hydrogen within Temkin-Poet model and compared to the cross
sections obtained using the collinear s-wave model. Results from these two calculations
are shown in Fig. 4.2. The pseudo resonances in our RMPS data were smoothed out using a
least squares fit to obtain the smooth RMPS curves in Fig. 4.2. As expected, there is better
agreement between the DW and RMPS for the collinear s-wave model potential than in the
case of the Temkin-Poet model potential. Cross sections for electron-impact ionization ob-
tained using the Temkin-Poet potential are roughly a factor of 3 larger than those obtained
using the model potential.
4.3.2 Electron-impact ionization of hydrogen
Fig. 4.3 shows electron-impact ionization cross sections from fully quantal collinear
s-wave TDCC, DW, and CTMC calculations for up to n = 4 of the hydrogen atom for
the singlet symmetry of the initial wave packet. The CTMC result for Esc = 1:5 starts
out ? 44% off the TDCC result at n = 1, then converges to the TDCC result by n = 4.
Note that the agreement between the CTMC and the TDCC methods gets better with the
increasing electron-impact energy.
The collinear s-wave DW results are in good agreement with the collinear s-wave
TDCC for the ground state ionization. For ionization out of states with higher n, DW
61
1 1.5 2 2.5 3Scaled Energy E
sc
0
2
4
6
8
10
12
? 2e-
(Mb)
Figure 4.2: Ionization probability out of n = 1 for H versus the scaled energy of the
incoming electron from collinear s-wave DW (solid curve), Temkin-Poet DW (dot-dash
curve), collinear s-wave RMPS (dash curve) and Temkin-Poet RMPS (dot-dot-dash curve)
calculations. Note that the agreement between the DW and RMPS is better for the weaker
collinear s-wave model potential. The data are fitted using least squares method to obtain
the smooth curves.
62
0
1
2
3
4
5
? 2e-(Mb)
TDCCDW
CTMC
02
46
810
1214
? 2e-(Mb)
2 4 6 8 10Scaled Energy E
sc
0
5
10
15
20
25
30
? 2e-(Mb)
2 4 6 8 10Scaled Energy E
sc
0
10
20
30
40
50
? 2e-(Mb)
n=1 n=2
n=3 n=4
Figure 4.3: Ionization Cross sections for e?-H(1S) scattering from n up to 4 calculated
in collinear DW, CTMC, RMPS, and s-wave TDCC methods as a function of the scaled
energy Esc of the incoming electron.
results slowly diverge from both TDCC and CTMC cross sections. This observation is
consistent with what has been seen from the fully three dimensional calculations [40].
We have further carried out collinear s-wave RMPS calculations to serve as a bench-
mark for our TDCC results. We have observed reasonable agreement between the collinear
s-wave RMPS and TDCC cross sections for all n we show in Fig. 4.3. This agreement can
be considered to be an independent confirmation of the convergence of the TDCC results
since both results are obtained via completely different quantal methods.
63
4.3.3 Electron-impact ionization of B4+
Electron impact ionization cross sections as a function of the scaled energy of the
incoming electron from collinear s-wave DW, TDCC, and CTMC calculations are plotted
for B4+ for initial states of n = 1;2;4 and 8 in Fig. 4.4 for the 1S symmetry. Cross sections
from the CTMC calculations at Esc = 1:5 start out as ? 74% larger than the TDCC cross
sections at n = 1, then very slowly converge to the TDCC results. For instance, at n = 8
the CTMC result differs from the TDCC result by ? 60% for scaled impact energy of
9:5. The agreement between the collinear s-wave TDCC and DW methods is excellent
for all n plotted in Fig. 4.4. Despite being perturbative, DW works very well even for the
highly excited states studied here because the interaction potential is weak and the transition
energy is large. The inelastic cross section to nearby n?s are not as accurate due to the small
energy spacing between Rydberg levels. Contrary to the case of hydrogen, both TDCC and
DW results are below the classical cross sections.
It is expected that the results of the CTMC calculation would not be as close to the
wave packet results as the H cross sections since higher nuclear charge manifests less clas-
sical behavior. Therefore, to see how high in n one should go for B4+ to reach agreement
with the CTMC results, we have performed TDCC calculations for n = 1;2;4;8;12;16;20
and 25 for the 1S symmetry of the initial wave packet. The ionization probabilities versus
the scaled energy of the incoming electron are plotted for all n in Fig. 4.5. The agreement
between the CTMC and TDCC results improves substantially slower than it does for hy-
drogen as the principal quantum number n increases. For instance for n = 25, the CTMC
results differ from the TDCC results by ? 17% at Esc = 5:5 and by ? 38% at Esc = 9:5.
64
0
2
4
6
8
? 2e-(10
-3 Mb)
TDCCCTMC
DW
0
1
2
3
? 2e-(10
-2 Mb)
2 4 6 8 10Scaled Energy E
sc
0
3
6
9
12
? 2e-(10
-2 Mb)
2 4 6 8 10Scaled Energy E
sc
0
1
2
3
4
5
? 2e-(10
-1 Mb)
n=1 n=2
n=8n=4
Figure 4.4: Ionization Cross sections for e?-B4+(1S) scattering from n = 1;2;4 and 8
calculated in collinear DW, CTMC, and s-wave TDCC methods as a function of the scaled
energy of the incoming electron.
65
2 4 6 8 10Scaled Energy E
sc
0
1
2
3
4
5
6
7
10-3
Ionization Probability n=1n=2n=4
n=8
n=12n=16
n=20n=25
CTMC
Figure 4.5: CTMC versus s-wave TDCC method for the electron impact ionization of B4+
for up to n = 25 for the 1S symmetry of the initial wave packet.
66
2 4 6 8 10Scaled Energy E
sc
0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
10-3
Ionization Probability
n=1n=2
n=4
n=8n=12
n=16n=20
n=25
CTMC
Figure 4.6: CTMC versus s-wave TDCC method for the electron-impact ionization of B4+
for up to n = 25 for the 3S symmetry of the initial wave packet.
We have performed the same set of TDCC calculations for the 3S symmetry of the
initial wave packet and the results are seen in Fig. 4.6. Ionization probabilities for the 3S
symmetry are smaller than those for the 1S symmetry as expected since the 3S wave packet
has a node along the r1 = r2 ridge. The 3S result from the TDCC calculation for n = 25 at
Esc = 9:5 is about a factor of 2 smaller than that from the CTMC calculation.
4.3.4 Effect of the ion stage
Since going from Z = 1 to Z = 5 drastically decreased the speed of convergence to
the CTMC results as a function of n, we repeated the collinear s-wave TDCC calculation
67
for an incoming electron with scaled energy of Esc = 9:5 for Z = 1 through 6 for initial
states with n = 1;2;3;4;8;12;16;20 and 25 to see the effect of the ion stage, Z, on the
convergence speed of the quantal ionization probabilities to the classical ones as a function
of n. In Fig. 4.7, we have plotted the ratio PQM(n;Z)=PCM(Z) for the 1S symmetry
of the initial wave packet where PQM(n;Z) is the ionization probability from the TDCC
calculation and PCM(Z) is the CTMC result. As noted before, for the case of Z = 1 the
TDCC method quickly converges to the CTMC result by n = 4 within ? 4% and by n = 8
within . 1% for Esc = 9:5. The oscillations about the CTMC result for higher n are at the
couple percent level and they may be due to the overall numerical accuracy of the TDCC
method, although they may also be due to quantum interference.
Fig. 4.8 shows ln[PQM(n;Z)] versus ln(Z) for n = 1;4;16 and 25 at the scaled
electron-impact energy of Esc = 9:5. On top of the data points we have plotted lines of the
form 2ln(Z) + C where C is constant for each n line and evaluated so that the last data
point for each n is fixed on its respective line. Note that for n = 1 and 2 all the points lie
on their lines whereas for the highest two ns the low Z substantially deviates from their
respective lines. The fact that the slope of the lines is 2 means that for low n the ionization
probability scales ? 1=Z2 and for high n it scales ? 1=Z2 only for the high Z.
4.3.5 Node structure near the scattering center
Looking at the classical trajectories that give rise to ionization, we can roughly esti-
mate the size of the region where the energy exchange between the electrons takes place
at the time of scattering from the core. To gain physical insight about the discrepancy
68
0 5 10 15 20 25n0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
PQM
(n,Z) / P
CM (Z)
Z=2
Z=1
Z=3
Z=4Z=5
Z=6
Figure 4.7: Ratio of the quantal and classical ionization probabilities,
PQM(n;Z)=PCM(Z), as a function of n for Z = 1?6.
69
0 0.5 1 1.5 2ln[Z]-7.0
-6.5
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
ln[PQM
(n,Z)] n=1
n=4
n=16n=25
Figure 4.8: ln[PQM(n;Z)] versus ln(Z) for various n at Esc = 9:5. Note that the straight
lines have a slope of 2 which implies ? 1=Z2 scaling of the ionization probability for n = 1
and 4 and for higher Z of n = 16 and 25.
70
between the classical and quantal results even for such high n, we may draw analogy to
geometrical-wave optics correspondence. According to the geometrical-wave optics corre-
spondence, light exhibits classical behavior when the wavelength is much smaller than the
size of the region with which it interacts. In other words, we recover geometrical optics as
the wave number gets larger within the interaction region. Applying the same idea to the
matter waves, we can expect the system to behave more classically as the number of nodes
increase in the region where classical energy exchange between the electrons takes place.
The classical size of the scattering region that give ionization is ?c ? 0:1 in scaled units
(which is Lc = 2n2?c) for Z = 5 when the scaled energy of the incoming electron is 9:5.
Using the Z-scaled potential, this characteristic size can be shown to be Lc = 51:2 (a.u.)
for n = 16. For the lowest energy component in the wave function, the kinetic energy
is approximately equal to the Z-scaled potential energy, i.e. k2=2 ? 1=r, from which
k(r) ? p2=r. We can use half wavelength in this energy regime to obtain a typical
length scale of the region where the scattering takes place from the core. Half wavelength
corresponding to k(r) is
? ?
Z Lq
0
k(r)dr = p8Lq (4.22)
from which Lq ? ?2=8 ? 1:2. Comparing this with the classical size of the scattering
region yields an intriguing result since the size of the classical scattering region is about
a factor of 43 larger than the quantal length scale; therefore one would expect the quantal
results to be in good agreement with the classical result at n = 16.
71
In Fig. 4.9 we have plotted the absolute value square of the wave function for both
Z = 1 (top panel) and Z = 5 (bottom panel) at n = 16 for scaled energy Esc = 9:5.
Various classical trajectories that give ionization are also plotted on top of each probability
distribution. In these plots, we can define regions by using the points of the trajectories at
which the coordinate of one electron vanishes, which we call bounce points. Note that for
Z = 1, the number of nodes in the region bound by the last two bounces of the outmost
classical trajectory is considerably larger than in the case of Z = 5. For Z = 5, Eq. (4.22)
with Lq ! Lc yields the number of nodes in the classical scattering region as ?p8Lc=? ?
6 which roughly equals the number of nodes in the bottom panel of Fig. 4.9 bounded by
the last two bounces of the outmost classical trajectory.
4.4 Conclusions
We have performed collinear s-wave time-dependent close-coupling (TDCC), classi-
cal trajectory Monte Carlo (CTMC), and distorted-wave (DW) calculations for electron-
impact ionization of H-like ions with Z = 1 ? 6 for various principal quantum numbers
up to n = 25. We have observed good agreement between the s-wave TDCC and the
CTMC methods for hydrogen as the principal quantum number reached n = 4. The good
agreement between the collinear s-wave TDCC and the classical results for hydrogen in the
high-n limit is in accord with the expectations raised by the Bohr correspondence principle.
Repeating the same set of calculations for B4+, we have found that one has to go to much
higher n for the collinear s-wave TDCC and the CTMC methods to agree. We showed that
the TDCC results converge to the classical results very slowly with increasing n, which is
72
Figure 4.9: Contour plots for j?n=16(r1;r2)j2 for H (top panel) and B4+ (bottom panel)
at the time of scattering from the nucleus at Esc = 9:5. The lines in each figure represent
classical trajectories that lead to ionization.
73
in agreement Griffin et al. [40] have observed in their full three dimensional calculations
up to n = 4. At n = 25 and for the 1S symmetry of the initial wave packet, the TDCC
method yields ionization probability which is ? 35% less than the classical result at the
highest electron-impact energy we considered. For the same n and electron-impact energy,
the TDCC result is about a factor of 2 less than the CTMC result for the 3S case.
The collinear s-wave DW calculations for hydrogen resulted in cross sections which
are in good agreement with the collinear s-wave TDCC calculations for the ground state,
and gradually got worse as n is increased. In the case of B4+, we have shown that the
agreement between the DW and TDCC is excellent up to n = 8. This suggests that, for
the low lying excited states, the DW is just as good as the exact TDCC method, which
makes it useful for the plasma modeling and diagnosing purposes. One would expect a
perturbative method to work when only a small fraction of the initial state ionizes, leaving
most of the initial population in the initial state. The fact that the collinear s-wave DW
method works so well even though it is perturbative may be due to the small interaction
potential compared to the ionization energy. The energy difference between the adjacent
states decreases as ? 1=n3 for high n and the incoming electron can knock the atom in a
nearby n state instead of ionizing it. This is even more probable for the model potential we
used, which is weaker than the actual 1=j~r12j potential.
To illustrate the effect of the ion state Z on the ionization probabilities, we have carried
out collinear s-wave TDCC calculations for Z up to 6 and for n up to 25 at the scaled
electron-impact energy Esc = 9:5. We found that for low-n the ionization probability
scales like ? 1=Z2 and for high-n it behaves the same only for high Z.
74
In order to understand the physical size scales involved and gain some insight on
the dynamics behind this slowly converging behavior, we drew analogy to geometrical-
wave optics correspondence and estimated the number of nodes in the region near the
core outlined by the classical trajectories that give ionization. We have found that the
size of the classical scattering region is about a factor of 50 larger than the quantal length
scale; therefore, one would expect the TDCC and CTMC methods to agree, which is not in
accord with what we have observed. The peculiar dynamics behind this slow convergence
is intriguing and may be due to an inherent characteristic of the line-land potential or a
quantum interference effect.
75
CHAPTER 5
RADIATIVE CASCADE OF HIGHLY EXCITED HYDROGEN ATOMS IN STRONG
MAGNETIC FIELDS
5.1 Introduction
In this chapter, we investigate the radiative cascade from the Rydberg states with en-
ergies effectively corresponding to hydrogenic n-manifolds of up to n = 35 in magnetic
field strengths of up to 4:0 T. We have also followed the radiative cascade from these initial
states in the presence of the black body radiation since the positron and anti-proton plasmas
have a temperature of 4 K in case of the ATRAP experiment.
High-jmj Rydberg atoms in the presence of strong magnetic fields were also investi-
gated by Guest and Raithel [55]. They found that in case of the high-jmj states in strong
magnetic fields the motions that are parallel and transverse to the magnetic field are adia-
batically separable. They studied the break down of the adiabaticity as they decreased jmj
from 200 to 40 and observed that the motion in states with jmj ? 200 ? 80 are adiabati-
cally separable to the extent where no more than 50% of the energy levels are overtaken by
the non-adiabatic couplings. As they decreased jmj further down to jmj ? 40 the energy
spectrum experienced a transition from adiabaticity to non-adiabaticity. They have also
calculated the natural and thermally enhanced decay rates for the states in which the mo-
tion is adiabaticaly separable [56]. In this paper we perform calculations for the jmj. 40
region which has not yet been treated.
76
In this study, we also exploit the fact that the size of the positron and anti-proton
plasmas are small compared to the size of the trap. As a result, once an anti-hydrogen
atom is formed, it quickly leaves the plasma and moves into the vacuum of the trap. If
the anti-atom is trapped then it spends its time in this vacuum radiatively decaying without
positron?anti-atom collisions. Therefore this study only considers radiative cascade in
vacuum.
The radiative cascade from high Rydberg states exhibit interesting details. Reference
[57] studied the phase space trajectory of an initial n;? state cascading to the ground state
in zero magnetic field. In principle, a similar treatment using the quantum numbers in the
magnetic field could be performed but is beyond the scope of this study.
We also give a semiclassical treatment for the radiative cascade from the highly excited
circular Rydberg states for the sake of physical transparency. We have shown that the
population time for the ground state strongly depends on the fraction of circular high-jmj
states in the initial distribution. We considered the relative effect of the cyclotron motion
with respect to the magnetron motion in circular high-jmj states on the decay rates and
draw conclusions on the affinity of these states to magnetic field gradients.
All our calculations are for regular hydrogen rather than anti-hydrogen but the results
apply equally well to anti-hydrogen. In this study, atomic units are used throughout unless
stated otherwise. The work presented in this chapter has been published as a paper in
Physical Review A [58].
77
5.2 Theory
Our results are obtained using an approximate Hamiltonian of a hydrogen atom placed
in a magnetic field,
H = (~p?
~A=c)2
2 ?
1
r (5.1)
where ~A is the vector potential. In the Appendix we discuss the limitations of this Hamil-
tonian and show that it serves our purposes. With the choice of symmetric gauge ~A =
?12~r?~B and a uniform magnetic field of magnitude B0 along the z direction, i.e. ~B = B0^z,
the Hamiltonian in spherical coordinates becomes,
H = Hatom + 2Lz +
2
8 r
2 sin2 (5.2)
where Hatom is the atomic Hamiltonian in the absence of the magnetic field,
Hatom = p
2
2 ?
1
r (5.3)
and = B0=(2:35 ? 105 T) is the magnetic field strength in atomic units when B0 is in
Tesla. Note that the total Hamiltonian H has rotational symmetry about the z axis like the
unperturbed atomic Hamiltonian.
In the next subsection, we describe the method used to compute the eigenenergy spec-
trum of the hydrogen atom in the magnetic field where we diagonalize the total Hamiltonian
H in the basis spanned by the eigenstates of the atomic Hamiltonian Hatom. In the second
subsection, we evaluate the dipole matrix elements in the magnetic field by rotating the
78
dipole matrix elements calculated in the fn;?;mg basis into the dipole matrix elements in
the fm;?g basis. The radiative decay rates in the magnetic field is then evaluated by using
these rotated dipole matrix elements. In the last subsection, we solve the time dependent
rate equation for an initial probability distribution of the eigenstates of H, which tells us
how the initial probability distribution evolves in time.
5.2.1 Calculation of the Energy Spectrum in the Magnetic Field
To calculate the new energy spectrum in the magnetic field, we diagonalized the matrix
representation of the full Hamiltonian H in the basis spanned by the eigenstates ?n?m of the
unperturbed atomic Hamiltonian. The matrix elements of the Hamiltonian H = Hatom +
Hmagnetic in this basis are
< n?mjHjn0?0m0 > =
h
?n? + 2m
i
?n;n0??;?0?m;m0 (5.4)
+
2
8 < n?mjr
2 sin2 jn0?0m0 >
where ?n? are the eigenvalues of Hatom, i.e. energies of the hydrogen atom in the absence
of the magnetic field. Noting that sin2 = (2=3)[1?p4?=5 Y 02 ( ;`)], the angular part of
the last integral in Eq.(5.4) can be evaluated as
< ?mjsin2 j?0m0 >= 23??;?0?m;m0 ? 23(?1)m
?
p
(2?+1)(2?0 +1)
0
B@? 2 ?0
0 0 0
1
CA
0
B@ ? 2 ?0
?m 0 m0
1
CA: (5.5)
79
Selection rules for these matrix elements can be deduced from the fact that the 3-j symbols
must satisfy the triangle relations (Ref. [62]) simultaneously in order to survive. For the 3-j
symbols involved in Eq. (5.5) this condition is realized as j???0j ? 2. Also the survival
of the first 3-j symbol in the Eq. (5.5) requires that the sum of the elements on the first row
must be an even integer. This condition along with j???0j ? 2 implies that the magnetic
field will only induce transitions for which ?? = 0 or ?? = ?2. Therefore with the
definition
Rn;?n0;?0(?) =
Z
Rn;?(r)r?Rn0;?0(r)dr (5.6)
where Rn;?(r) is the radial part of the eigenfunction ?n?m, the matrix elements of the full
Hamiltonian H in the fn?mg basis become,
< n?mjHjn0?0m0 > =
?
?n? + 2m
?
?n;n0??;?0?m;m0 (5.7)
+
2
12
"
?m;m0??;?0 ?(?1)m
p
(2?+1)(2?0 +1)
?
0
B@? 2 ?0
0 0 0
1
CA
0
B@ ? 2 ?0
?m 0 m0
1
CA
#
Rn;?n0;?0(2):
Here Rn;?(r) is generated on a square root mesh by direct integration of the Schr?odinger
Equation in a box. All orbitals satisfy the boundary condition such that Rn;?(rf) = 0 where
the box size rf is chosen to be larger than the size of the physical states in our simulations.
By choosing this boundary condition, we greatly increase the rate of convergence with n
80
because eigenenergies ?n? increase rapidly with n for large n [59]. Thus our basis has
substantial continuum character.
We have checked the convergence of our results with respect to the box size rf, the
number of radial grid points N and the nmax and ?max describing the basis set used in the
diagonalization of H. For example, in a 4:0 T field, decay times obtained for n = 30 in
a 2000 (a.u.) box with 8000 radial points using a basis with (nmax;?max) = (45;40) differ
from the decay times obtained in a 3000 (a.u.) box with 12000 points using a basis described
by (nmax;?max) = (50;45) at a few percent level. The energy of the state (nmax;0) in
the 2000 (a.u.) box is ?45;0 = 4:1138 ? 10?4 (a.u.) and the energy of the state (30;0) is
?30;0 = ?5:5526 ? 10?4 (a.u.). For the 3000 (a.u.) box the state with (nmax;0) has the
energy ?50;0 = 3:8628 ? 10?5 (a.u.) whereas energy of the state (30;0) is the exact value
?5:5556?10?4 (a.u.).
Diagonalization of the matrix with the matrix elements given by Eq. (5.7) gives the
new energy spectrum f~?jg of the hydrogen atom in the magnetic field. Note that in the pres-
ence of the magnetic field the good quantum numbers are the magnetic quantum number
m and the z-parity ? rather than the principal quantum number n and the orbital quantum
number ?.
As a check on our codes, we have compared our energies for hydrogenic n = 23
manifold in a 4:7 T field for states with m = 0;1;2 listed in Ref. [60] and found very good
agreement except at a few particular energy values. The difference might be due to the
fact that usage of non-orthogonal basis set methods, like the one used in Ref. [60], can be
unstable, e.g. their energies may not be converged for those particular states.
81
5.2.2 Calculation of the Dipole Matrix Elements
To calculate the dipole matrix elements in the magnetic field, we first calculated the
dipole matrix elements in the fn?mg basis and rotated those matrix elements obtaining the
dipole matrix elements in the fm?g basis of the full Hamiltonian H.
Dipole matrix elements for the linearly polarized states in the fn?mg basis are
< n?mjzjn0?0m0 >=< n?jrjn0?0 >< ?mjcos j?0m0 > : (5.8)
Again noting that cos = p4?=3 Y 01 ( ;`) one can deduce
< n?mjzjn0?0m0 > = (?1)m
p
(2?+1)(2?0 +1) (5.9)
?
0
B@ ? 1 ?0
0 0 0
1
CA
0
B@ ? 1 ?0
?m 0 m0
1
CARn;?
n0;?0(1):
In the same manner, for the left and right hand circularly polarized states one obtains
< n?mjx?iyp2 jn0?0m0 > = currency1(?1)m
p
(2?+1)(2?0 +1) (5.10)
?
0
B@ ? 1 ?0
0 0 0
1
CA
0
B@ ? 1 ?0
?m ?1 m0
1
CARn;?
n0;?0(1):
respectively. Selection rules for the magnetic field induced transitions can be deduced from
the properties of the 3-j symbols in Eqs.(5.9) and (5.10) as before. Since the second row of
the second 3-j symbols in both of the expressions need to add up to zero for the survival of
82
the term, we can immediately conclude that there is no m change for the linearly polarized
states while the circularly polarized states change m by 1. Also the first 3-j symbols in the
terms tell us that j???0j? 1 and j?+?0j must be an odd integer implying that ? must change
by one for the term to survive. Since ?min = m + ? and ? increases from ?min to n ? 1
in integer steps for a given m it can be concluded that j?mj + j??j = 1 for an allowed
transition.
To obtain the dipole matrix elements in fm?g space, we will simply rotate the matrix
elements in Eqs.(5.9) and (5.10) computed in the fn?mg basis. Let us index the state (n?m)
by j and the state (m?) by fi. Denoting the new rotated dipole matrix element by dfi;fi0 we
have
dfi;fi0 =
X
j
X
j0
< ?jjd j?j0 > U?j;fiUj0;fi0 (5.11)
where d is the dipole operator which is z for linearly polarized states and (x?iy)=p2 for
the left and right hand circularly polarized states respectively. The rotation matrix U is the
matrix whose columns are the eigenvectors of the full Hamiltonian H in the fn?mg basis
which is obtained by direct diagonalization of H,
X
j
HkjUjfi = Ukfi~?fi (5.12)
where ~?fi is the fith eigenenergy of the Hamiltonian H with the eigenstate Ufi.
Having obtained the dipole matrix elements we now can calculate the partial decay
rates of the states of the hydrogen atom in the magnetic field. Partial decay rate from initial
83
state i to final state f in length gauge is given by
?fi = 43c3(~?i ?~?f)3j < ~?fj~r j~?i > j2 (5.13)
where ~?fi and ~?fi are the energy and the eigenfunction of state fi of the hydrogen atom in
the magnetic field and 4=(3c3) = 5:181?10?7 in atomic units. Note that the usual velocity
gauge expression derived in the absence of a magnetic field becomes
?fi = 43c3(~?i ?~?f)j < ~?fj~p? ~A=c j~?i > j2 (5.14)
with the inclusion of the magnetic field because the physical momentum of the electron is
now (~p? ~A=c) as being different from the canonical momentum ~p of the electron.
5.2.3 Solution of the Rate Equation
Having obtained the partial decay rates ?fi;fi0 in the magnetic field, we solved the time
dependent rate equation to simulate the flow of initial probability distribution among the
states. The rate equation for the probability }fi of finding the atom in state fi is
d}fi
dt = ??fi}fi +
X
fi0>fi
?fifi0}fi0 (5.15)
where ?fi is the total decay rate of state fi,
?fi =
X
fi0 fi0 implies the constraint ~?fi0 < ~?fi. The first term on the right
hand side of the rate equation represents the total flow of probability out of the state fi while
the sum in the second term is due to the total flow of probability into state fi from all higher
states fi0. The ?fi;fi0s span several orders of magnitude making Eq. (5.15) difficult to solve.
Assume the solution to be of the form }fi = }(0)fi +}(1)fi where }(0)fi satisfies
d}(0)fi
dt = ??fi}
(0)
fi : (5.17)
The time propagation of this equation by one time step ?t for a given }(0)fi at a time t is
}(0)fi (t+?t) = e??fi?t}fi(t): (5.18)
Substituting }fi = }(0)fi +}(1)fi into the rate equation and noting thatPfi }fi(0) = Pfi }(0)fi (0) =
1 , one obtains,
}(1)fi (t+?t) =
X
fi0>fi
Bfifi0(1?e??fi0?t)}fi0(t): (5.19)
where Bfifi0 ? ?fifi0=?fi0 is the branching ratio. Combining Eq.(5.18) and Eq.(5.19) }fi
becomes,
}fi(t+?t) = e??fi?t}fi(t)+
X
fi0>fi
Bfifi0(1?e??fi0?t)}fi0(t): (5.20)
For an initial distribution of probability among the states one can propagate the probability
distribution along time with the knowledge of the decay rates. We have checked the conver-
gence of our results with respect to the maximum effective cut off quantum numbers nc;?c
of the basis states used to describe the radiative decay from a given initial distribution. The
85
states that make up this initial distribution have effective quantum numbers that are smaller
than these cut off nc;?c quantum numbers. Convergence with respect to the time step ?t
was also checked.
5.3 Results
We will start by making some rough estimates for the effect of the magnetic field on
the classical velocity of the electron in a circular orbit to obtain an idea of the extent to
which the decay rates from these states change from their field free estimates. The second
subsection contains results from calculations for the radiative cascade in 1:0 T, 2:0 T, 3:0 T
and 4:0 T fields where the initial distribution of states with energies corresponding to var-
ious principal quantum numbers up to n = 35 are completely ?;m mixed. These results
are compared with some results obtained using an analytical rate formula. In the following
subsection we present results from calculations for which the initial states are localized in
the highest jmj regions of an n manifold and again some results obtained using an analyt-
ical formula for the decay rates. The effect of the black body radiation will be taken into
account to see if the results of the previously presented radiative cascade calculations are
affected by a 4 K [5, 6] radiation field. Finally we present a semiclassical treatment for
the calculation of the radiative decay rates of circular orbits and compare them to those
calculated quantum mechanically.
86
5.3.1 Rough Estimates
For a classical electron in a closed circular orbit in a Coulomb field, Newton?s laws
give
mev
2
0
r = k
e2
r2 (5.21)
where we have employed the SI units. Solution of this equation for the classical velocity
immediately yields v0 = ?pke2=mer. In the presence of a uniform magnetic field of
strength B0 directed perpendicular to the plane of the circular orbit, Eq. (5.21) becomes
mev
2
r = k
e2
r2 +evB0 (5.22)
whose direct solution for v now yields
v = eB0r2m
e
?
r?eB
0r
2me
?2
+v20: (5.23)
Note that the ? sign in front of the square root term in Eq. (5.23) implies the fact that
electron either speeds up or slows down depending on the sign of Lz of the orbit for a given
direction of the magnetic field. Expanding the square root to the 4th order in binomial
series for the case eB0r=(2me) < v0, one approximates the classical velocity in magnetic
field v by
v ? eB0r2m
e
?
"
v0 + 12v
0
?eB0r
2me
?2
+ 18v3
0
?eB0r
2me
?4#
: (5.24)
87
Noting that k = me = e = 1 in atomic units and r ? n2 for a circular classical orbit with
energy ? ??1=(2n2), we can infer
vcln ??1n
"
1currency1 n
3
2 +
1
8( n
3)2 + 1
128( n
3)4
#
(5.25)
where we have recalled that = B0=(2:35 ? 105 Tesla) is the magnetic field in atomic
units when B0 is in Tesla.
In the field free case, Eq. (5.25) predicts vcln ? 1=n for all n which is exactly the
quantum mechanical velocity vn = 1=n. For n = 10, the zero field velocity differs from
the B0 = 4:0 T value by ? 1%. At n = 20 the difference between the velocities for 2:0 T
field and the zero field is ? 3% and ? 7% at 4:0 T. When we are up to n = 30, vcln differs
from zero field value by ? 12% for B0 = 2:0 T, by ? 19% for B0 = 3:0 T and by ? 26%
for B0 = 4:0 T. This suggests that magnetic field strengths of interest in the anti-hydrogen
experiments has little effect on the circular orbits with n < 10 while starting to effect the
velocities in the orbits with n ? 20 within ? 10% and in the orbits with n ? 30 within
? 30%. Noting that the radiative decay rate scales like a2 ? v4=r2, we can infer that
the magnetic field affects the decay rates more strongly than it affects the velocity of the
electron in the orbit.
In the limit where eB0r=2me v0 Eq. (5.23) can be expanded to 2nd order as
vcln ? eB0r2m
e
? eB0r2m
e
?
1+ v
2
0
2(eB0r=2me)2
!
(5.26)
88
yielding vcl+ ? eB0r=me + v20=(eB0r=me) for the orbit in which the electron is sped up,
and vcl? ? ?v20=(eB0r=me) for the orbit in which the electron is slowed down by the
magnetic field. The case where the electron is slowed down corresponds to the guiding
center approximation since the Larmor period is much smaller than the orbital period of
the electron. In 4:0 T field, the absolute value of the ratio of the velocities in orbits with
same energy but of opposite helicities, i.e. jvcl+=vcl?j, is ? 1:0 for orbits with n up to 25,
? 1:2 at n = 30 and ? 2:2 at n = 40.
5.3.2 Energy Spectrum in the Magnetic Field
From the diagonalization of the full Hamiltonian H, we have found that the energies
of the states with negative magnetic quantum numbers decrease whereas the energies of the
states with positive magnetic quantum numbers increase with the increasing magnetic field
strength. In the anti-hydrogen experiments, magnetic field will be deliberately generated
in such a way that it is stronger near the walls of the cylindrical container in which anti-
proton and positron plasmas are mixed. Therefore the states with negative m?s are attracted
by the increasing magnetic field strength and states with positive m?s are repelled by the
increasing magnetic field. Hence we can expect the states with negative m that are close to
the magnetic field gradient to plummet to the wall of the container faster than those with
positive m. Therefore atoms that are formed with high field seeking character will need to
decay especially rapidly if they are not to be accelerated by the magnetic field gradients. For
extremely negative m ? ?200 of [56], the atoms could have low field seeking character
because the positive magnetic moment from the electron?s cyclotron motion is larger in
89
magnitude than the negative magnetic moment from the guiding center character of the
electron?s motion around the proton.
5.3.3 Radiative Cascade Starting from Completely Random (?;m) Distribution
Since n is not a good quantum number when magnetic field is present, the initial
probability distribution was chosen such that the probability of finding the atom in a state
whose energy is between energies of hydrogenic n and n ? 1 manifolds is equally parti-
tioned between the states that lay in this energy range. Initial distributions that correspond
to n = 10;15;20;25;30 and 35 in magnetic fields of strengths 1:0 T, 2:0 T, 3:0 T and 4:0 T
have been explored. Furthermore, instead of monitoring the flow of probability of finding
the atom within a particular state, we have monitored the probability of finding the atom
in a state whose energy lies in a range which embraces 5-10 n manifolds. Evolution of the
probability distribution with time when started out in states with effective n = 35 in a 4:0 T
field are plotted in Fig. 5.1. As time progresses states of lower energies become populated,
eventually with all population ending up in the ground state. At early times the fast de-
cay is mostly coming from the low-jmj states which cascade by large ?n steps. The slow
decay at later times is due to the states with large-jmj which cascade through ?n ? ?1
transitions.
In the anti-hydrogen experiments, the formed anti-hydrogen atoms move through a
magnetic field that is spatially dependent, e.g. stronger near the walls of the cylindrical con-
tainer. Since the magnetic force on the center of mass of the anti-hydrogen atom depends
on its internal energy, it is important to know the time scale for the variation of the internal
90
0 0.01 0.02 0.03 0.04 0.05Time (s)0
0.2
0.4
0.6
0.8
1
Probability
E < En=1.5E
n=1.5 < E < En=5.5E
n=5.5 < E < En=15.5E
n=15.5 < E < En=25.5E
n=25.5 < E < En=35.5
Figure 5.1: Flow of probability in the completely ?;m mixed distribution for n = 35 in
a 4:0 T field as a function of time. Each curve represents the total probability of finding
the atom in a state whose energy is in one of the particular energy ranges indicated on the
legend.
91
energy of the anti-hydrogen atom to estimate the time it has before colliding with the wall
of the container and get annihilated. A practical percentage of the anti-atoms should decay
to the ground state before they get destroyed to be used in the CPT and Lorentz violation
experiments. Therefore we have calculated the time it would take to populate the ground
state by 10%, 20%, 50% and 90% if everything starts from states that are in aforementioned
n regimes. In Figs. 5.2(a) and (b) the time it takes 10% and 50% of the hydrogen atoms to
decay down to the ground state as a function of the given initial state n is plotted for four
different magnetic field strengths. It can be seen that for n . 25 the magnetic field makes
no important difference in terms of populating the ground state by 10% and when we are
up to n = 35 we see ? 0:4 ms separation between 1:0 T and 4:0 T values. This implies
that the increase in the time it takes to populate the ground state by n is mainly a zero field
effect. When we look at the 50% population plot it can be noted that for n . 25 there
is no change coming from increasing the field strength but by n = 30 there is ? 0:8 ms
and by n = 35 there is ? 3:5 ms separation between the lowest and highest magnetic field
strengths.
The thermal speed, pkBT=M, of an anti-hydrogen atom at 4 K [5, 6] is ? 260 m/s
which is ? 26 cm/ms and higher by a factor of 2 at 16 K [4]. Noting that the diameter of
the cylindrical container is approximately a couple of centimeters, we can conclude that in
the field regime of the experiments there is not enough time to populate the ground state
by even 50% before the anti-hydrogen atoms hit the walls and get destroyed. According to
Fig. 5.2(a), for n = 35 the time needed to populate the ground state by 10% is about 0:2
ms in which the thermal anti-hydrogen atoms can move ? 6 cm. Hence we can conclude
92
0.05
0.10
0.15
0.20
0.25
0.30
t 10%(ms)
B = 1 TB = 2 T
B = 3 TB = 4 T
10 15 20 25 30 35n0
2.0
4.0
6.0
8.0
t 50%(ms)
(a)
(b)
Figure 5.2: Time required to populate the ground state of the hydrogen atom by (a) 10%
and (b) 50% as a function of the effective hydrogenic principal quantum number n for 1:0
T, 2:0 T, 3:0 T and 4:0 T fields. Initial distribution of states is completely ?;m mixed.
93
that at 4 K and in magnetic fields up to ? 4 T the ground state population will not be able
to exceed 10%. Since ATHENA experiment runs at a higher temperature, this population
percentage would be even lower in their experiment. It seems clear that in order to trap
substantial amounts of anti-hydrogen, the anti-atoms need to be trapped while in Rydberg
states.
Since the lifetime of the completely ?;m mixed case increases ? n5 for a highly
excited hydrogenic state in the absence of a magnetic field, time scaled by n5 would stay
roughly constant as n increases. Therefore we scaled our results for the decay times by n5
to see the n range for which decay times are practically unaffected by the magnetic field.
These scaled plots can be seen in Fig. 5.3 where we have plotted scaled time needed for
the ground state to be populated by 10%, 50% and 90% versus initial n. In all the plots
scaled time stays approximately constant up until n ? 25 meaning that decay rates of these
states are not affected by the magnetic field compared to the field free case; this is due to
the fast direct decay to low n-states. Scaled time then starts to spread out noticeably with
growing n there on. The states with higher-jmj decay more slowly and have rates that are
more strongly affected by the magnetic field. This is the reason for the larger magnetic
field effect in Figs. 5.3(b) and (c) than in Fig. 5.3(a).
For comparison, we have also done the same calculations for zero field case using
decay rates calculated via an analytical formula which assumes complete ? mixing of the
states populated in the initial distribution when there is no magnetic field [62]. In the actual
decay process, states with low-jmj can make bigger ?n jumps towards the ground state
than the ones with high-jmj, resulting in the pile up of population in states with jmj ? n
94
0
1.0
2.0
3.0
4.0
5.0
t 10%/n
5 (ps) B = 1 T
B = 2 TB = 3 T
B = 4 T
0.05
0.10
0.15
0.20
t 50%/n
5 (ns)
10 15 20 25 30 35n0
1.0
2.0
3.0
4.0
5.0
t 90%/n
5 (ns)
(a)
(b)
(c)
Figure 5.3: Scaled time required to populate the ground state of the hydrogen atom by (a)
10%, (b) 50% and (c) 90% as a function of the effective hydrogenic principal quantum
number n for 1:0 T, 2:0 T, 3:0 T and 4:0 T fields. Initial distribution of states is completely
?;m mixed.
95
over time. Since the analytical formula assumes complete ? mixing within individual n-
manifolds, it does not simulate the actual behavior after the piling up of population in
high-m states. Therefore the applicability of this analytical formula is limited to short time
periods from the beginning. In this picture, partial decay rate from state ni to state nf goes
like
?ni!nf = 8fi
3
3p3?
1
n5inf
1
1?(n2f=n2i) (5.27)
where fi is the fine structure constant. The total decay rate is again calculated as usual,
?ni =
ni?1X
nf=1
?ni!nf: (5.28)
For n = 20, time needed for population of the ground state to reach 10% is calculated using
the rates from Eq. (5.27) is about 27% bigger than our quantal result while 90% population
result is off by about 60%. Going up to n = 30 we found that time for 10% population of
the ground state is longer by ? 14% and 90% population time is longer by ? 76% than our
quantum mechanical result.
5.3.4 Radiative Cascade Starting from High-jmj Distribution
The dominant process in the anti-hydrogen formation is believed to be three body
recombination. It is also known that the atoms formed in this process are likely to be in
a high m state [12]. This is clearly a different situation than the completely ?;m mixed
case where every state within a prescribed energy range is equally likely to be populated.
To simulate the case of the three body recombination we restricted our initial states not just
96
being in the [?n?1;?n] energy range but also their m values to be extreme to qualify as one
of the initial states. We investigated both extreme cases in which m ? n and m ? ?n
although it seems that the m ? ?n case is more likely. High-jmj states are chosen such
that there are only a few states that make up the initial distribution.
For comparison, we also calculated the time it takes the atom to reach to its ground
state in zero magnetic field by an approximate formula for the decay of circular states:
?n!(n?1) = 23(n? 1
2)
2n3c3: (5.29)
The decay rate estimates we made using Eq. (5.29) for zero magnetic field are in agreement
with the exact quantum results within about 5% for the states with n = 2 and get better
with increasing n reaching agreement with the quantal result within 0:5% by n = 5 for
B = 0.
The results of the quantum mechanical calculations for 10% population of the ground
state are plotted in Fig. 5.4 for magnetic field strengths of 1:0 T and 4:0 T for highest and
lowest m states in their respective n manifolds with the zero field results for comparison.
Both 1:0 T and 4:0 T curves for high m lie just below the zero field results throughout the
whole n range from 10 to 30, while for the states with lowest m?s 1:0 T field lifts the 10%
population time just above the zero field curve and 4:0 T field stretches the required time
for 10% population of the ground state from the zero field value by a factor of ? 13 for
n = 30. Note that the largest effect is for the states with m ??n, only becoming important
above n ? 25. The fact that the 4:0 T field has a more pronounced effect for the jmj??n
97
10 15 20 25 30n0.0
20.0
40.0
60.0
80.0
t 10% (ms)
B = 1 T (low m)B = 1 T (high m)
B = 4 T (low m)B = 4 T (high m)
B = 0 T
Figure 5.4: Time required to populate the ground state of the hydrogen atom by 10% as a
function of the effective hydrogenic principal quantum number n for 1:0 T and 4:0 T fields.
Initial distribution of states only involves states with high jmj.
mixed distribution than for the completely ?;m mixed distribution is because now the states
in the distribution are mostly circular and most of the transitions are of ?n = ?1 type.
In Fig. 5.5, the times required for the population of the ground state by 10%, 50% and
90% are scaled by n6 since now initial distribution involves only circular states. Scaled de-
cay times are plotted versus n for the high-jmj states again in magnetic fields of strengths
1:0 T and 4:0 T. In all three plots, both of the m ? n cases lie just below the zero field
98
curve. For the states with m ??n, 1:0 T field cannot alter the scaled time of 10% popula-
tion by more than ? 30% from the field free case but for the 4:0 T field scaled population
time is larger than zero field case by a factor of ? 2:5 at n = 25 and by a factor of ? 11 by
n = 30. Again starting from the states with m ? ?n, 50% population of the ground state
in 1:0 T field does not differ from the field free value by more than ? 25% but for the 4:0 T
field scaled population time is larger than zero field case by a factor of ? 3 at n = 25 and
by a factor of ? 11 by n = 30. For the 90% population of the ground state, 1:0 T field again
does not do a good job in lifting the scaled time by n ? 25 from the zero field value and the
4:0 T field is again gives higher scaled time than zero field by a factor of ? 2:5 at n = 25
and by a factor of ? 15 at n = 30. The fact that the decay time for the negative m-states
is much longer than the positive m-states can be understood from classical arguments. The
classical radiation is proportional to the square of the acceleration of the electron which
goes ? v2=r. Since the electrons in states with m ? ?n are slowed down by the magnetic
field, they have lower acceleration than the electrons in states with m ? n. Therefore the
electrons in states with m ??n radiate less intensely and decay more slowly compared to
the electrons in states with m ? n.
5.3.5 Effect of Black Body Radiation
One concern was whether the experimentally reported black body temperatures of
16 K [4] and 4 K [5, 6] could strongly affect the radiative decay rates. The effect would be
stimulated emission and photoabsorption fueled by the black body radiation. Expressing
the decay rates in terms of Einstein A and B coefficients, the rate of radiative decay from
99
0.025
0.050
0.075
0.100
0.125
t 10%/n
6 (ns)
B = 1 T (low m)B = 1 T (high m)
B = 4 T (low m)B = 4 T (high m)
B= 0 T
0.05
0.10
0.15
0.20
t 50%/n
6 (ns)
10 15 20 25 30n0
0.050.10
0.150.20
0.250.30
t 90%/n
6 (ns)
(a)
(b)
(c)
Figure 5.5: Scaled time required to populate the ground state of the hydrogen atom by (a)
10%, (b) 50% and (c) 90% as a function of the effective hydrogenic principal quantum
number n for 1:0 T and 4:0 T fields. Initial distribution of states only involves states with
jmj? n.
100
state i to state f would be
Ai!f +Bi!fu(?if) (5.30)
where u(?if) is the Planck distribution
u(?if) = ~!
3
?2c3
1
eh?if=kBT ?1: (5.31)
Einstein Bi!f coefficient is evaluated from the Ai!f coefficient in the following manner.
Bi!f = ?
2c3
~!3 Ai!f (5.32)
In terms of evaluating the effect of the black body radiation, an important quantity is the
ratio of the stimulated emission rate to the spontaneous emission rate, (eh?if=kBT ? 1)?1.
At 4 K, the n ! (n ? 1) transitions up to about n = 20 the ratio is of the order of 10?6
and jumps up to about 5 ? 10?2 at n = 30 and becomes ? 0:4 at n = 40. Substantial
contribution to this ratio comes from the transitions between the lowest energy states of
different m manifolds. To see the effect of the black body radiation on the decay times,
we used the rate in Eq. (5.30) and found that the time needed to populate the ground state
changed by less than 10% for n < 40. We conclude that the black body radiation will not
qualitatively affect the radiative cascade for n < 40.
101
5.3.6 Semiclassical Treatment
The time averaged classical power over one orbit emitted from an electron in a circular
orbit with radius r in SI units is given by [63]
I = 23 e
2
4??0c3j
?~rj2
= 23 e
2
4??0c3
?v2
r
?2
(5.33)
where v is given by Eq. (5.23) and r is given by the Bohr-Sommerfeld quantization condi-
tion:
m~ = mevmrm ? eB02 r2m
= ?rm
s
eB0rm
2
?2
+ ke
2me
rm : (5.34)
The canonical angular momentum is Lz = m~and ? indicates the sign of m.
Energy of the orbit is obtained by
E = 12mev2m ? e
2
4??0
1
rm (5.35)
and the semiclassical decay rate is evaluated as ?m = Im=~(v(?)m =rm) where the ? again
indicates the sign of m. Note that this semiclassical treatment is only for the circular states,
i.e. the high-jmj states of a given n-manifold. Therefore the only allowed transitions are
n ! (n?1) transitions and the ground state does not decay.
102
Comparing the rates calculated semiclassicaly with the fully quantal rates in a 4:0 T
field we saw that at n = 10 the quantal rate is larger by ? 11%, at n = 20 by ? 4% and at
n = 30 by less than 1%. Similar errors were found for other field strengths.
We can use this semiclassical method to estimate the relative size of the decay rate
due to the cyclotron motion compared to the decay rate due to the magnetron motion. For
example, the cyclotron decay rate in a 1:0 T field is ? 1 Hz for the n = 3 state which
corresponds to 4 K. The decay rate of the m = ?85 state due to the magnetron motion is
also of the same size.
An important question is whether guiding center atoms with m << ?1 can be trapped
using magnetic field gradients. The change in the energy of the circular states as a function
of the magnetic field strength is an indication of the affinity of the state to a magnetic field
gradient. For the purposes of trapping Rydberg atoms using magnetic field gradients the
internal energies of the atoms must increase with increasing magnetic field strength. To
see if the atom is a high field or low field seeker we have approximated the motion of the
electron in the orbit as a superposition of a cyclotron motion and a magnetron motion. The
force on the center of mass of an atom is ~F = ?(~rB)(dE=dB) when it is slowly moving
and has internal energy E. For an electron in a central potential, dE=dB = (e=2) ^B?(~r?~v).
We computed this quantity for the circular states in this semiclassical approximation and
compared it to the dE=dB for pure cyclotron motion which is E=B. In a 1:0 T field,
dE=dB is ?24 K/T for m = ?85 and is ?20 K/T for m = ?125. The comparable values
in a 3:0 T field are ?14 K/T and ?12 K/T. The magnetron motion leads to high field
seeking character.
103
To trap the anti-hydrogen atoms in a spatially dependent magnetic field which is
stronger near the wall of the container, one needs the atoms to be low field seekers. The
dE=dB for the magnetron motion can be counteracted by the cyclotron motion. The
dE=dB for pure cyclotron motion at 4 K in a 1:0 T field is 4 K/T; this is a factor of 6
smaller in magnitude than for the magnetron motion at m = ?85 and of opposite sign.
Thus the contribution from the cyclotron motion into the low field seeking character of the
atom is much smaller than the contribution into the high field seeking character of the atom
from the magnetron motion. Hence we can conclude that either a lot of energy must be put
into the cyclotron motion if the atoms are in circular negative m states or the atoms cannot
be in nearly circular negative m states to be able be trapped by the spatially dependent
magnetic field.
5.4 Conclusions
We have investigated the radiative cascade of highly excited hydrogen atoms in strong
magnetic fields up to 4:0 T where we have considered states up to effective n ? 30?35.
We have considered two different cases of initial distributions. In one case, the states were
equally populated within a small energy range and in the other case only the states with
jmj ? n are populated. We have found that in the completely random ?;m case the time
needed to populate the ground state by 10% is longer than the time it would take the anti-
hydrogen atoms in recent experiments to hit the walls of the container due to their thermal
speed. We have also shown that the experimentally reported black body radiation at 4 K
did not affect the rates and the times needed to cascade down to the ground state by more
104
than a few percent which suggests that the black body radiation will not be of much help
in getting the anti-hydrogen atoms to the ground state faster. This also means that it cannot
slow down the cascade process by means of photoabsorption either.
Both in the completely random (?;m) distribution and the jmj ? n distribution we
see that states up to about n ? 20 are not affected much by the magnetic field as would
be expected considering the atomic unit of the magnetic field being of the order of 105 T,
compared to the external field of a few Teslas.
When the initial distribution only involves states with jmj ? n, time needed to pop-
ulate the ground state is found to be longer than the completely random ?;m case. This
results from the initial states being high-m states, which are circular or almost circular, and
the fact that in completely random ?;m case these states comprise a smaller fraction of
the population. The only allowed transitions for the circular states in the presence of the
magnetic field are either ?m = 0 or j?mj = 1 transitions depending on the parity change
in the transition. Therefore the states with high-jmj need to cascade down through smaller
?n transitions than those with smaller jmj. This greatly increases the decay time to the
ground state from the highly excited states in the case where the cascade initiated from the
negative large m states when the magnetic field points in the z-direction. When started
from positive large m states the magnetic field actually speeds up the cascade process since
in these states the electron is sped up by the magnetic field, radiating more intensely and
decaying faster. In the case of the negative large m states the same argument can be drawn
noting that this time the magnetic field slows down the electron which radiatively loses its
energy at a slower rate than in the field free case. We have shown that when started from
105
a distribution of extreme negative m states, it takes about 800 times longer for the ground
state to reach 10% population level than it does for the completely random ?;m distribution
at n = 30 in 4:0 T magnetic field.
We have also found that when the radiative cascade starts from a circular state, the
semiclassical treatment we have given here does a very good job in terms of yielding the
decay rates, especially for the states with effective n & 20. In cases where the large
fraction of the atoms are in high-jmj states, the old quantum theory can be used to evaluate
the decay rates instead of the full quantum theory. We used this semiclassical method to
estimate the decay rate due to the cyclotron motion of the electron and compare it to the
rate originating from the magnetron motion of the electron in a circular orbit. We evaluated
the force on the center of mass of the atom due to the magnetic field gradient and found that
for the atoms to be trapped with a spatially dependent magnetic field they should be either
in highly eccentric orbits or a large amount of energy must be imparted into the cyclotron
motion of the electron.
APPENDIX
The exact Hamiltonian for a Hydrogen atom in an external magnetic field ~B = B^z is
H =
?
~p?q~A(~r)
?2
2? +
B2
2M(x
2 +y2)+ BK
M x?
1
r2 +
K2
2M
where M = me + mp, ? = memp=M is the reduced mass, K is the pseudomomentum
assumed to be in the y direction, and q = e(mp ? me)=M [64]. The Hamiltonian (5.2)
106
can be obtained by making four approximations to the exact Hamiltonian. (1) We use
the mass of the electron, me, instead of the reduced mass, ?. This gives a change at the
? 1=1000 level and thus is not interesting for our purposes. (2) We do not include the last
term K2=2M since this a constant and does not affect the composition of the states or their
relative energies. (3) We do not include the term B2(x2 + y2)=2M. This term is ? 500
times smaller than the comparable diamagnetic term in our Eq. (5.2) and thus does not
have an important effect our results. (4) We do not include the term BKx=M. The effect
of this term is the motional Stark effect since the atom has a velocity perpendicular to the
magnetic field ~B. The pseudomomentum
~K = ~P + e
2
~B ?~r (5.36)
where ~P = M~V is the linear momentum of the proton and ~r is the distance from the
proton to the electron. In a 4 T field and with a kinetic energy of 4 K the first term in
~P has magnitude ? 4:3 ? 10?25 kg m/s while the second term e~B ?~r=2 has magnitude
? 1:5 ? 10?26 kg m/s. Thus we can take K ? MV in our estimates. To see that the
Stark shift does not have an important effect on our results, we note that the magnitude of
the paramagnetic splitting of the levels due to the magnetic field is ?E(1)P = (B=2)j?mj
and is ? 2:13?10?6 (a.u.) in a 1 T field for the adjacent levels. The thermal speed of an
anti-hydrogen atom at 4 K [5, 6] is ? 260 m/s. An anti-hydrogen atom moving with a
velocity ~V in a magnetic field ~B experiences an electric field of magnitude E ? j~V ? ~Bj.
The correction to the energy levels in the first approximation in an n-manifold due to the
107
linear Stark effect resulting from this field is E(1)S = 32En(n1 ? n2) where n1 and n2 are
the parabolic quantum numbers [61]. The separation of the adjacent Stark levels would be
3En which is ? 4:5?10?8 (a.u.) in n = 30 manifold for an atom moving at 260 m/s in a
1 T field. Note that the paramagnetic splitting is larger than the Stark splitting by roughly
two orders of magnitude, therefore the linear Stark effect is negligible.
Actually, the eigenvalues and eigenvectors of VBx + BLz can be found analytically
within an n-manifold. We computed the average spread in m for each state in an n-manifold
which we defined to be ?m2 ?hL2zi?hLzi2. We averaged ?m2 over all of the states and
found it to be 3V 2n3(n + 1)=4 which is ? 9 ? 10?3 for n = 30. Again this shows that
most states hardly mix with other m?s and the main mixing is with m?1. Since each state
is mixing with states of similar radiative properties, our results on radiative cascade should
be accurate to better than a few percent.
Clearly, the extent to which the Hamiltonian of Eq. (5.2) is accurate depends on the
application. For investigations of large scale and/or averaged quantities, several different
estimates including comparison of the sizes of MV and eBr=2 in the pesudomomentum,
comparison of the proton cyclotron frequency to the frequency of the electron in the guiding
center atom and the size of the motional Stark field to the magnetic field suggest that the
Hamiltonian (5.2) is good enough for accuracy better than ? 10% for n . 60 for B . 4 T
and kinetic energy less than 4 K.
In addition to the strong magnetic fields, anti-hydrogen experiments also involve spa-
tially dependent electric fields of ? 10 V=cm which is roughly a factor of ? 1-4 times
larger than the motional Stark field. Since the configurations of the traps are not known
108
at this time, we will not speculate on the effect of these electric fields. In the light of the
estimates in this appendix, it seems unlikely that these electric fields will have a substantial
effect on our results.
109
CHAPTER 6
CHAOTIC IONIZATION OF A HIGHLY EXCITED HYDROGEN ATOM IN PARALLEL
ELECTRIC AND MAGNETIC FIELDS
6.1 Introduction
Within the last two decades, systems whose underlying classical mechanics exhibit
chaos attracted much attention both from the theoretical and the experimental point of
view. Considerable effort has been made to observe and explain chaotic behaviour within
the purely quantum mechanical framework. These efforts resulted in observations of a
rich variety of signatures of chaos in the energy level statistics [65], wave functions [66],
absorption-ionization spectra [67, 68, 69] etc., of quantum systems whose classical coun-
terparts exhibit chaos. Unfortunately, attempts to explain the physical origins of such irreg-
ular behaviour in quantum mechanics have failed and it was found that even though quan-
tum systems imitate classical systems for a finite time, the classical chaos is suppressed in
quantum systems in the long time limit. In the sense of positive Liapunov exponents, chaos
does not exist in quantum mechanics, mainly due to the linearity of the Shr?odinger equa-
tion. Due to its lack of origin in the quantum mechanics, much work went into developing
semiclassical and statistical quantum mechanical theories such as random matrix theory
[70] to study manifestations of classical chaos in quantum systems [71]. Unbounded irreg-
ular scattering has attracted particular attention to generalize certain signatures of chaos to
110
a wide variety of systems since Bl?umel and Smilansky [72] showed that Ericson fluctua-
tions are a universal phenomenon. These fluctuations were originally proposed by Ericson
[73] as a signature of chaos in the photoabsorption spectrum of compound nuclear reactions
before its experimental observation [74].
In this chapter, chaotic ionization of Rydberg systems have been investigated in the
energy domain both classically and semiclassically for different regimes of microwave ion-
ization [75] characterized by a scaled frequency. Furthermore, classical theories such as
closed orbit theory have been developed to analyze long range chaotic fluctuations in the
photoabsorption spectra to interpret spectra of atoms in external fields. But there is a lack
of effort in the exploration of chaotic ionization dynamics in the time domain. Recently
Mitchell et al. performed classical calculations to investigate ionization of a hydrogen atom
in parallel electric and magnetic fields as an example of chaotic escape dynamics in an open
system in the time domain [76, 77]. In their simulations, they excited a hydrogen atom to
a Rydberg state with n ? 80 and studied the time-dependent structure of the ionization rate
for a distribution of launch angles and energies. They observed distributions of electrons
ionizing via a train of pulses - a consequence of the classical chaos induced by the external
magnetic field. They looked at the ionization rate and studied the initial launch angular
distribution of the classical trajectories as a function of the time it takes the electrons to
reach their detector. They observed a self-similar structure in the launch angle versus time
distribution which they were able to link to a fractal structure. They showed that each es-
cape segment in the launch angle-time distribution corresponds to a particular pulse in the
pulse train of the ionizing electrons and therefore the fractal structure of the launch angle
111
versus time distribution is directly reflected in the time-dependent ionization-rate. They
also argued that the true quantum mechanical treatment of this problem may bring in some
interesting effects due to interference and the Heisenberg uncertainty principle. The pur-
pose of this paper is to investigate these purely quantum mechanical effects and to explore
quantum signatures of the underlying classical chaos for this system in the time domain.
We will discuss results from our non-perturbative quantum mechanical and classical
trajectory Monte Carlo (CTMC) calculations and observe ionization pulse trains induced
by chaos. We present results on time-dependent ionization of a hydrogen atom excited to
high Rydberg levels around n = 40, n = 60 and n = 80 by a short laser pulse in parallel
electric and magnetic fields. We have further extended our calculations up to n = 120
and observed very similar behaviour in the time-dependent dynamics that we have seen
for the lower n values. We look at the current of ionizing electrons as a function of time
through a spherical detector placed just outside the peak of the potential. We compare
these results with those from our CTMC calculations to see the extent to which the Bohr
correspondence principle holds. We look at the wave function of the system as a function
of time and investigate angular momentum distribution of the escaping electrons as they
hit the detector. We compare our results with the classical calculations of Mitchell et al.
[76, 77] where possible. We study the time evolution of the interference patterns on the
detector. We observed dependence of the escape time of a trajectory on its classical launch
angle which incited us to manipulate the pulse train by placing nodes at certain angles into
the angular distribution of the initial wave packet in our quantum wave packet calculations.
At this point we introduce core scattering into the problem via a model core potential
112
and explore the coupling between two different mechanisms that produce ionization pulse
trains. Then we turn to the question of the onset of chaos in our system and introduce
scaled energy Ec to examine the transition from chaotic ionization dynamics to regularity
as we vary the magnetic field strength. We also study the time-dependent autocorrelation
function and the corresponding spectral autocorrelation function to look for any effects of
Landau quantization, such as the quasi-Landau oscillations. We perform our quantal and
classical calculations in parallel in each of these sections to see if the Bohr correspondence
principle applies in each case. The work presented in this chapter has been published as a
paper in the Journal of Physics B [78].
113
6.2 Theory
In the following two subsections we discuss methods of non-perturbative quantum
mechanical wave packet and the classical trajectory Monte Carlo (CTMC) calculations.
We use atomic units throughout this chapter unless we state otherwise.
6.2.1 Quantum mechanical time-dependent wave packet method
The total Hamiltonian of a hydrogen atom in parallel electric and magnetic fields is:
H0 = ?12 d
2
dr2 ?
1
r +
l(l +1)
2r2 +
2Lz
+
2
8 r
2 sin2 +Frcos (6.1)
where = B=(2:35?105 T) is the magnetic field strength in atomic units when B is in T,
and F = F ?(1:94?10?10 a.u./(V/cm)) is the electric field strength in atomic units when
F is in V/cm. The external fields are taken to be in the z-direction such that ~B = B^z and
~F = F^z, respectively. A detailed account of the approximations made in the Hamiltonian
(6.1) can be found in Ref. [58]. We break up the Hamiltonian H0 into two pieces; the
atomic Hamiltonian Ha, the static field Hamiltonian Hf:
H0 = Ha +Hf (6.2)
114
and introduce a laser term Hl to excite the hydrogen atom to a Rydberg state with energy
E,
H = H0 +Hl (6.3)
with
Ha = ?12 d
2
dr2 ?
1
r +
l(l +1)
2r2 (6.4a)
Hf = 2Lz +
2
8 r
2 sin2 +Frcos : (6.4b)
Hl = ?(t)e?i(E?Eg)trcos (6.4c)
where ?(t) = expf?t2=t2wg is the Gaussian laser pulse and Eg is the energy of the initial
ground state that is being excited. We use first order time-dependent perturbation theory
to evaluate the time-dependent wave function of the system. The total first order wave
function ?(~r;t) within the time-dependent perturbation theory is
?(~r;t) = ?(~r;t)e?iEt +?g(~r)e?iEgt: (6.5)
Here ?g(~r) is the wave function of the initial ground state that is being excited to the
Rydberg state with energy E. ?(~r;t) is the wave packet excited by the laser. Within this
approximation, the time-dependent Schr?odinger equation for ?(~r;t) reads [27]
i @@t ?(H0 ?E)
?
?(~r;t) = S0(r;t): (6.6)
115
Note that the energy E of the Rydberg state that is being excited is subtracted from the
diagonal elements of the Hamiltonian to reduce the numerical errors due to phase accu-
mulation. We choose the source term S0(r;t) to be spherically symmetric and spatially
compact. It is launched to a Rydberg state with energy E by a short laser pulse;
S0(r;t) = ?l;0 ?(t) r2e?r=2: (6.7)
The laser pulse peaks at t = 0 and we start time propagating Eq. 6.6 at t = ?6tw where tw
is the width of the laser pulse. In our calculations we have chosen tw to be ?Ryd=15 where
?Ryd = 2?(?2E)?3=2 is the Rydberg period of the state being excited. Integrating Eq. 6.6
over one time step and using the trapezoidal rule for the integrations, one obtains
i??(t+?t)??(t)? = ?t2
?
(H0 ?E)??(t+?t)+?(t)?+?S0(t+?t)+S0(t)?
?
(6.8)
which yields the quantum propagator,
?(t+?t) =
1?i(H
0 ?E)?t=2
1+i(H0 ?E)?t=2
?
?(t)? i?t1+i(H
0 ?E)?t=2
S0(t+?t2 )+O[?t3]: (6.9)
We further make the approximation
i?t
1+i(H0 ?E)?t=2 ? i?t+
?t2
2 (H0 ?E)+O[?t
3] (6.10)
116
for the second term on the right hand side which preserves the order of the propagator. The
operator acting on ?(t) in Eq. 6.9 is the lowest order Pad?e approximation to the time evolu-
tion operator e?iH0t. We therefore evaluated the first term on the right hand side of Eq. 6.9
by means of the split operator technique without altering the order of the propagator. We
used a higher order split operator technique than the one employed on pg.2 in Ref. [80]. We
have found that going to higher order in the split operator technique significantly reduced
the phase errors.
?(t+?t) =
1?iH
f?t=4
1+iHf?t=4
?" 1?i(H
a ?E)?t=4
1+i(Ha ?E)?t=4
?" 1?i(H
a ?E)?t=4
1+i(Ha ?E)?t=4
?
?
"
1?iHf?t=4
1+iHf?t=4
?
?(t)? 14
i?t+ ?t
2
2 (H0 ?E)
?
S0(t+ ?t2 )
#
?12
i?t+ ?t
2
2 (H0 ?E)
?
S0(t+ ?t2 )
#
?14
i?t+ ?t
2
2 (H0 ?E)
?
S0(t+ ?t2 )
#
+O[?t3]: (6.11)
117
The effect of the external fields on the solutions of the field free hydrogen atom can simply
be deduced by noting that [see Ref. [79], pg. 250]
cos Y ml =
s
(l +1)2 ?m2
(2l +1)(2l +3) Y
m
l+1 +
s
l2 ?m2
(2l +1)(2l?1) Y
m
l?1 (6.12)
sin2 Y ml = ?
s
[(l +1)2 ?m2][(l +2)2 ?m2]
(2l +1)(2l +3)2(2l +5) Y
m
l+2 (6.13)
+
?
1? (l +1)
2 ?m2
(2l +1)(2l +3) ?
l2 ?m2
(2l +1)(2l?1)
?
Y ml
?
s
[l2 ?m2][(l?1)2 ?m2]
(2l +1)(2l?1)2(2l?3) Y
m
l?2:
From this it can be seen that the electric field mixes state l with states l?1 and the magnetic
field mixes state l with states l, l?2. A natural representation of the wave function can be
realized on an (l;r) grid. On this grid, operator Hf acts only along the l-direction and it is
pentadiagonal. The atomic Hamiltonian Ha acts along the r-direction and involves the usual
kinetic energy operator and the Coulomb potential. We describe the second derivative in the
kinetic energy within the three point differencing scheme and therefore Ha is tridiagonal in
r (see Ref. [80]) and diagonal in l. The number of points Nl in the l-direction are chosen
to be 150, 200, and 250 for n = 40, n = 60, and n = 80 cases respectively. The number of
points Nr in the r-direction are 1000, 1500, and 2500 for n = 40, n = 60, and n = 80 and
the results we present are converged with respect to Nl and Nr.
We use square root mesh along the r-direction of the grid as described in Ref. [80].
The radial extent of the spatial direction on the grid, i.e., box size rf, is chosen to be twice
the radial extent of the peak of the potential, i.e., rf ? 2rp. The number of points in
118
this direction is chosen such that the maximum r-mesh spacing ?rmax is less than 0:5=pmax
where pmax is the largest possible linear momentum an ionized electron can attain, i.e.,
pmax =
p
?2(E ?V(rf)) =
q
(1=n2)+5pF: (6.14)
The time step we use for the time propagation of Eq. 6.9 is chosen such that it is
smaller than the time scales involved in our problem. The time scale set by the electric
field can be deduced from the electric field part of the propagator in Eq. 6.11;
expf?izF?t=2g? 1?izF?t=41+izF?t=4 (6.15)
which is ? ?i for zF?t=4 ? 1, giving a ?=2 phase shift. This constricts the appropri-
ate time step below 4=(Frf). In the same manner, the magnetic field part of the propa-
gator contributes another ?=2 phase error when expf?iB2?2?t=16g ? 1 which implies
?t . 32=(B2r2f ). The time step is also chosen to be much less than the Rydberg period
of the state corresponding to the energy E, specifically, ?t ? ?Ryd=800. We have per-
formed convergence checks with respect to both ?rmax and ?t and the results we present are
converged within a percent.
Since we are using a finite size box to time propagate the wave function, we use a
mask function to absorb current that reaches the end of the box to prevent reflections from
the box edge. The form of the mask function is M(r) = 1?s ?t((r?rm)=(rf ?rm))2 and
it spans the radial range from rm = (1=pF + 2rf)=3 to rf. The parameter s quantifies the
strength of the absorption. To remove the part of the wave function that reaches the end of
119
the box, we simply multiply the wave function with M(r) for each l on the grid after each
time step during the time propagation.
Since we are interested in time-dependent current of ionizing electrons, we calculate
the probability current density through a spherical detector which we then integrate over
the surface to obtain the current as a function of time. We place the detector such that its
surface is located just outside the peak of the potential. When the electron goes beyond
this peak in space it is considered ionized. Since the peak of the potential is located at
rp = 1=pF, we chose the radius of our spherical detector at rs = (rp + 2rf)=3. The total
current through the surface of the detector is therefore
' = ?Re
"
i
X
l
??(rs;l;t)@?l(r;l;t)@r
flfl
flfl
r=rs
#
: (6.16)
To check our codes, we have also calculated the total current as the rate of change of the
total probability that lies within the spherical region enclosed by the detector after the laser
pulse is turned off. We find total agreement between the two methods as expected since
@}(r;t)
@t
flfl
flfl
r=rs
= ?@j(r;t)@r
flfl
flfl
r=rs
(6.17)
where } is the survival probability density within the spherical region and j is the proba-
bility current density through the surface of the detector.
120
6.2.2 Classical trajectory Monte Carlo method
To compare results of the fully quantal time-dependent formalism to the underlying
classical dynamics, we have also performed classical calculations in the framework of clas-
sical trajectory Monte Carlo (CTMC) method. We solved Newton?s equations for the tra-
jectory of the hydrogenic electron in presence of the external parallel electric and magnetic
fields using fourth order adaptive time step Runga-Kutta algorithm described in Ref. [81].
We worked in cartesian coordinates and time propagated the trajectories until certain crite-
ria are met. Time propagation of a trajectory is terminated when the electron passes through
the spherical detector or when a predefined final propagation time tf is reached. When the
propagation of the Newton?s equations is terminated, it is binned in time as an escaped tra-
jectory or discarded depending on whether the electron has reached at the detector or not.
After a large number of trajectories are calculated, what is left is a probability distribution
of survival in time within the spherical region inside the detector. Each bin in time contains
a fraction of probability to escape within the corresponding time interval. Integrating this
probability distribution over time up to a particular bin ? gives the probability of survival
P(?) inside the spherical region up until the time ?. Widths of the time bins in our calcula-
tion are d? = tf=200. From this survival probability, current through the spherical detector
is evaluated as (P(?)?P(? ?d?))=d?. To have good statistics, we had to sample a large
number of trajectories and we have checked convergence with respect to the number of
trajectories to insure good statistics. The results we present here have been obtained with
3?106 classical trajectories.
121
To closely match the fully quantal case, each classical trajectory is launched in a spher-
ically symmetric fashion, i.e., the distribution of the cosine of the angle at which a tra-
jectory is launched is chosen from a flat distribution to mimic the angular distribution of
the initial quantum wave packet. The azimuthal angle ? is a cyclic coordinate due to the
cylindrical symmetry of the problem. There are three random initial parameters for each
launched trajectory in our calculations: the cosine of the launch angle of the trajectory,
launch time distribution F(t), and initial energy distribution G(?) of the trajectory. The
launch time distribution is given by the squared absolute value of the Gaussian laser pulse
?(t) in Eq. 6.4c, and the energy distribution G(?) is evaluated as the squared absolute value
of the Fourier transform of ?(t).
F(t) = e?2t2=t2w
G(?) = e?(??E)2t2w=2 (6.18)
Here E is the central energy of the laser pulse, and tw is the width of the laser pulse in time.
We have generated these Gaussian distributions of launch time and initial energy using the
methods described in Ref. [81]. The laser pulse used to excite the Rydberg states is exactly
the same as the one used for the quantum wave packet calculations.
122
6.3 Results
6.3.1 Flux of ionizing electrons
Results from the calculations for the current of ionizing electrons through the spherical
detector for n = 40, 60, and 80 can be seen in Fig. 6.1(a), (b), and (c) respectively. Fig-
ures show both quantum mechanical (solid curve) and classical (dotted curve) calculations
to contrast the quantum mechanical current with its classical counterpart which exhibits
chaos. The electric and magnetic field strengths for the n = 40 calculations are B = 3:92 T
and F = 304 V/cm. For the cases of n = 60 and 80, we scaled the magnetic field we used
for n = 40 by 1=n3 and the electric field by 1=n4 to stay at the same dynamical regime.
The external field strengths for these calculations are chosen such that the system is in the
n-mixing regime and the energy is above the classical ionization threshold. The box size
used in each of the quantum mechanical calculations are 8000, 18000, and 32000 (a.u.) for
n = 40, n = 60, and n = 80 respectively and the width of the laser pulse is tw = ?Ryd=15.
As pointed out in Ref. [77], the actual resolution of the pulses in the pulse train is deter-
mined by the interplay of ?t and ?E. Therefore, since both ?t and ?E contribute to the
broadening of the pulses, there exists an optimal set of (?t;?E) for which the ionization
pulses along the time axis are the sharpest. We have found that a longer pulse duration than
we defined above actually gives a noticeably sharper pulse train than that seen in Fig. 6.1.
The number of angular momenta included in each calculation scales roughly as ? n and
is 150 for n = 40. We time propagate the wave function ? for (9=2)?Ryd in each case of
123
n. We have also extended our calculations up to n = 120 and did not find any significant
shifts from the time-dependent dynamics we describe below for lower n values.
Whether a classical electron will ionize or not, and the time of its ionization, depend
on the initial launch angle of the classical trajectory. The trajectories that are launched
directly downhill, i.e., at = ?, will be the first ones to ionize. Ionization current due to
these electrons makes up the first prominent pulse in each of the figures in Fig. 6.1, which is
called the direct pulse. Classically, this corresponds to the first prominent escape segment
in the self-similar pattern of the classical launch angles versus time seen in Fig. 2(a) and
(b) of Ref. [77].
A remake of Fig. 2(a) and (b) of Ref. [77] can be seen in Fig. 6.2, which is gener-
ated using the data kindly supplied to us by K. A. Mitchell. Fig. 6.2(a) shows two time-
dependent ionization rate curves as a function of scaled time t, which is related to the real
physical time ^t through t = ^tn3. The calculation is for n = 80 in parallel electric and
magnetic fields of F = 19 V/cm and B = 0:49 T. The ionization rate depicted by the solid
line is calculated using an ensemble of trajectories with a precise energy and launch time,
whereas the smooth solid line is calculated using a Gaussian wave packet (see Ref. [77]
for more detail). Fig. 6.2(b) shows the time it takes for a trajectory to ionize as a function
of its launch angle. This launch angle distribution is referred to as the escape-time plot in
Ref. [77]. Each of the icicles in this plot marks out a range of launch angles such that when
a trajectory is launched at an angle from within this range, it ionizes through a particular
pulse in Fig. 6.2(a). These icicles are called escape segments and every escape segment
gives rise to a particular pulse inthe ionization rate. The dotted lines in Fig. 6.2(b) matches
124
Figure 6.1: Current-time plots for ionization with excitation to (a) n = 40, (b) n = 60,
and (c) n = 80 states of hydrogen via a short laser pulse in parallel electric and magnetic
fields. Results from quantum mechanical wave packet (solid curve) and CTMC (dotted
curve) calculations are plotted on top of each other along with the pure electric field case
(dash-dotted curve). (d) shows just the quantum mechanical current-time plots in (a), (b),
and (c) plotted together for comparison.
125
each escape segment with its corresponding pulse in the ionization pulse train. The escape
segments of Fig. 6.2(b) show a self-similar structure and Ref. [77] connects this structure
to fractal structure in the classical mechanics of the escape dynamics.
In the self-similar pattern of launch angles seen in Fig. 6.2(b), the trajectories launched
from within the angular range ? ? ?0:94 rad make up the direct ionization pulse. This
escape segment is by far the widest compared to the rest of the escape segments in the
pattern. Also notice that all the direct pulses seen in Fig. 6.1 terminate with a knee structure.
This knee is due to the electrons that were not launched directly downhill but at a slightly
smaller or larger angle. They are turned around by the force from the electric field before
they can scatter from the nucleus and reach the detector before the direct pulse of electrons
has completely passed through the detector. This knee is absent in the previous classical
calculations of Mitchell et al. where they used a minimum uncertainty Gaussian wave
packet for their ensemble of trajectories to obtain a smooth ionization rate curve, as the
one seen in Fig. 6.2(a). In some of their calculations [76], the direct pulse is smaller than
the second peak in the pulse train whereas our calculations and the calculations presented
in Fig. 6.2(a) yield a larger prominent direct pulse compared to the rest of the peaks in
the pulse train. This is due to the fact that Ref. [76] uses a px angular distribution for the
initial launch angles of the trajectories while Fig. 6.2 and we use an s-wave distribution.
Irregular peaks that come after the direct pulse in the pulse train are the electrons that were
launched from angles outside of ? ??0:94 rad. Their ionization is delayed for different
durations of time depending on their launch angle. The trajectories perform a cyclotron
motion due to the magnetic field while going down the potential hill as seen from the wave
126
Figure 6.2: Classical calculations of Ref. [77] for the ionization of hydrogen in parallel
electric and magnetic fields of F = 19 V/cm and B = 0:49 T for n = 80. (a) shows the
ionization rate as function of scaled time, and (b) shows the scaled time for a trajectory
to escape as a function of its launch angle. The solid curve in (a) is calculated using an
ensemble of trajectories with a precise launch time and energy whereas the smooth solid
curve is calculated using a Gaussian distribution for the energy and the launch time. Each
of the escape segments seen in (b) correspond to a particular pulse in the ionization rate
and are matched with their pulse via the dotted lines. The scape segment that gives rise to
the direct pulse peaks at = ? and most of it falls outside the range plotted. (We thank
K. A. Mitchell for kindly supplying the data for this partial replot of Fig. 2 of Ref. [77].
We also thank K. A. Mitchell and J. B. Delos for their permission to include this replot in
this paper.)
127
function plots of Fig. 6.3. Classically, these trajectories originate from the various smaller
escape segments marked as A1, A2, A3, etc., in Fig. 6.2(b). As the pulse train grows
longer, more escape segments contribute to a given pulse in the pulse train. Therefore
one would expect that these later pulses in the pulse train would show quantum mechanical
interference effects most prominently. This is indeed displayed in the time-dependent wave
functions seen in Fig. 6.3. The fact that the launch angle of a classical trajectory determines
its place in the pulse train suggests that by manipulating the launch angle distribution of
the trajectories, we can tailor the pulse train to a desired shape (see Sec. III D).
Fig. 6.3 shows snapshots of the probability distribution j?j2 in time for n = 80 in
the ?-z plane. The nucleus is positioned at the origin and the force from electric field is
pointing in the negative z-direction. At t = 58 ps (Fig. 6.3(a)), the spherically symmetric
outgoing wave is seen just when the current is near the peak of the direct pulse in Fig. 6.1(c).
This outgoing wave gives rise to the direct ionization pulse. At t = 117 ps (Fig. 6.3(b)),
the first pulse of ionizing electrons reaches the spherical detector and gets absorbed by
the spherical mask just after it crosses the detector surface at r = 2:68 ? 104 (a.u.). The
part of the wave function that corresponds to the ionizing electron current density (labeled
as fl in Fig. 6.3(b)) goes downhill as it performs a cyclotron motion due to the magnetic
field parallel to the electric field. This fl part of the wave function is the direct pulse of
Fig. 6.1(c), at the time when the current reaches the knee of the direct pulse along the pulse
train. Note that this part of the wave function does not yet show any interference patterns.
This is due to the fact that it corresponds to the classical trajectories launched from within
the first prominent escape segment in Fig. 6.2(b). The part of this escape segment that
128
makes up the direct pulse has no angular overlap with the rest of the escape segments.
This means that there is only one way for a trajectory to escape during the direct pulse and
that is to be launched from within this first escape segment. The lack of alternative escape
paths results in no interference patterns in the wave function. The part of the wave function
marked by is relatively localized between 0 < z < ?1 (a.u.) and shows the Stark
oscillating part of the wave function. Nodes of this part of the wave function corresponds
to the nodes of the Stark sublevels. Fig. 6.3(c) shows the wave function at t = 175 ps,
near the peak of the second pulse in Fig. 6.1(c). The wave function shows two types of
scattering: the fl part seen in Fig. 6.3(b) and a part that is going directly down the potential
hill (marked as ?). The fl part makes up the skirt of the direct pulse in Fig. 6.1(c) onto
which the second pulse sits. The ? part of the wave function gives rise to the second pulse
in the pulse train. Note that there is an interference pattern on this part of the wave function
which is due to the interference between the components of the wave function moving in
opposite direction along the ? dimension as they perform cyclotron motion rolling down the
potential hill. The classical escape segment responsible for the second pulse has an angular
overlap with the first prominent escape segment of Fig. 6.2(b), which is causing interference
between the ? and the fl parts of the wave function. Fig. 6.3(d) shows the wave function at
t = 233 ps, around the third pulse in Fig. 6.1(c). The fl and the ? parts of the wave function
has now completely blended into each other such that the fl part of the wave function is
hardly recognizable. This is so since the escape segment that gives rise to the third pulse in
Fig. 6.2(b) now overlaps in angle with both the first prominent escape segment as well as
the escape segment A1 which is responsible for the second pulse. The increased number of
129
ways to escape during the third pulse results in interference between these channels, leading
to interference between the fl and the ? parts of the wave function. The interference patterns
formed show two standing wave patterns perpendicular to each other with roughly the same
wavelength suggesting that both components have similar momentum. Figs. 6.3(e) and (f)
sample the wave function further along the pulse train t = 292 ps and t = 350 ps. In
these cases, the interference patterns in the wave function are noticably more complicated
than those of the earlier times. The classical escape segments corresponding to these later
times overlap with all the escape segments that gave rise to the previous pulses, resulting
in the ever increasing complexity of the interference patterns. The interference patterns
are dynamic, and resemble scars of the classical periodic orbits going downhill as they
perform a cyclotron motion about the z-axis. Such scars are commonly observed in bound
state eigenfunctions of classically chaotic Hamiltonian systems [66] and are well known
signatures of chaos in quantum systems.
The classical chaotic dynamics of this open system can be attributed to the presence
of the external magnetic field, without which the dynamics would be regular. To check
this, we have extended our quantum mechanical wave packet calculations to exclude the
magnetic field. This should result in a single prominent pulse of ionization in the absence
of core electrons, and this is indeed in accord with our results (dash-dot-dashed curves in
Fig. 6.1(a), (b), and (c)).
Fig. 6.1(d) shows all three quantum mechanical calculations for n = 40, 60, and 80
on top of each other to see if there is any effect due to the 1=n3 decreasing of the Rydberg
130
Figure 6.3: Snapshots of the absolute value squared wave function of hydrogen in parallel
electric and magnetic fields for n = 80 case at (a) 58 ps, (b) 117 ps, (c) 175 ps, (d) 233
ps, (e) 292 ps, and (f) 350 ps. In (b), marks the part of the wave function that shows
Stark oscillation whereas fl shows the one that gives rise to the direct ionization pulse in
Fig. 1(c). In (c), ? shows the part of the wave function that is responsible for the second
peak in the pulse train.
131
level spacings as n is increased. We see no significant differences within the n range we
investigate.
6.3.2 l-distributions of ionizing electrons
Fig. 6.4 shows the snapshots of the time-dependent angular momentum distribution
of j?j2 within the region bounded by the spherical detector for n = 80. The main plot in
each panel is for the case of the parallel electric and magnetic fields whereas the inserts are
for the pure electric field case. Fig. 6.4(a), (b), and (c) are snapshots of the l-distribution
during the direct ionization pulse of Fig. 6.1(c) at t = 44 ps, t = 58 ps, and t = 85 ps,
respectively. The angular momenta of the initially s-wave electrons spread to higher l as
the wave function expands outward (Fig. 6.3(a)). At t = 58 ps, the l-distribution shows two
distinct types of behavior: the part of the distribution that propagates out to higher l in the
same manner as the pure electric field case (marked as ?), and the irregular part that remains
at low l (marked as ?). Given the fact that the ? part of the distribution almost coincides
with that of the pure electric field case, and in the pure electric field case one ends up with
a single prominent ionization pulse, we can conclude that the ? part of the distribution
represents the electrons that are escaping through the direct pulse of Fig. 6.1(c). The ? part
of the distribution however classically represents the electrons that stay bound in the atom
and escape at a later time through another pulse in Fig. 6.1(c). The extent of the ? part of
the distribution reaches l ? 150 at t = 85 ps. By this time the direct pulse has well passed
its peak and ? 13% of the electrons have escaped. Fig. 6.4(d) shows the l-distribution at
t = 129 ps, just before the knee of the direct pulse reaches the detector. Note that the ? part
132
Figure 6.4: Time evolution of the angular momentum distribution of the electrons within
the space bounded by the spherical detector at (a) 44 ps, (b) 58 ps, (c) 85 ps, (d) 129 ps, (e)
234 ps, and (f) 338 ps for n = 80. The inserts show the angular momentum distributions
of the electrons at corresponding times for the pure electric field case for which the time
dependent dynamics is regular.
133
of the distribution has almost entirely diminished, leaving behind the ? part from which the
second peak in Fig. 6.1(c) is about to emerge. Fig. 6.4(e) and (f) are at t = 234 ps and
t = 338 ps, are the snapshots of the l-distribution further along the chaos induced pulse
train in Fig. 6.1(c). Note that these electrons do not go up as high in l as the population that
ionized through the direct pulse.
The distribution starting from small l propagates into larger l?s and a small part of it
reflects back from l ? 200 and moves back to the smaller l. This oscillation of l between
the small and the large angular momenta continues until after the second pulse and the
entire population more or less localizes to l . 100. Also notice that in the pure electric
field case the irregular features of the l-distribution is absent and the ionization is regular.
6.3.3 Angular distribution of the ionizing pulse trains on the detector
Fig. 6.1 shows very good agreement between our classical and quantum mechanical
calculations. Note that they show the current that reaches the detector integrated over the
surface of the spherical detector. A more detailed look at the distributions of the electrons
at the detector surface reveals differences between the classical and quantal pulse trains.
Fig. 6.5 shows the time evolution of j?j2 on the surface of the spherical detector as a
function of . The angle spans the range [? ? ?=6;? + ?=6] for better visualization of
the current through the spherical cap. The figure is for n = 80 and shows results obtained
by both quantum mechanically (solid curves) and classically (dotted curves). Fig. 6.5(a)
shows the angular distribution on the detector at t = 85 ps, just when the electrons are
escaping through the direct pulse in Fig. 6.1(c). Note that all the probability density is
134
Figure 6.5: Time evolution of the classical (dotted curve) and quantum (solid curve) angular
distributions of the electrons hitting the detector at (a) 85 ps, (b) 107 ps, (c) 151 ps, (d) 173
ps, (e) 289 ps, and (f) 347 ps for n = 80.
135
localized around = ? since these are the electrons that were originally launched directly
downhill. Figs. 6.5(b) and (c) show the angular distribution on the detector at t = 107 ps
and t = 151 ps, at the beginning and the end of the knee of the direct pulse. In Fig. 6.5(b),
there are no electrons at = ? and all of the wave function is hitting the detector at
? ??0:2 rad, forming a ring that is centered at the point where the z-axis goes through the
spherical detector. In Fig. 6.5(c), the two peaks in (b) come together at = ?. The quantum
mechanical peaks interfere whereas the classical peaks simply rejoin without interference.
The fact that the peaks in In Fig. 6.5(b) come together is a manifestation of the cyclotron
motion the electrons perform as they approach the detector. In Fig. 6.5(d) is at t = 173 ps,
during the second peak in the pulse train. Note that again the wave function arrives at the
detector in two peaks as in (b), but this time quantum mechanical peaks show interference
even though the peaks are separated. The classical distribution in this case is sharper than
that of Fig. 6.5(b) meaning that the trajectories that give rise to the second pulse have a
sharper distribution of launch angles.
Finally, Figs. 6.5(e) and (f) show the angular distribution on the detector at t = 289 ps
and t = 347 ps. These are the distributions of electrons on the detector that escape after the
second pulse. Note the interference patterns of the quantum mechanical distributions are
much more complicated and intricate than those of the direct and the second pulses. This
points out that the quantum mechanical distribution shows more and more interference
as the pulse train grows longer which is in agreement with our discussions regarding the
time-dependent wave function in Sec. III A. Even though the quantum mechanical and the
classical distributions depicted in Fig. 6.5 show fine structural differences due to quantum
136
interference, the effect of the interference in the total current through the sphere is washed
out when the current density is integrated over the surface of the detector. This results in
the very well agreeing classical and quantum mechanical time-dependent currents seen in
Fig. 6.1.
6.3.4 Effect of the angular distribution of the source term
In the quantum mechanical calculations we have presented so far, the initial electron
wave packet has been chosen to be spherically symmetric. Furthermore, the CTMC results
have also assumed a flat distribution of the launch angle of the trajectories. This allowed
us to include all the possible trajectories that can be launched in our simulations. As dis-
cussed in Sec. III A, previous classical calculations [77] and Fig. 6.2 suggest that the
angle at which a trajectory is launched ultimately defines its time of escape. Therefore, the
launch-angular distribution of the trajectories in the classical calculations, or the angular
distribution of the initial wave packet in our quantum mechanical calculations determine
the shape of the ionization pulse train.
In this section, we explore the effect of the angular distribution of the initial wave
packet in the quantum mechanical calculations on the structure of the ionization pulse
trains. In the classical case, the launch angle distribution of the trajectories determine
the relative heights of the peaks in the pulse train which means that the angle at which a
trajectory is launched is directly related to its ionization dynamics. For example, the elec-
trons launched from = ? would be expected to ionize via the direct pulse whereas an
electron launched initially uphill, i.e. < ?=2, would escape at a relatively later time via
137
one of the later pulses in the pulse train. This suggests that eliminating certain trajectories
from the CTMC calculations that give rise to ionization through a certain pulse in the pulse
train may give us the ability to manipulate the structure of the pulse train. This can be
realized classically by finding the angular range which spawns the trajectories that escape
via a given pulse and excluding these angles from the classical launch angle distribution.
In our quantum mechanical calculations, this would correspond to placing a node in the
angular distribution of the initial wave packet.
We have particularly attempted to eliminate the second pulse in Fig. 6.1(c) for n = 80.
To eliminate this second pulse we needed to know classically at what angle a trajectory
must be launched to contribute to this pulse in the train. This was done by binning the
trajectories that reached the detector between 158 ps and 213 ps in angle. The result is seen
in Fig. 6.6(d) as a function of the classical launch angle. The angular range from which
a trajectory must originate to contribute to the second pulse is a fairly sharp distribution
with a smaller secondary peak. The main peak is centered at ? 2:0 rad whereas the
secondary peak is located at ? 2:2 rad. Comparing with Fig. 6.2, we realize that these
peaks correspond to the escape segments A1 and A2 in the escape-time plot, which are
responsible for the second and the third peaks in the classical calculations of Mitchell et
al. Restricting the launch angles of the classical trajectories to outside of this range in
the CTMC calculations should therefore eliminate the second peak in the pulse train. The
same should also be true in our quantum mechanical calculations if we place a node at the
peak of the angular distribution of Fig. 6.6(d) in the angular distribution of the initial wave
packet.
138
Placing a node in the angular distribution of the source term in our quantum me-
chanical calculations physically corresponds to superposing a p-wave onto our otherwise
s-wave initial wave packet or source term. The radial part of the zero energy solution to
the Schr?odinger equation for the hydrogen atom is [82]
R1;l(r) = Ap2rJ2l+1(p8r) (6.19)
where J2l+1 is the Bessel function of order 2l + 1, and A is a normalization constant.
Therefore, to introduce a node in the wave packet, we superpose a p-wave source S1 =
?l;1r(1?r=2)e?r=2 with an s-wave source S0 (see Eq. (6.7)) via a mixing coefficient a.
Z 1
0
S0J1(r)dr
?
J1(r)Y 00 +a
Z 1
0
S1J3(r)dr
?
J3(r)Y 01 = 0: (6.20)
The zero of this superposition for a given angle 0 in Y 01 yields the mixing coefficient a.
The integrals in Eq. (6.20) can be evaluated by standard techniques to give an exact result.
For the Bessel functions J1(r) and J3(r) outside the integrals we have used the asymptotic
form of the Bessel function retaining only its outgoing wave part, i.e.,
Jm(z) ?
r 2
?ze
?i(z?m?=2??=4) (z >> jm2 ?1=4j): (6.21)
Solving Eq. (6.20) for a, we have found that the mixing coefficient for which there is a
single node at = 0 is a = ?1:47=cos( 0).
139
Superposing higher angular momentum components can help making the node sharper.
Such a wave packet may be experimentally tailored by first exciting a ground state hydro-
gen to a low-n state in an external electric field. The fact that the electric field mixes states
with l ? 1 gives rise to Stark oscillations in the l-distribution. This forms a Stark wave
packet, which then can be excited to a Rydberg wave packet via a second laser pulse. Hit-
ting this Stark wave packet at the right time would enable one to control the l-distribution
of the initial wave packet at the time of its launch to the Rydberg wave packet. For in-
stance, exciting the ground state up to a wave packet within the n = 2 manifold results in
a wave packet composed of l = 0, 1, and 2 angular momentum components, mixed in a
time-dependent fashion. Hitting this wave packet with the second laser pulse at the right
moment would launch it with a desired superposition of l = 0, 1, and 2 states to a high-n
Rydberg wave packet. Since the Stark period would be longer compared to the period of
the laser pulse, with the proper timing of the second laser pulse, a desired nodal pattern can
be achieved for the initial wave packet.
To eliminate the second pulse in Fig. 6.1(c), we have excluded the launch angles in
the range [1.8,2.4] rad in our CTMC calculation and placed a node at = 1:9 rad in the
angular distribution of the source term in the quantum mechanical wave packet calculation.
The results are the dotted curves in Fig. 6.6(a) and (b) for quantum and classical calcula-
tions respectively. Clearly excluding the entire range of angles that make up the peaks in
Fig. 6.6(d) does not only remove the second pulse in Fig. 6.1(c), but knocks out the entire
pulse train after the direct ionization pulse. This means that the excluded angular range
does not only give rise to the electrons that escape through the second pulse of electrons
140
but also includes almost all the other trajectories that escape afterwards within the time
range we explore. This is in accord with the previous classical calculations of Mitchell et
al. [76, 77] since the angular range we excluded is actually larger than the entire range
they explored and resolved for particular peaks in the pulse train. The fact that the direct
pulse remains intact in this case is expected since it is mainly risen by the trajectories that
are launched directly downhill, i.e. = ?.
Instead of removing the entire angular range that make up the peak in Fig. 6.6(d), we
tried to remove smaller ranges of the classical launch angle within this peak. Results of
these classical calculations can be summarized by the set of curves in Fig. 6.6(b).
a. Allowing the full [0;2?] range for the launch angles in the classical calculations
gives the solid curve in Fig. 6.6(b). This is the case investigated previously.
b. Excluding a thin slice from the center of the angular distribution depicted in Fig. 6.6(d),
i.e. the range [1:998;2:035] rad, gives the dashed curve of Fig. 6.6(b). In this case,
the second peak has gotten smaller and the rest of the pulse train remains intact.
c. Excluding a large slice of launch angles from the peak in Fig. 6.6(d) such that only
the small secondary peak is left gives the dash-dotted curve of Fig. 6.6(b). Note
that the second peak has now completely disappeared along with the later pulses in
the pulse train except a large fraction of the third pulse. Comparing with the case
in which the entire peaks in Fig. 6.6(d) were excluded (dotted curve), this suggests
that the small secondary peaks in the launch angular distribution is almost entirely
responsible for the third pulse in the pulse train.
141
Figure 6.6: (a) Current-time plots from quantum mechanical wave packet calculations for
various angular distributions of the source term. All the angular distributions we tried have
at most a single node in the angular range [0;?] and are plotted as a function of in (c).
The solid curve has no nodes whereas the dash-dotted curve has node at = ? rad, dashed
curve at = 1:28 rad, and the dotted curve at = 1:9 rad. Figure (b) shows ionization
current from the CTMC calculations for various exclusions from the angular range of the
launch angle shown in (d). The solid curve allows this entire range whereas the dotted
curve completely excludes it from its allowed launch angles. The dash-dotted and dashed
curves are obtained by excluding the angular ranges [1:8;2:123] rad and [1:998;2:035] rad
from the distribution shown in (d).
142
To simulate the same effect in our quantum calculations, we placed nodes at various
angles in the angular distribution of the source term, i.e. the initial wave packet. Fig. 6.6(c)
shows several angular distributions we have tried and the resulting current-time plots make
up Fig. 6.6(a). Same line-styled curves in Fig. 6.6(a) and (c) correspond to each other.
a. The solid curve in Fig. 6.6(a) corresponds to the angular distribution depicted by the
solid curve in Fig. 6.6(c) with no nodes and a maximum at = ? rad. This angular
distribution reproduces the current-time curve in Fig. 6.1(c) almost exactly.
b. The dash-dotted curve in Fig. 6.6(a) has an angular distribution with a node at = ?
rad as seen in Fig. 6.6(c). This knocks out the direct pulse in Fig. 6.1(c) as expected
since this is classically equivalent to removing the first prominent escape segment in
Fig. 6.2(b).
c. The dashed curve of Fig. 6.6(a) is calculated using an angular distribution with a
node at = 1:28 rad which corresponds to the angular distribution in Fig. 6.6(c)
with the same line style. Notice the direct pulse in this case remains unchanged while
the second pulse is clearly reduced. This is because the node is not placed exactly at
the center of the main classical peak in Fig. 6.6(d) but it is close enough such that it
partially knocks it out.
d. When we place a node at = 1:9 rad, at the center of the classical peak in Fig. 6.6(d),
we get the dotted curve in Fig. 6.6(a). This kills off the entire pulse train following
the direct pulse as in the classical case shown with the dotted curve in Fig. 6.6(b).
143
Classically this is same as killing off the escape segments labeled as A1 and A2 in
Fig. 6.2(b).
To sum up, we find that we can diminish the pulse train as much as we want by sliding
the node within the classically calculated peaks of Fig. 6.6(d). We were also able to remove
the direct ionization pulse by placing a node at ?. This is physically identical to killing off
the trajectories that were classically launched directly downhill. Elimination of the pulses
by inserting nodes in the angular distribution of the quantum mechanical initial wave packet
would be more clear and isolated from the neighboring pulses if the corresponding classical
escape segments in Fig. 6.2(b) were to span wider angular ranges.
6.3.5 Effect of core scattering
We have already pointed out that in the presence of a pure electric field, the hydrogen
atom ionizes only via a direct pulse due to the absence of a core. In the presence of a core,
the direct ionization peak is followed by an irregular train of ionizing electrons due to the
part of the wave function which did not directly ionize but initially headed uphill, turned
around by the uphill potential and then scattered in all directions by the core electrons [83].
Non-hydrogenic atoms in external electric fields have also been found to show statistical
properties in the nearest neighbor spacings that is neither regular nor chaotic suggesting
that the dynamics is somewhere in between [84]. Therefore introducing a core into our
problem may reveal interesting features attributable to the coupling between these two
different dynamics.
144
We incorporated the core electrons into our calculations through a model potential.
Instead of modeling a real atom using a model potential with proper quantum defects,
it is more informative to use a model potential of the form Ce?r=a=r. Playing with the
parameters C and a we can control the range of the core potential. This allows us to
control the extent of the core penetration by the electron.
Fig. 6.7(c) and (d) shows potentials for C = ?1 and two different values of a along
with the field free potential of the hydrogen atom for l = 0, 1, and 2. Values used for a are
0:05 and 0:5 for Fig. 6.7(c) and Fig. 6.7(d) respectively. The model potentials are plotted
as the solid curves whereas the field free hydrogenic potentials are the dotted, dashed,
and dash-dotted lines for l = 0, 1, and 2, respectively. Notice that in case of a = 0:05
the model core potential is buried deep below the angular momentum barrier even for the
lowest angular momentum considered, i.e. l = 0. It is natural to expect the core potential to
have no effect on the ionization dynamics in which we are interested, and this is indeed the
case. The fully quantal ionization pulse train for this case is the solid curve in Fig. 6.7(a).
Same case within our CTMC model is again the solid curve in Fig. 6.7(b). Both classical
and fully quantal computations yield exactly what was seen for hydrogen in Fig. 6.1 due to
the inability of the electron to penetrate into the core region. Increasing a to 0:5 we extend
the range of the core potential beyond the l = 1 potential barrier and obtain dotted curves
in Fig. 6.7(a) and Fig. 6.7(b) using quantum mechanical and CTMC methods respectively.
Fig. 6.7(a) also shows the pure electric field case for a = 0:5 (dashed curve). Contrasting
the solid curve with the dotted curve in Fig. 6.7(a) we can tell that the addition of the core
potential that extends beyond the l = 1 potential barrier enhances the pulses that come after
145
Figure 6.7: Quantum mechanical (a) and classical (b) current-time plots with the inclusion
of core scattering. In (a) the solid curve is for C = ?1 and a = 0:05 and the dotted
curve is for C = ?1 and a = 0:5. The dashed curve shows the pure electric field case in
the absence of chaos. In (b) again the solid curve is for C = ?1 and a = 0:05 and the
dotted curve is for C = ?1 and a = 0:5. The difference is less prominent compared to the
quantum mechanical case in (a). (c) and (d) show the model core potentials curves (solid
lines) and the field free atomic potential curves for l = 0, 1, and 2 for a = 0:05 and a = 0:5
respectively. All the calculations are for n = 80.
146
the second pulse in the train with an additional pulse of electrons dwelling on the skirt of
the second pulse (marked as P). This new pulse is absent in the classical calculation in
Fig. 6.7(a) suggesting that it is due to interference between the parts of the wave function
that scatters from the core and the part that is escaping due to chaotic nature of the time-
dependent dynamics.
6.3.6 Transition to regularity - the case of high magnetic fields
Transition from regular to chaotic dynamics in open atomic systems has been exten-
sively studied in the energy domain for the diamagnetic Kepler problem and ionization
of Rydberg atoms in external microwave fields. In the latter system, as the microwave
frequency is varied from low to high, the dynamics pass through three distinct dynamical
regimes of ionization. With the definition of a scaled frequency ?0 = ?n3 (the ratio of
the microwave frequency ? to the Kepler frequency of the Rydberg state 1=n3) microwave
ionization in these regimes are categorized as the tunneling, chaotic and the multiphoton
ionization regimes. In the diamagnetic Kepler problem, introduction of a scaled param-
eter ? = E= transforms this problem into a problem of two coupled two-dimensional
harmonic oscillators where the coupling strength is quantified by ?. By changing ?, or
equivalently changing magnetic field strength for fixed energy E, transition from regu-
larity to chaos is observed by studying the statistical distributions of the nearest neighbor
energy level spacings [85].
Due to the scaling properties of the classical Hamiltonian of the diamagnetic Kepler
problem, the dynamics of the electron does not depend on the energy of the electron and
147
the magnetic field independently, but rather depend on a scaled energy Ec = E= 2=3 [86].
Several properties of the diamagnetic Kepler problem has been studied by fixing the scaled
energy Ec such as the quasi-Landau modulations in the photoabsorption spectra of diamag-
netic hydrogen. So far in our study of the ionization of diamagnetic hydrogen in uniform
electric field, we have only considered a single ionization regime and observed signatures
of chaos in the current, wave functions, and angular momentum distributions of the ion-
ized electrons. This dynamical regime corresponds to Ec ? 0:48 for all the energies and
magnetic field strengths we have considered in previous sections.
We now will study different field regimes with different scaled energies while keeping
the energy E and the electric field strength fixed. We particularly consider the n = 80 case
and the scaled energies Ec = fi?2=3(E= 2=3) with fi = 0:5, 1:0, 2:0, and 4:0 where fi = 1:0
being the case we have studied in Sec. III A. These scaled energies simply correspond to
magnetic field strengths of fi , i.e. B = 0:245 T, B = 0:49 T, B = 0:98 T, B = 1:96 T
respectively, and E = 1=(2n2) = 7:8?10?5 (a.u.) and F = 19 V/cm. The current-time
plots we have obtained are plotted in Fig. 6.8(a), (b), (c), and (d) in increasing magnetic
field strength, i.e., fi. Each plot shows quantum mechanical wave packet and CTMC results
on top of each other for comparison. In case of fi = 0:5 of Fig. 6.8(a), the time-dependent
ionization dynamics appears regular. This is expected since for a weak enough magnetic
field the dynamics is dominated by the external electric field and the magnetic field has
little effect. The case of the pure electric field gives a single ionization pulse as discussed
before leading to the prominent direct pulse in Fig. 6.8(a) with no further escape at later
times. The regularity in this case is a direct consequence of the integrability of the system
148
Figure 6.8: Quantum mechanical (solid curves) and classical (dashed curves) current-time
plots for four different scaled energies quantified by fi; (a) fi = 0:5, (b) fi = 1:0, (c)
fi = 2:0, and (d) fi = 4:0. All the calculations are for n = 80.
due to the separability of the classical equations of motion and the agreement is perfect
between the CTMC and the quantum mechanical calculations. Fig. 6.8(b) is the case for
which fi = 1:0 which is what we have considered in Sec. III A and the ionization is chaotic
with very good agreement between the classical and the quantum mechanical current-time
plots.
149
Figs. 6.8(c) and (d) shows the time-dependent current of ionizing electrons for fi = 2:0
and fi = 4:0. Notice that the irregular structure of the ionization pulses from the CTMC
calculation is getting washed out compared to the fi = 1:0 case. This is expected since
higher the magnetic field closer we are to the Landau regime where the energy levels are
becoming similar to those of the harmonic oscillator. Therefore the diamagnetic Kepler
problem in a uniform parallel electric field is integrable in this regime since a harmonic
oscillator in uniform parallel electric field is integrable. The fact that the Landau regime is
integrable leads to regularity in the classical dynamics in this regime as displayed by the
dashed CTMC calculations in Figs. 6.8(c) and (d). Transition towards regularity displayed
in the CTMC calculations can also be understood in the semiclassical grounds. Volume of a
unit phase-space cell is proportional to the inverse of the effective Planck constant divided
by 2?, i.e. 1=~eff which scales like ? 1=n [87]. Instead of increasing the magnetic field
strength , we might just as well decrease n and maintain the same scaled energy Ec. This
would simply increase the volume of the unit phase-space cell, giving rise to blurring and
hence washing out of the fine structures in the phase-space, suppressing chaos. In contrast
to the classical case, the fully quantum mechanical results show more erratic fluctuations
relative to the fi = 1:0 case. This situation is even more pronounced for the case of fi = 4:0
in Fig. 6.8(d). Even though the structure of the pulse train in this case gives a strong
impression of chaos, these enhanced oscillations in the quantum case does not necessarily
point to enhanced chaos. Actually, given the fact that the CTMC calculations suggest
transition to regularity in the underlying classical dynamics as fi is increased, it can be
150
concluded that the enhanced fluctuations in the quantum mechanical current-time plot are
not due to enhanced chaos but rather are results of a different physical mechanism.
The reason for the enhanced fluctuations for the higher magnetic fields can be under-
stood if one considers how the number of open ionization channels change as the magnetic
field is increased. Since we are closer to the Landau regime in the high fi (or magnetic
field strength) limit, the energy levels are becoming more like the Landau levels. In this
case, the energy levels are almost equally spaced by !c as for a harmonic oscillator and the
separation increases linearly with increasing magnetic field strength. For a fixed electric
field strength, the classical ionization threshold is fixed at ?2pF and this gives a fixed
opening for ionization since the azimuthal symmetry is the dominant symmetry in the sys-
tem in the Landau regime. Any escaping trajectory must pass through this opening. For
n = 80 at F = 19 V/cm, the energy is E = ?1=(2n2) ? ?7:8 ? 10?5 (a.u.) and the
classical ionization threshold is E = ?12:2?10?5 (a.u.). Therefore the size of the opening
is ?W = E ? E = 4:4 ? 10?5 (a.u.). On the other hand, the Landau level spacing !c
is 2:09 ? 10?6(a.u.) and 4:18 ? 10?6(a.u.) for fi = 2:0 and fi = 4:0 respectively. The
ratio ?W=!c roughly gives the number of channels available for ionization and is ? 21 for
fi = 2:0 and ? 10 for fi = 4:0. In the quantum mechanical case, the number of ionization
channels that fit into this opening therefore decrease with the increasing separation of the
equally spaced energy levels. This is what gives rise to the enhanced oscillations in the
quantum mechanical current-time plot in Fig. 6.8(d).
One of the interesting phenomena observed for the diamagnetic Kepler problem is the
quasi-Landau modulations in the photoabsorption spectrum at the zero-field threshold [88].
151
These modulations were originally observed to be separated by 1.5 times the Landau spac-
ings for a free electron in magnetic field and are associated by the resonance states which
are fairly localized in the plane perpendicular to the external magnetic field. Later wave
packet calculations pointed out other Landau resonances with energy spacings of 0:64!c
[89]. Such quasi-Landau fluctuations were also seen at lower energies and explained in
terms of the individual magnetic field states [90]. It was found that it was the highest
energy diamagnetic states of the n-manifolds evolved into the Landau states [91]. To see
whether such effects occur in our problem, we have evaluated the time-dependent autocor-
relation function C(t), and the corresponding spectral autocorrelation function C(E) for
the fi = 2:0 and fi = 4:0 cases of Fig. 6.8. The real part of C(t) for each case is plotted as
the solid curves in Figs. 6.9(a) and (b) respectively. The solid curves are obtained from the
autocorrelation functions by smoothing them via boxcar averaging. Inserts in the figures
show C(t) near t = 0. Notice in the fi = 2:0 case there are clearly two modes of os-
cillation in the autocorrelation function; the thick oscillation modulated by much narrower
fluctuations in time. Both types of oscillations correspond to revivals of particular pieces of
the wave function and manifest themselves in the spectral autocorrelation functions C(E)
in Figs. 6.9(c) and (d). The spectral autocorrelation function C(E) in our case roughly
corresponds to the photoabsorption spectrum. In the spectral autocorrelation functions the
frequency axis is shifted by 1=(2n2) such that 0 frequency corresponds to the n = 80
component.
Both types of the oscillations depicted in Fig. 6.9(a) for n = 80 are due to the local-
ized part of the wave function labeled as in Fig. 6.3(b), which is the Stark oscillating
152
Figure 6.9: Time-dependent autocorrelation functions for (a) fi = 1:0 and (b) fi = 4:0.
(c) and (d) are the spectral autocorrelation functions that correspond to (a) and (b) respec-
tively. The time-dependent autocorrelation functions clearly show two types of oscillations
whereas the spectral autocorrelation functions show equally spaced resonances which indi-
cates Landau quantization.
153
part of the wave function. In particular, the revival time of the sharper oscillations modu-
lating the oscillations depicted by the thick curve in Fig. 6.9(a) is t(1)rev ? 40 ps as can be
seen from the insert. The revival time of the oscillation shown by the thick curve for the
same scaled energy is t(2)rev ? 600 ps. These revival times correspond to energy spacings of
?E(1)rev = 2?=t(1)rev ? 3:8 ? 10?6 (a.u.) and ?E(2)rev = 2?=t(2)rev ? 0:25 ? 10?6 (a.u.). The
spacings between the pronounced resonances of Fig. 6.9(c) have a spacing of ? 0:2?10?6
(a.u.) which roughly corresponds to ?E(2)rev . We do not see any resonances with spacings
corresponding to ?E(1)rev for this is almost as wide as the frequency range spanned by the
resonances in Fig. 6.9(c). The spectral autocorrelation functions seen Figs. 6.9(c) and (d)
clearly show equally spaced resonances, much like the energy levels of a harmonic oscil-
lator. This is due to the Landau quantization at the scaled energies we consider in Fig. 6.9.
Number of such equally spaced resonances in Figs. 6.9(c) and (d) can be counted to be
? 15 and ? 8 respectively, which roughly agrees with the number of the ionization chan-
nels we have estimated above. The decrease in the number of Landau states is roughly a
factor of two as would be expected, since the level spacing !c in the Landau regime linearly
scales with the magnetic field strength, and the scaled energies considered in Figs. 6.9(a,c)
and (b,d) are fi = 2:0 and fi = 4:0. Although we observe Landau states in the spectral
autocorrelation function, we do not see any clear sign of quasi-Landau oscillations.
6.4 Conclusions
In this paper, we examined ionization of a hydrogen atom excited to a Rydberg state
via a short laser pulse in parallel electric and magnetic fields. The field regime we have been
154
interested in is the one for which the classical dynamics of this system displayed chaos in
the time domain as reported in Refs. [76, 77]. Since physical roots of irregular scattering
is obscure and ill defined on purely quantum mechanical grounds, we tried to explore the
time-dependent dynamics of ionization from this system using a non-perturbative quantum
mechanical wave packet method along with a CTMC method.
Comparing quantum mechanical and classical results from these calculations, we ob-
served quantum mechanical effects, which would be absent in the classical picture, such
as interference effects. We have been able to reproduce chaos induced pulse trains of
ionization, previously observed in classical calculations [76, 77]. We found very good
agreement between our quantum mechanical and classical calculations of the ionization
current in the time domain for high Rydberg states with n = 40, 60, and 80. Looking at
the time-dependent wave function of the system, we have observed flux of electrons that
ionize through different pulses in the ionization pulse train and connected certain parts
of the wave function with the classically observed escape segments in the classical time-
dependent launch angle distribution of Ref. [77], which we have included in our paper as
Fig. 6.2. We have found that as the ionization pulse train grows longer, the number of es-
cape segments that contribute to a given pulse in the pulse train increases which ultimately
leads to interference between different escape channels in the quantum mechanical wave
function. We have also observed these interference patterns on the surface of our spheri-
cal detector which confirmed the observation of previous classical calculations suggesting
that the launch angle of a classical trajectory determines the time of the electron?s escape,
giving rise to a train of pulses in the time dependent ionization current. Although the total
155
ionization current, which is obtained by integrating the current density through the surface
of the spherical detector, from the classical and the quantum mechanical calculations agree
very well, this does not mean that the classical and the quantum mechanical pictures does
not differ from one another. The main difference is the quantum interference discussed in
Sec. III C, whose effect on the current gets washed out when the current density is inte-
grated through the surface of the detector. Another difference between the classical and
the quantum mechanical pictures is that the classical picture lacks the phase information
which is incorporated into the quantum calculations through the time propagation opera-
tor. A semi-classical modeling of this problem could shed light into the question of how
important the the phase information omitted is in the classical calculations.
This angular dependence of the shape of the suggested that it was possible to tailor the
shape of the pulse train by using a proper angular distribution of the initial wave packet.
In an attempt to manipulate the structure of the ionization pulse train, we ran CTMC sim-
ulations to figure out in which angular range a trajectory must be launched to contribute
to the second pulse in the pulse train. Excluding this range of angles from the launch an-
gular distribution in the CTMC calculations, we have eliminated pulses from our classical
time-dependent ionization current. We further placed nodes at corresponding angles in the
angular distribution of the initial wave packet in our quantum mechanical wave packet cal-
culations, and successfully removed the second and the prominent direct pulses from the
pulse train.
156
We have also investigated the angular momentum distributions of the electrons inside
the region bounded by our spherical detector. We were able to recognize the part of the l-
distribution that corresponds to the direct ionization pulse which extended up l ? 200. This
part of the distribution almost completely diminishes after the direct pulse terminates and
the small fraction of the remaining decays away slowly since the prominent escape segment
in the classical escape-time plot seen in Fig. 6.2(b) contributes to all the later pulses in the
pulses train. We found that the angular momenta of the electrons escaping through the later
pulses are confined to smaller l?s than those escape through the direct pulse.
Inclusion of core scattering into our problem allowed us to study the extent to which
the core penetration of the electron can affect the structure of the chaos-induced pulse train.
We used a model core potential to allow l = 0 and l = 1 core penetrations of the electron.
We reproduced our previous hydrogenic results for the l = 0 case as would be expected,
and observed enhancements in the pulses that came after the second pulse in the pulse train
for the l = 1 case. This enhancement of the pulses was more prominent in our quantum
mechanical simulations than in our CTMC simulations.
To study the transition from chaotic dynamics to regularity in this system, we varied
scaled energy of the system and found that for high scaled energy, i.e., high magnetic
field, the approach to the Landau regime resulted in the transition to regular regime in
the classical dynamics implied by the suppression of chaotic structure of the ionization
pulse train. On the other hand, our quantum mechanical calculations showed enhanced
oscillations in the current-time plot in this field regime. We have concluded that this was
due to the decrease in the number of open ionization channels in the quantum case. Looking
157
at the time-dependent autocorrelation function, we were able to see revivals of the Stark
oscillating part of the time-dependent wave function and the resonances that correspond to
them in the spectral autocorrelation function. We observed equally spaced resonances in
the spectral autocorrelation function which we believe to be due to the Landau quantization
in the strong magnetic field strengths we consider. Despite the apparent existence of the
Landau quantization in the spectral autocorrelation function, we see no clear signatures of
quasi-Landau oscillations.
158
CHAPTER 7
SUMMARY
In this work, we have studied various fundamental atomic processes regarding highly
excited one- and two-electron atoms. These processes include photo-double ionization,
electron-impact ionization, and radiative cascade and chaotic ionization in static external
fields. The primary tools we have used in these studies were based on solving the time-
dependent Schr?odinger equation in a non-perturbative manner, and we have made compar-
isons with perturbative calculations as well as purely classical results where possible.
After developing fundamentals of the numerical time-dependent techniques we use
throughout this work in Chapter 2, we have first studied the photo-double ionization of he-
lium atom from ground and the first excited state 1s2s(1S) near double ionization thresh-
old to investigate the threshold behavior of the double ionization probability just above
the threshold. We have contrasted our results with the predictions of classical theories of
Wannier and Temkin, which predict a smooth power law and an oscillating but monotoni-
cally increasing law for the threshold behavior of the double ionization cross section near
threshold. We have found excellent agreement with our results and the prediction of the
Wannier law and did not observe any oscillations modulating the ionization probability as
were predicted by the Temkin?s Coulomb-dipole theory and observed in the experiments of
Wehlitz et al [24] and Luki`c et al [25].
Then in the fourth chapter we have turned our attention to the electron-impact ion-
ization of hydrogen-like Rydberg atoms, where we have reported results from collinear
159
time-dependent close-coupling (TDCC), distorted-wave (DW), and R-Matrix with pseudo
states (RMPS) calculations for the electron-impact ionization of H-like ions up to Z = 6
within an s-wave model. We compared the results of these calculations with those from a
collinear classical trajectory Monte Carlo (CTMC) calculation to investigate the correspon-
dence between the quantal and classical ionization probabilities as the principal quantum
number of the initial state increases. We studied the ionization probability from the ground
state and highly excited states up to n = 25 as a function of incident energy and the charge
of the ion. We have found that the fully quantal ionization probability converged to the
classical results rapidly for hydrogen whereas the higher ion stages exhibited much slower
convergence with respect to n. We have observed good agreement between the DW and
TDCC for the n range we have considered for B4+. For hydrogen, we have found fairly
good agreement between the DW and TDCC for the ground state, but the worsening dis-
agreement with increasing n. There was reasonable agreement between the results from
the collinear R-Matrix calculations and the results from the TDCC calculations confirming
the convergence of the TDCC results.
In the fifth chapter, we have studied the radiative decay of atomic hydrogen in strong
magnetic fields of up to 4 T, which is believed to be an important process taking place in
the anti-hydrogen experiments. We have followed the radiative cascade from completely
?;m mixed distributions of highly excited states as well as from distributions that involve
highly excited states with jmj ? n. We have found that the time it takes to populate the
ground state is not affected by the magnetic field for the initial states with n . 20. For
higher n-manifolds, the electrons in the most negative m states were substantially slowed
160
down by the magnetic field resulting in a much longer lifetime. We have showed that less
than 10% of the anti-hydrogen atoms with n ? 35 generated in anti-hydrogen experiments
at 4 K would decay into their ground states before they hit the wall of the vacuum container
unless they are trapped. We have also found that the decay time was mainly determined by
the fraction of the atoms that were initially in the highest negative m states due to the fact
that only j?mj + j??j = 1 transitions are allowed in the magnetic field. We have given
a semiclassical method for calculating the decay rates for circular states and showed that
when the initial states have high-jmj, semiclassical decay rates agree with the full quantum
mechanical rates within couple of percent for states with effective n & 20.
Finally, we presented results of our simulations of the ionization of a hydrogen atom
excited to a Rydberg wave packet in the presence of external parallel electric and mag-
netic fields. This is an example of an open, quantum system whose classical counterpart
has been shown to display chaos in the time domain. Within the framework of classical
mechanics, electrons escape through chaos induced pulse trains. We reproduced such pre-
viously observed signatures of classical chaos in the time-dependent current of ionizing
electrons and studied the interference effects between the outgoing pulse trains which is
absent in the classical picture. Our attempts at manipulating the ionization pulse trains and
the effect of core scattering coupled with the chaotic ionization were also discussed. We
further investigated the onset of chaos as a function of the scaled energy of the system.
We found that for relatively high magnetic fields, quantum mechanical ionization current
shows erratic fluctuations in contrast with the classical current which shows transition to
regularity. We concluded that the oscillations result from the decrease in the number of
161
the ionization channels for the higher magnetic field strengths. We further studied the
time-dependent autocorrelation function and its Fourier transform to look for the effects
of the Landau quantization in the photoabsorption spectrum and did not observe any clear
signatures of the quasi-Landau oscillations. Our results included calculations via classi-
cal trajectory Monte Carlo method to compare our non-perturbative quantum mechanical
results with the underlying chaotic classical dynamics.
162
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