Download quantum few-body physics with the configuration interaction

quantum few-body physics with
the configuration interaction approach
—
method development and
application to physical systems
Jonas C. Cremon
Division of Mathematical Physics, LTH, 2010
c
2010
Jonas C. Cremon
c
Paper I 2007
by the American Physical Society
c
Paper II 2008
by IOP Publishing and Deutsche Physikalische Gesellschaft
c
Paper III 2009
by the American Physical Society
c
Paper V 2010
by the American Physical Society
Printed in Sweden 2010
ISBN 978-91-7473-043-2
quantum few-body physics with
the configuration interaction approach
—
method development and application to physical systems
Jonas C. Cremon
Dissertation for the degree of
Doctor of Philosophy in Engineering
Thesis Advisor: Stephanie M. Reimann
Faculty Opponent: Jainendra K. Jain
Division of Mathematical Physics, LTH
Lund University, Sweden
Academic dissertation which, by due permission of the Faculty
of Engineering at Lund University, will be publicly defended
on Friday, December 17th, 2010, at 13.15 in lecture hall B,
Sölvegatan 14A, Lund, for the degree of Doctor of
Philosophy in Engineering.
Contents
List of publications
1
Popular summary of the thesis
5
Populärvetenskaplig sammanfattning
11
1 Introduction
17
2 The configuration interaction method
19
2.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 The Lee-Suzuki approximation . . . . . . . . . . . . . . . . . 21
3 Numerical implementation
3.1 The Hamiltonian . . . . . . . . . . . .
3.2 The many-particle basis set . . . . . .
3.3 Matrix representation of operators . .
3.4 Diagonalization . . . . . . . . . . . . .
3.5 Interpreting the wavefunction . . . . .
3.6 Optimization and memory usage . . .
3.7 Numerical Lee-Suzuki transformation .
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4 Cold atomic quantum gases and nanostructures
4.1 Cold and dilute atomic or molecular gases . . . . . . .
4.2 Solid-state nano-structures . . . . . . . . . . . . . . . .
4.3 Quantum many-particle phenomena: Vortices, Wigner
tals and the Tonks-Girardeau gas . . . . . . . . . . . .
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crys. . . . 34
0
CONTENTS
5 Summary of results
5.1 Papers I and II – Rotating two-component Bose-Einstein condensates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Paper III – The Lee-Suzuki method for trapped bosons . .
5.3 Paper IV – Electrons in a nanowire quantum dot . . . . . .
5.4 Paper V – Dipolar bosons in one dimension . . . . . . . . .
5.5 Paper VI – Wigner states with dipolar atoms or molecules .
5.6 Additional results – Bilayer excitons . . . . . . . . . . . . .
6 Outlook
A An
A.1
A.2
A.3
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example
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The configuration interaction method . . . . . . . . . . . . . 47
The Lee-Suzuki approximation . . . . . . . . . . . . . . . . . 50
Second quantization formalism . . . . . . . . . . . . . . . . . 52
B Quantum harmonic oscillator
55
B.1 One dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 55
B.2 Two dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Bibliography
61
Acknowledgments
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Paper I
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Paper II
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Paper III
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Paper IV
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Paper V
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Paper VI
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List of publications
This thesis is based on six publications. They are reprinted at the end,
and listed on the following page where I also state my contribution to each
paper. For summaries of the included papers, see chapter 5. Please note
that I changed my name during my graduate studies, from Christensson to
Cremon.
1
2
LIST OF PUBLICATIONS
I
Mixtures of Bose Gases under Rotation
S Bargi, J Christensson, G M Kavoulakis, S M Reimann
Physical Review Letters 98, 130403 (2007)
I participated in the analysis, discussion, and manuscript preparation, but
not in the Bogoliubov analysis. I and S Bargi independently did all configuration interaction calculations. I did not establish the exact form of the
analytical expressions for e.g. the energies.
II
Rotational properties of a mixture of two Bose gases
J Christensson, S Bargi, K Kärkkäinen, Y Yu, G M Kavoulakis,
M Manninen, S M Reimann
New Journal of Physics 10, 033029 (2008)
I participated in the analysis, discussions, and manuscript preparation.
I and S Bargi independently did all configuration interaction calculations.
III
Effective-interaction approach to the many-boson problem
J Christensson, C Forssén, S Åberg, S M Reimann
Physical Review A 79, 012707 (2009)
I did all method development and calculations, except the scattering lengths.
I did most of the conceptual development and writing.
IV
Signatures of Wigner Crystallization in Epitaxially
Grown Nanowires
L H Kristinsdóttir, J C Cremon, H A Nilsson, H Q Xu,
L Samuelson, H Linke, A Wacker, S M Reimann
(arXiv: 1010.5147)
Apart from taking part in the analysis, interpretation and writing, I developed the numerical software needed to find the few-particle eigenstates
for this model, and to compute various quantities needed in the transport
calculation. I did not participate in the experiments.
V
Ground-state properties of few dipolar bosons in a quasi-onedimensional harmonic trap
F Deuretzbacher, J C Cremon, S M Reimann
Physical Review A 81, 063616 (2010)
I and F Deuretzbacher did all numerical calculations. I contributed to the
analysis and interpretation.
VI
Tunable Wigner states with dipolar atoms or molecules
J C Cremon, G M Bruun, S M Reimann
(arXiv: 1009.3119)
I did all calculations and method development, except the derivation of the
in-plane interaction. I did most of the conceptual development and writing.
3
Other papers, not included in this thesis:
Metastability of persistent currents in trapped gases of atoms
K Kärkkäinen, J Christensson, G Reinisch, G M Kavoulakis, S M Reimann
Physical Review A 76, 043627 (2007)
Universality of many-body states in rotating Bose and Fermi systems
M Borgh, M Koskinen, J Christensson, M Manninen, S M Reimann
Physical Review A 77, 033615 (2008)
Two-component Bose gases under rotation
S Bargi, K Kärkkäinen, J Christensson, G M Kavoulakis, M Manninen,
S M Reimann
AIP Conference Proceedings, Nuclei and Mesoscopic Physics 995, 25 (2008)
Vortices in small Bose or Fermi systems with repulsive interactions
J Christensson, M Borgh, M Koskinen, G M Kavoulakis, M Manninen,
S M Reimann
Few-Body Systems 43, 161 (2008)
The Ab Initio No-core Shell Model
C Forssén, J Christensson, P Navrátil, S Quaglioni, S Reimann, J Vary,
S Åberg
Few-Body Systems 45, 111 (2009)
Vortices in rotating two-component boson and fermion traps
S Bargi, J Christensson, H Saarikoski, A Harju, G M Kavoulakis,
M Manninen, S M Reimann
Physica E 42, 411 (2010)
Coreless vortices in rotating two-component quantum droplets
H Saarikoski, A Harju, J C Cremon, S Bargi, M Manninen, S M Reimann
Europhysics Letters 91, 30006 (2010)
Transport and interaction blockade of cold bosonic atoms
in a triple-well potential
P Schlagheck, F Malet, J C Cremon, S M Reimann
New Journal of Physics 12, 065020 (2010)
Popular summary of the thesis
In the research I have participated in, one investigates various physical
effects which may appear in systems with interacting quantum mechanical
particles. Some of the results are described below, but first a short physics
background is given.
Quantum physics and many-particle systems
Quantum mechanical particles do not behave in the way we expect everyday
particles to. When we say “particle”, we often think of a small dot or ball.
But in reality, a particle (for example an electron) is more like a small cloud.
One cannot say exactly where the particle is, but one can say that it exists
more where the cloud is denser, and less where the cloud is thinner.
Quantum physics rule for particles’ clouds:
A diffuse cloud means less
energy, lower speed and/or
smaller mass
A concentrated cloud means
more energy, higher speed
and/or larger mass
This means that it can be hard to compress a particle’s cloud – just like it
can be hard to compress a spring.
In a system with many interacting particles, it is often difficult to predict
how they will move. A pool game is a typical example. With quantum
mechanical particles the situation becomes even more complicated since
one has to take into account that all the particles may be everywhere at the
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6
POPULAR SUMMARY OF THE THESIS
same time. The consequence is that also systems with only a few particles
can give rise to complex and unexpected phenomena.
Bose-Einstein condensate
Even if most atoms consist of several electrons, protons and neutrons, it
is sometimes possible to think of an atom as a single particle, which can
be described with a single small cloud. This simplified picture works if the
atom is in a gas, such that it flies around without being attached to other
atoms.
If one takes a container filled with gas – that is, with many atoms flying
around – and then cools the gas considerably, odd phenomena may appear
for some atom types, a group called bosons. When an atom in a gas becomes
cooler it moves slower, which means that its cloud becomes more and more
diffuse.
Room temperature: Every atom is an individual small cloud flying around.
Very low temperature: The atoms become so slow that their
clouds start to overlap.
Close to zero temperature: All atoms are so slow and diffuse
that one cannot tell them apart – the whole gas seems to be one
single, large atom. This is called a Bose-Einstein condensate.
The research in this dissertation
Papers I & II – Rotating double Bose-Einstein condensates
If one rotates a Bose-Einstein condensate a vortex may appear. But if one
rotates faster, one might not get a larger vortex. Instead, there may be a
number of separated small vortices.
Pictures from an experiment with a rotating Bose-Einstein condensate.
Modified figure from reference [67], reprinted with permission,
c
courtesy of J Dalibard, 2000
American Physical Society.
7
We investigated, theoretically, what would happen if one would mix two different Bose-Einstein condensates and then rotate them together. It turned
out that the two condensates would form vortices around each other, instead of just simply being two condensates on top of each other. This is an
example of how quantum mechanical systems can behave completely differently than things we are used to – the motions of regular pool table balls
are independent of their colors.
Same number of red and blue atoms:
The two condensates form vortices around
each other, and the emerging structures are
similar to those of a single rotating condensate.
More red than blue atoms:
First the fewer blue atoms form a vortex
around the red ones, which are still, and then
the other way around. For even faster rotation, the red atoms form a larger vortex than
before, instead of two small vortices.
Paper IV – Wigner localization with electrons in small tubes
Short tube:
Long tube:
Three electrons (green clouds)
are trapped in the tube. Their
clouds overlap here, since the repulsive electrical forces between
the electrons are not strong
enough to compress them into
individual clouds.
Here there is more space, and
the electrical forces now manage
to keep the electrons separated
without the need to compress
their clouds very much. This is
called Wigner localization.
Wigner localization is a phenomenon which does not appear spontaneously
in nature, but may be created artificially. We thought that it could occur if
one confines a few electrons in a very small tube. Our calculations showed
that if one tries to drive an electrical current (of electrons) through the tube,
8
POPULAR SUMMARY OF THE THESIS
currents of different strengths would be obtained depending on whether the
electrons overlap or not.
We compared our calculations with experiments, where electrons had
been trapped in small tubes (of indium-antimonide). We could then observe that in a tube with a length of 70 nanometers, the electrons’ clouds
overlapped, while in a tube with a length of 160 nanometers they were just
about to separate. Unfortunately it was too difficult to build a long enough
tube (of about 300 nanometers) to really observe localized electrons, but
otherwise our calculations agreed with the experiments.
Paper III – The Lee-Suzuki method
Often, it becomes very difficult to predict how the particles will behave.
Then we make computer programs which help us. But also computers have
limitations, they can only do calculations for systems with few particles.
Some scientists have, in their attempts to describe how nuclei work,
invented a way to ease the workload of the computers, such that they can
manage to describe more particles. In my research group, we were interested
to see if this method could be applied for other systems than nuclei. It
turned out that it can be useful also to describe some cold atom gases.
Difficult to predict movements when there are many particles:
If there are too many particles it becomes very difficult to
predict how they will move, also for our computers. Especially if the particles interact strongly, maybe the computers
can only give an approximate description.
Easier to describe just two particles:
The idea with the Lee-Suzuki method is to first carefully
study just two interacting particles, which is much easier
to do.
Modified interaction:
Then, one creates a new fake interaction between the particles, based on the knowledge about the two-particle system.
Now, the work of the computers may be enough to give a
detailed description about the motions of all particles, since
one is accurately considering each pair of particles separately.
9
What’s really the point of all this?
One may wonder what this research is actually good for. Actually, often no
one really knows. But historically it has turned out that research pays off
– even if one cannot see any benefits of it for the time being.
What one can say, however, is that this research aims to increase the
understanding about how quantum mechanical particles (such as electrons
and atoms) interact. For example, computers, mobile phones, mp3 players,
the internet, and other things, are all dependent on the knowledge about
how to send around electrons in efficient ways. And since electronic components become smaller and smaller, it may become of interest to know how
electrons behave in very small tubes.
In many research areas there is a need to predict how quantum mechanical particles behave together. Therefore it is important to try to improve
the computational methods we have.
Populärvetenskaplig
sammanfattning
I forskningen jag deltagit i undersöks olika fysikaliska effekter som kan
förekomma i system med växelverkande kvantmekaniska partiklar. Några
av resultaten beskrivs nedan, men först ges en kort fysik-bakgrund.
Kvantfysik och mångpartikelsystem
Kvantmekaniska partiklar beter sig inte som vi är vana vid att partiklar gör.
När vi säger ”partikel” tänker vi oss oftast en liten punkt eller boll. Men
egentligen är en partikel (t ex en elektron) snarare ett litet moln. Det går
inte att säga exakt var partikeln är, men man kan säga att den finns mer
där molnet är tätare, och mindre där molnet är tunnare.
Kvantfysik-regel för partiklars moln:
Utspritt moln betyder mindre
energi, lägre fart och/eller
mindre massa
Koncentrerat moln betyder
större energi,
högre fart
och/eller större massa
Det här innebär att det kan vara jobbigt att pressa ihop en partikels moln
– ungefär som att det kan vara jobbigt att pressa ihop en spiralfjäder.
I ett system med flera växelverkande partiklar blir det ofta svårt att
förutsäga hur de kommer att röra sig. Ett biljardspel är ett typiskt exempel.
Med kvantmekaniska partiklar blir det ännu mer komplicerat eftersom man
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POPULÄRVETENSKAPLIG SAMMANFATTNING
måste ta hänsyn till att alla partiklarna kan vara överallt samtidigt. Detta
gör att även system med bara ett fåtal partiklar kan ge upphov till komplexa
och oväntade fenomen.
Bose-Einstein-kondensat
Även om de flesta atomer egentligen består av flera elektroner, protoner och
neutroner, så kan man ibland tänka sig en atom som en enda partikel, som
kan beskrivas med ett enda litet moln. Den här förenklade bilden fungerar
om atomen är i en gas, så att den flyger omkring utan att sitta ihop med
andra atomer.
Om man tar en behållare fylld med gas – alltså med många atomer
som flyger omkring – och sedan kyler ner gasen ordentligt så kan underliga
fenomen uppstå för vissa atom-typer, en grupp som kallas bosoner. När en
atom i en gas blir nedkyld så rör den sig långsammare, vilket innebär att
dess moln blir mer och mer utspritt.
Rumstemperatur: Varje atom är sitt eget koncentrerade moln
som flyger omkring.
Väldigt låg temperatur: Atomerna blir så långsamma att deras
moln börjar överlappa varandra.
Nära absoluta nollpunkten: Alla atomer är så långsamma och
utsmetade att man inte kan skilja dem åt – hela gasen blir som
en enda stor atom. Detta kallas för ett Bose-Einstein-kondensat.
Forskningen i den här avhandlingen
Papers I & II – Roterande dubbla Bose-Einstein-kondensat
Om man roterar ett Bose-Einstein-kondensat kan en virvel uppstå. Men om
man roterar snabbare är det inte säkert att man får en större virvel, det kan
istället uppstå flera virvlar på olika ställen.
Bilder från ett experiment med ett roterande Bose-Einstein-kondensat.
Modifierad figur från referens [67], återgiven med tillstånd,
c
med tack till J Dalibard, 2000
American Physical Society.
13
Vi undersökte, teoretiskt, vad som skulle hända om man blandade två olika
Bose-Einstein-kondensat och sedan roterade dem tillsammans. Det visade
sig bland annat att de två kondensaten bildar virvlar runt varandra, istället
för att bara vara två kondensat ovanpå varandra. Det här är ett exempel
på hur kvantmekaniska system kan bete sig helt annorlunda än vad vi är
vana vid – vanliga biljardbollar rör sig på samma sätt oavsett vilken färg
de har.
Lika många röd som blå atomer:
De två kondensaten bildar virvlar runt varandra, strukturerna som uppstår liknar de som
kan ses då ett ensamt kondensat roterar.
Fler röda än blå atomer:
Först bildar de färre blå atomerna en virvel
runt de röda, som är stilla, och sedan tvärtom. För ännu snabbare rotation så bildar de
röda en större virvel än tidigare, istället för
två olika små virvlar.
Paper IV – Wigner-lokalisering med elektroner i små rör
Kort rör:
Långt rör:
Tre elektroner (gröna moln) är
fångade i röret. Deras moln
överlappar varandra här, eftersom den elektriska repulsionen
mellan dem inte är tillräckligt
stark för att trycka ihop elektronerna till separerade moln.
Här finns det mer plats, så de
elektriska krafterna mellan elektronerna orkar hålla isär molnen utan att de måste bli väldigt
koncentrerade. Det här kallas
för Wigner-lokalisering.
Wigner-lokalisering är ett fenomen som inte förekommer spontant i naturen,
men kan framställas på konstgjord väg. Vi tänkte att man skulle kunna få
fenomenet att uppstå om man stänger in ett par elektroner i ett väldigt litet
rör. Våra beräkningar visade, att om man försöker få en elektrisk ström
14
POPULÄRVETENSKAPLIG SAMMANFATTNING
(av elektroner) att gå genom ett sådant rör, så fås olika starka strömmar
beroende på om elektronernas moln överlappar varandra eller inte.
Vi jämförde våra beräkningar med experiment, där man stängt in några
elektroner i små rör (av indium-antimonid). Vi kunde då med säkerhet
säga att i ett 70 nanometer långt rör överlappade elektronmolnen varandra,
medan i ett 160 nanometer långt rör började de precis dela upp sig. Tyvärr
blev det för svårt att bygga ett tillräckligt långt rör (ca 300 nanometer)
för att verkligen kunna se lokaliserade elektroner, men i övrigt stämde våra
beräkningar överens med experimenten.
Paper III – Lee-Suzuki-metoden
Ofta blir det väldigt svårt att förutsäga hur partiklarna kommer bete sig,
då tillverkar vi datorprogram som hjälper oss. Men datorer har också begränsningar, de klarar bara av att räkna på system med några få partiklar.
Några forskare har, i sina försök att beskriva hur atomkärnor fungerar,
uppfunnit ett sätt att förenkla datorernas arbete, så att de klarar av att
beskriva fler partiklar. I min forskargrupp var vi nyfikna på om metoden
kunde vara användbar även för andra system än kärnor. Det visade sig att
metoden också kan fungera för att beskriva vissa kylda atomgaser.
Svårt att förutsäga hur flera partiklar rör sig tillsammans:
Om partiklarna är för många blir det väldigt jobbigt att
förutsäga hur de kommer röra sig, även för våra datorer.
Särskilt om partiklarna växelverkar väldigt kraftigt med
varandra orkar datorerna bara ge en ungefärlig beskrivning.
Enklare att beskriva endast två partiklar:
Lee-Suzuki-metoden går ut på att man först noggrannt studerar hur endast två partiklar beter sig tillsammans. Två partiklar är mycket enklare att beskriva.
Modifierad växelverkan:
Sedan skapar man en ny låtsas-växelverkan mellan partiklarna, baserad på kunskapen om två-partikel-systemet. Nu
kan datorernas arbete räcka för att ge en noggrann beskrivning av alla partiklarnas rörelser, tack vare att man noggrannt tar hänsyn till hur varje partikel-par beter sig för sig
självt.
15
Vad är allt det här bra för egentligen?
Man kan undra vad man egentligen ska ha den här forskningen till. Faktum
är att för det mesta är det ingen som vet. Men historiskt har det visat sig
att forskning lönar sig – även om man inte förstår nyttan av den just för
stunden.
Det man kan säga är dock att den här forskningen syftar till utökad
förståelse om hur kvantmekaniska partiklar (t ex elektroner och atomer)
samverkar. Till exempel så är datorer, mobiltelefoner, mp3-spelare, internet, med mera, beroende av att man förstår hur man ska skicka runt elektroner på bästa sätt. Och eftersom elektroniska komponenter blir mindre
och mindre så kan det bli intressant att veta hur elektroner uppför sig i t ex
väldigt små rör.
Inom en mängd forskningsområden behöver man kunna räkna ut hur
kvantmekaniska partiklar beter sig tillsammans. Därför är det viktigt att
försöka förbättra de beräkningsmetoder som finns.
Chapter 1
Introduction
The research field of many-particle physics deals with effects which arise
because several particles interact with each other. That is, physical effects
which can not be explained by the properties of individual separated particles, but only if all particles and their interactions are considered together.
This may result in systems with great complexity, even if they at first appear simple, and it can give rise to various interesting phenomena. The
research I have participated in has had the aim to theoretically model some
particular systems of this kind and try to predict some of their properties.
There are various naturally existing physical systems which can be considered as isolated few- or many-particle systems. Nuclei are a typical example – the nucleons interact strongly with each other, but much less with
their surroundings. The electrons in a single atom or molecule can be another example. However, the systems studied in this thesis are artificially
created, specifically designed to put a number of particles together, and keep
them fairly isolated from the outside. An overview of related experimental
research, and some of the interesting phenomena that are investigated, is
presented in chapter 4.
This thesis includes six papers, which are reprinted at the end. Short
summaries of them are given in chapter 5, together with some additional
unpublished results.
Most of the research I have participated in has involved use of the configuration interaction method, described in chapter 2. It is a numerical
method with which one may solve the Schrödinger equation for a system
with a few interacting particles. This method has many advantages, but
its main drawback is that it can only handle very small systems, say five
17
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CHAPTER 1. INTRODUCTION
particles or so. For larger systems, the workload becomes much too large
for nowadays computers. But few-particle physics can be interesting on its
own, and in many cases, precursors of phenomena usually discussed for very
large systems can be seen also with only a few particles.
After some initial experience with the configuration interaction method,
I designed an implementation intended to take full advantage of the generality of the method – with an emphasis on decreased development times when
new physical systems are to be considered (see chapter 3). The versatility
of the resulting library is reflected in the list of published papers.
The research presented here has also involved further development of the
configuration interaction method itself. In the field of nuclear structure theory, the Lee-Suzuki approximation is used as a way to extend the method’s
applicability to larger nuclei than would be possible otherwise. As demonstrated in Paper III, this approximation can be useful also for trapped cold
bosonic atom gases – systems that are seemingly very different from nuclei.
Chapter 2
The configuration interaction
method
The configuration interaction method is a way to find approximate solutions
of the Schrödinger equation for a system with interacting particles. The
name configuration interaction is the one most commonly used in the field of
quantum chemistry, see e.g. reference [23]. In condensed matter physics, one
more often uses the name exact diagonalization, which may be misleading
since the obtained solution is always a numerical approximation. Nuclear
physicists typically use the name shell model calculations, or no-core shell
model calculations [76].
The – a priori unknown – wavefunction is expanded as a linear combination of several many-particle basis states (“configurations”), and the states
are then allowed to mix (“interact”) with each other to minimize the energy.
In theory, the method provides solutions to the Schrödinger equation to arbitrary numerical precision. In practice, however, the numerical complexity
limits the usage to systems with very few particles. Even if the capabilities
of our computers continue to increase at a similar rate to that of the last
decades, the configuration interaction method will most likely not become
an all-purpose method for quantum many-particle physics. But it has some
advantages compared to other methods. It is very general, as it is applicable for both fermions and bosons, and is not limited to a single type of
interaction. It gives both the ground state as well as the excited states. It
typically also makes it possible to obtain an estimate of the associated numerical error. For a chosen basis space, the actual ground state within that
space is obtained. This is in sharp contrast to many variational procedures
19
20 CHAPTER 2. THE CONFIGURATION INTERACTION METHOD
for energy minimization, which may be hindered by the presence of local
minima in the energy functional. Furthermore, the configuration interaction method – in its pure formulation – is a variational method, such that
an obtained approximative ground state energy is always an upper bound
of the true value – a property not shared by all methods. (This, however,
no longer holds when the Lee-Suzuki approximation is used.)
There are several other quantum many-particle methods; for example
the so-called coupled-cluster method [23] may allow for treatment of larger
particle numbers. Recent studies have applied this method to e.g. quantum dot systems [81, 108] and trapped bosonic gases [17], demonstrating
that the coupled-cluster method can be a powerful alternative to the pure
configuration interaction method, at least for the considered systems.
Quantum Monte Carlo methods may treat much larger particle numbers
(see e.g. references [47, 80]). However, a disadvantage is that they in general
do not give excited states of a system.
Below, in section 2.1 the configuration interaction method is formally
described, and section 2.2 presents the Lee-Suzuki approximation which
may be used to improve the convergence of the calculations. These two
sections are kept rather brief. An explicit example is given in appendix A.
2.1
The method
We wish to find the eigenstates of a Hamiltonian describing an isolated
system of N interacting particles, such as
Ĥ = Ĥ
(1)
+ Ĥ
(2)
=
N 2
X
p̂
N
1X
+ U (ri ) +
V (ri , rj )
2m
2
i
i=1
(2.1)
j6=i
where Ĥ (1) is a one-particle operator, the kinetic energy and e.g. a trapping
potential U , while Ĥ (2) is a two-particle operator with the interaction V
between the particles.
Here, the wanted state with N particles will be denoted by |Ψ(N ) i. There
are in general many possible solutions, so the state should have an index,
but this will be omitted here. The basis functions, which are Fock states
(N )
(see section A.3), will be denoted by |Φν i and the set of them by P (N ) . If
the single-particle orbital set (see section A.3) is infinite, then the number
of possible many-particle states based on this set is also infinite. Then
the expansion must be truncated somehow. A commonly used approach is
2.2. THE LEE-SUZUKI APPROXIMATION
21
to introduce an energy cutoff for the Fock states, so that only basis states
(N )
|Φν i with low enough energy are included in the sum. (The energy referred
(N )
to here is the kinetic and potential energy of |Φν i, ignoring the interaction
between the particles.) This can make it possible to describe at least the
lowest states in the energy spectrum with reasonable accuracy. A slightly
different approach is to introduce the cutoff in the single-particle space
instead. Either way, better approximations can be obtained by increasing
the basis space, to the point where the resulting energies are practically
independent of the cutoff – and converged solutions are obtained.
In matrix formulation, the eigenvalue problem is


(N )
(N )
(N )
(N )
hΦ1 |Ĥ|Φ1 i hΦ1 |Ĥ|Φ2 i · · ·

 (N )
)
(N )
(N )
(N )
 hΦ(N

i = E (N ) |Ψ(N ) i
2 |Ĥ|Φ1 i hΦ2 |Ĥ|Φ2 i · · ·  |Ψ

..
..
..
.
.
.
(2.2)
Diagonalization of this matrix gives approximations to the eigenstates of
the Hamiltonian, expanded in the basis P (N ) . See also the explicit example
in section A.1.
2.2
The Lee-Suzuki approximation
The major problem with the configuration interaction method is that the
many-particle basis becomes very large, even for small particle numbers, and
it becomes practically impossible to diagonalize the corresponding Hamiltonian matrix. And if the basis is kept too small, the obtained approximative
eigenvalues are too inaccurate. The Lee-Suzuki approximation is an approach which can make it possible to keep a small basis, and still get eigenvalues which are close to those of the full (infinite) matrix [102]. A heuristic
description of the method is that one modifies the particle-particle interaction, such that correlations between the particles are accounted for even if
the chosen subspace is insufficient to describe them. The Lee-Suzuki method
has been used successfully in ab-initio calculations for nuclei [74, 75, 76, 102].
We were interested to see if it could be useful also for other systems, such
as trapped ultra-cold atoms. In Paper III we demonstrated that it works
well for bosonic particles with a repulsive short-range interaction (see section 5.2). Apart from that paper, we have not yet exploited this method for
any of our published results, but plan to do so in future studies on trapped
22 CHAPTER 2. THE CONFIGURATION INTERACTION METHOD
cold gases (see chapter 6 for a short discussion). Recently, studies have
also been performed where the Lee-Suzuki method is used for calculations
on electrons in quantum dots [54, 57, 81]. A short discussion of how those
results relate to ours is given in section 5.2.
The Lee-Suzuki method is briefly described below. An explicit example
is given in section A.2. A rigorous presentation is given by Suzuki and
Okamoto in reference [102], and references therein, and in the earlier works
of references [95, 96, 97, 98, 99, 100, 101]. I also found the description given
in reference [75] helpful.
For a system with N particles, let us split the infinite many-particle
space into two parts, with the split defined by a cutoff energy. We will
call the finite-size low-energy part P (N ) and the infinite high energy part
Q(N ) . P (N ) is often referred to as the model space, and is the space used
for the actual calculations. The idea is to apply a unitary transformation
T̂ (N ) to the Hamiltonian, making it decoupled between the P (N ) and Q(N )
spaces [102]:
(N )
Ĥeff = T̂ (N ) (Ĥ (1) + Ĥ (2) )T̂ (N )†
(2.3)
The transformation T̂ (N ) should decouple the spaces, but otherwise change
the Hamiltonian as little as possible within them [102]. As the transforma(N )
tion is unitary, Ĥeff will have the same eigenvalues as the original Ĥ (N ) . In
general, the transformation will be an N -particle operator [75], and finding
it would require about as much work as diagonalizing the full matrix, so
this is where an approximation must be used.
For a system with only two particles, it is feasible to actually do the full
transformation and decouple the corresponding P (2) and Q(2) spaces. Then,
one can take the effective two-particle Hamiltonian in P (2) and subtract the
one-particle part of the Hamiltonian (it is assumed here that Ĥ (1) does
not couple the different spaces) [102]. This yields an effective two-particle
interaction, decoupled between the spaces:
(2)
Ĥeff = T̂ (2) (Ĥ (1) + Ĥ (2) )T̂ (2)† − Ĥ (1)
(2.4)
Now we can use this effective interaction in the N -particle system, and work
with a Hamiltonian that is decoupled as wanted.
The final effective two-particle Hamiltonian can be calculated without
the need to explicitely calculate the full T̂ (2) [75]. An equation for this, and
a stable numerical implementation, is given in section 3.7.
2.2. THE LEE-SUZUKI APPROXIMATION
23
The scheme described above corresponds to a two-particle cluster approximation for the effective interaction, as it is based on solutions for a
two-particle system. In principle, one may instead use a three-particle cluster approximation [75]: First diagonalize the full three-particle Hamiltonian,
and then use these solutions to create an effective three-particle interaction
to use in the N -particle system. As the full transformation is an N -particle
operator, the use of a larger particle cluster will give better approximations.
However, this would also mean more demanding calculations.
With the Lee-Suzuki approximation, the configuration interaction
method is no longer a variational approach, with the consequence that an obtained approximate ground-state energy is not necessarily an upper bound
to the correct one. This might not be a problem in practice, as long as
the calculations converge, but should be noted as an important difference
compared to the standard configuration interaction method.
Chapter 3
Numerical implementation
The configuration interaction method is very general, in the sense that it can
be applied to a large variety of different physical systems. It is not restricted
to a single type of interaction, nor to only bosons or only fermions. The
algorithms which need to be implemented to do the calculations with a
computer program can be designed to exploit this generality. Then, when
one wishes to investigate a physical system, one need only to focus on the
specifics of that particular system. In practice, this means determining the
(1)
(2)
one- and two-particle matrix elements hij and hijkl of the Hamiltonian, for
a given trapping potential and interaction, as defined in section A.3.
With some initial experience of using the configuration interaction
method, I designed an implementation intended to take full advantage of
the generality of the method. The implementation, written in the C++
programming language [94], can be seen as a toolbox with functions for the
different steps in the calculation.
Many other implementations of the configuration interaction method
exist, see e.g. references [16, 24, 56, 86]. But while they may be highly optimized for their particular model system, they are typically also restricted to
that model – with respect to basis orbitals, spatial dimensions, interactions,
and particle statistics.
3.1
The Hamiltonian
(1)
First, as mentioned above, the one- and two-particle matrix elements hij
(2)
and hijkl of the Hamiltonian (equations A.20 and A.21) need to be specified.
25
26
CHAPTER 3. NUMERICAL IMPLEMENTATION
Then the calculation is determined by simple parameters, e.g. the number
of particles and a cutoff energy.
The single-particle orbitals are often chosen as the eigenstates of Ĥ (1) ,
such that this part of the Hamiltonian is diagonal. Other choices are possible
however, e.g. one may use orbitals obtained from a Hartree-Fock calculation
for the system under consideration [23]. The one-particle matrix elements
(1)
hij are in many situations known analytically, or can otherwise usually be
computed with relatively small numerical effort. But the matrix elements of
(2)
the interaction, hijkl , are often more difficult to obtain. Typically, they are
multi-dimensional integrals with singular integrands, and with four different
indices there are many integrals to be calculated. Brute-force numerical
integration can be prohibitively expensive.
In some cases, a transformation to center-of-mass and relative coordinates can be employed to factorize the integrals. This is particularly useful
for the harmonic oscillator potential, as discussed in appendix B, but can
be helpful also in other situations. The integrals used in the example in
appendix A, e.g. those in equation A.5, were calculated with this approach.
For the special case of the electrostatic Coulomb interaction, the singularity can be expressed using a differential operator [105]. An integration
by parts then yields an integral without any singularity present, which simplifies a numerical evaluation.
3.2
The many-particle basis set
Given a number of single-particle orbitals, we need to generate all possible
combinations in which N indistinguishable particles can be placed in them.
For fermions, one simply avoids placing more than one particle in each
orbital. Figure 3.1 shows a way to iterate through all possible combinations,
for bosonic particles. At each level in figure 3.1, another particle is added, in
the different single-particle orbitals. At level N , many-particle states with
the wanted number of particles are found and can be saved in a list to be
used later. The list is P (N ) , the many-particle basis space. At this point, it
is straightforward to truncate the basis with a cutoff energy (as discussed
in section 2.1). Often, one also wants to exploit any block diagonality of
the Hamiltonian matrix, by only considering states with e.g. a certain total
angular momentum.
The algorithm can be implemented with a recursive function, with each
arrow in figure 3.1 representing a new function call. As also illustrated by
3.3. MATRIX REPRESENTATION OF OPERATORS
27
Figure 3.1: A scheme for how to iterate through all possible Fock states for
a given particle number, and a given number of orbitals. The particles are
here bosons – for fermions one need to avoid the branches where more than
one particle is placed in the same orbital. See discussion in section 3.2.
the figure, the number of many-particle basis states grows rapidly with N .
This is why the configuration interaction method in practice only works for
systems with few particles.
3.3
Matrix representation of operators
(N )
For the selected many-particle basis, P (N ) = {|Φν i}, a matrix representation of any operator has dimension dP (N ) × dP (N ) , where dP (N ) is
the number of many-particle states. Given an operator Ĥ, typically the
Hamiltonian, we thus need to determine d2P (N ) matrix elements of the form
(N )
(N )
Hµν = hΦµ |Ĥ|Φν i (cf equation 2.2). One way to do this for a oneP
(1)
particle operator i,j hij â†i âj is described below, in pseudo-code. A twoparticle operator can be treated in an analogous manner. Below, dP (1) is
(N )
the number of single particle orbitals, and each |Φν i is here considered as
an array with dP (1) integers, the occupation numbers.
28
CHAPTER 3. NUMERICAL IMPLEMENTATION
initially set Hµν = 0 for all µ and ν
for each ν from 1 to dP (N )
for each j from 1 to dP (1)
(N )
6 0 then
if âj |Φν i =
for each i from 1 to dP (1)
(N )
(N )
let β|Φα i = â†i âj |Φν i
(N )
if β 6= 0 and if |Φα i exists in P (N )
(1)
add hij β to Hαν
(N )
A detail when considering optimization is the search for |Φα i in P (N ) .
Brute-force linear searching would require a time proportional to dP (N ) for
this step. But this situation is well-suited for the use of a hash-table, which
makes the search time constant and independent of dP (N ) [52].
A typical Hamiltonian with a two-particle interaction can only couple
(N )
(N )
(N )
(N )
hΦµ | directly with |Φν i if one can transform |Φν i into |Φµ i by moving at most two particles. The consequence is that the matrix will be sparse,
i.e. most of the elements will be zero, and this sparsity is important to exploit for efficiency of the calculations. For example, the above algorithm
can be useful since it only spends time on the non-zero elements.
3.4
Diagonalization
Standard numerical algorithms for diagonalizing a matrix require a time
proportional to d3P (N ) [41]. Such algorithms are implemented in e.g. the
libraries gsl [37] or lapack [2]. But, as discussed above, the matrix under
consideration is typically sparse, making it possible to use algorithms developed specifically for this. The arpack library [65] contains implementations
of the so-called implicitely restarted Arnoldi iteration, which for Hermitian
matrices corresponds to a variant of the Lanczos method [41]. These algorithms do not directly consider the matrix itself, but only require iterative
multiplication of the matrix with arbitrary vectors. This is where the sparsity is useful, as calculation of a matrix-vector product then only require
a time proportional to the number of non-zero elements. Nearly all the
time required to find the wanted eigenvalues is spent on this matrix-vector
multiplication.
Algorithms for sparse matrices typically only produce a few selected
eigenvalues, e.g. the lowest ones [65]. For the configuration interaction
method, this need not be a problem since a truncated subspace can anyway
3.5. INTERPRETING THE WAVEFUNCTION
29
only be expected to accurately reproduce the lowest part of the energy
spectrum.
With repeated matrix-vector multiplications, the algorithms converge
towards the wanted eigenvalues. The number of iterations needed is usually
much smaller than dP (N ) [41], but can be very difficult to predict.
3.5
Interpreting the wavefunction
The full many-particle wavefunction can be difficult to visualize and interpret – it simply contains too much information. However, there are various
quantities derived from it that may give an insight. For example, the singleparticle probability density distribution is defined as
ρ(1) (x) = hΨ̂† (x)Ψ̂(x)i =
X
φ∗i (x)φj (x)hΨ(N ) |â†i âj |Ψ(N ) i
(3.1)
i,j
´
Its integral equals the number of particles, ρ(1) (x)dx = N .
More generally, with the known eigenstates, expectation values of various
operators can be calculated. For any one-particle operator, the corresponding expectation value can be expressed in terms of the reduced one-particle
(1)
density matrix, whose elements are defined as ρij = hΨ(N ) |â†i âj |Ψ(N ) i.
These matrix elements can be calculated with an algorithm very similar to
that in section 3.3. For example, they can then be used to evaluate the
single-particle density distribution for any position x.
The pair-correlated density is defined as
ρ(2) (x, x0 ) = hΨ̂† (x)Ψ̂† (x0 )Ψ̂(x0 )Ψ̂(x)i =
X
=
φ∗i (x)φ∗j (x0 )φk (x)φl (x0 )hΨ(N ) |â†i â†j âl âk |Ψ(N ) i (3.2)
i,j,k,l
This quantity gives the probability density to simultaneously find one particle at x and one at x0 , and can as such be normalized if needed. A possible
interpretation is that ρ(2) (x, x0 ) gives the density distribution of N − 1 particles, as a function of the position x, assuming that one particle has been
found at position x0 .
An efficient way to calculate the pair-correlated density distribution
ρ(2) (x, x0 )
is
to
first
compute
an
intermediate
quantity,
30
CHAPTER 3. NUMERICAL IMPLEMENTATION
P
Dik = j,l φ∗j (x0 )φl (x0 )hâ†i â†j âl âk i, for the chosen reference position x0 , after which the pair-correlated
distribution can be evaluated for any position
P
x as ρ(2) (x, x0 ) = i,k φ∗i (x)φk (x)Dik .
3.6
Optimization and memory usage
Typically, the main part of the memory used in a calculation is needed
for storing the matrix itself. (The diagonalization algorithms do require
some auxiliary memory, but the size of this is proportional to dP (N ) and
thus much smaller than the matrix itself [37, 65].) However, the matrix
itself is not really needed, only its product with an arbitrary vector. Thus,
it is possible to avoid saving the matrix, and instead re-construct it (e.g.
row-by-row) for each matrix-vector multiplication. Usually, this would slow
down the calculations considerably, but in some special cases one may exploit symmetries in P (N ) to factorize the entire matrix into smaller ones,
as discussed in reference [16]. An example where I used this approach is
the model system discussed in section 5.6, two coupled quantum dots with
electrons in one and holes in the other. The calculations then became limited by the memory needed to store just a few (∼ 10 [41, 65]) vectors of
dimension dP (N ) .
Typically, the diagonalization step requires most time in a calculation,
followed by the time for constructing the matrix itself. Both these steps
are slowed down not by the speed of the computer’s processor, but by the
transferring of data between the processor and the memory. Unfortunately,
this means that it is difficult to benefit from recent years’ development of
parallelized processors, as they typically share a common channel to the
memory.
3.7
Numerical Lee-Suzuki transformation
When implementing the Lee-Suzuki transformation, I encountered problems
with the numerical stability, due to reasons explained below. I resolved this
issue by using the singular-value decomposition for matrices [41], and later
found the same approach, with extensive related discussions and interpretations, in reference [55]. Following reference [75], the Lee-Suzuki transformation is given by the equation
3.7. NUMERICAL LEE-SUZUKI TRANSFORMATION
P̂ + P̂ ξˆ† Q̂ Q̂ξˆP̂ + P̂
Ĥeff = q
Ĥ q
P̂ + ξˆ† ξˆ
P̂ + ξˆ† ξˆ
31
(3.3)
where P̂ and Q̂ are projection operators onto the corresponding spaces, so
that P̂ + Q̂ = 1, and ξˆ is an operator acting as a mapping between the
spaces, satisfying ξˆ = Q̂ξˆP̂ . Here we only care about the part of Ĥeff acting
within the P space, of dimension dP . The corresponding formula needed
for actual calculations is (from equations 13–16 in reference [75]):
[Ĥeff ]P = ((A† )−1 A−1 )−1/2 (A† )−1 EA−1 ((A† )−1 A−1 )−1/2
(3.4)
with E being a diagonal matrix containing the lowest dP eigenvalues of
Ĥ, and A a matrix containing the P -components of the corresponding dP
eigenvectors of Ĥ (cf equation A.10).
Unfortunately, A is not always invertable. For example, if an eigenstate
in the lower spectrum consists only of basis states outside P , then a column
in A is zero. Also, two different eigenstates may have very similar expansions
within P , causing the matrix to lose rank. However, any matrix can be
factorized using the singular-value decomposition A = U ΣV † , where U and
V are unitary matrices, and Σ is a diagonal matrix containing the singular
values of A [41]. If all singular values are non-zero, A can be inverted as
A−1 = V Σ−1 U † . This can be inserted in equation 3.4, which can, eventually,
be reduced to
[Ĥeff ]P = U V † EV U †
(3.5)
The singular values themselves are no longer present in the expression, which
means that the effective Hamiltonian can be computed in a numerically
stable way.
Chapter 4
Cold atomic quantum gases and
nanostructures
4.1
Cold and dilute atomic or molecular gases
A dilute gas, where the particles are either atoms or molecules, may be
cooled to extremely low temperature [3, 22, 28, 42, 51, 82, 113]. This essentially eliminates any thermal motion, and the system may go into its
lowest energy state. These systems are typically very clean, with very few
impurities, and they are also very isolated from the surrounding world. A
nice feature is that, in many cases, optical imaging can be used, see e.g.
references [67, 77, 110], and also figures 4.1 and 4.2. The strength and type
of the interaction between the particles may be changed – e.g. via so-called
Feshbach resonances, the short-range scattering properties of some atoms
may be tuned by applying an external magnetic field [82]. For ions, it is
instead the electrostatic repulsion that dominates [110]. Furthermore, some
atoms and molecules have permanent dipole fields around them, allowing
for more complex correlations between the particles [4, 60]. Both bosonic
and fermionic particles have been cooled and trapped [3, 28, 42, 113]. The
first experiments typically involved clouds of 106 particles or so [82], but
with the use of optical lattices also systems with single or very few particles
may be realized [9, 26]. Since the systems are so clean and have minimal
coupling to their surroundings, it can be possible to directly implement
model Hamiltonians that are used for theoretical studies of other physical
systems [9].
33
34
CHAPTER 4. COLD ATOMIC QUANTUM GASES AND
NANOSTRUCTURES
Cold dilute gases are thus ideal for studying various aspects of quantum
many-particle physics. More information can be found in e.g. the reviews
[4, 8, 9, 25, 38, 60, 62, 71, 110] and the books [82, 84].
4.2
Solid-state nano-structures
Nano-structured materials present another interesting possibility for studies
of many-particle physics. The charge carriers in semi-conductors – electrons
and holes (see e.g. the textbook [49]) – can be manipulated in many ways.
Confining trapping potentials may be engineered to various shapes. The
effective strength of the particles’ electrostatic Coulomb interaction can be
altered by adjusting the particle density, or considering different materials.
So-called quantum dots are small confinements where a few charge carriers
may be trapped, providing a highly tunable few-particle system [85]. More
recently, the growing of heterostructured nanowires has allowed the creation
of well-defined and highly configurable small dot regions, see e.g. reference
[89]. In contrast to the ultracold gas systems mentioned above, solid-state
systems often need to be studied via indirect techniques, such as currentvoltage measurements, as the charge carriers are confined within a solidstate crystal.
Various quantum many-particle effects are studied in solid-state systems. The numerous technological applications involving semiconductors
and nano-structures, along with the rapidly developing possibilities for construction of such systems, also motivates further studies in this area. With
the shrinking sizes of electronic components, understanding of resulting
quantum few-particle systems can be expected to become increasingly important.
More information can be found in e.g. the reviews [11, 46, 85, 109] and
the books [19, 50].
4.3
4.3.1
Quantum many-particle phenomena: Vortices,
Wigner crystals and the Tonks-Girardeau gas
Bose-Einstein condensation and quantized vortices
If the particles in a system are bosons, they may all condense into the same
quantum state, a phenomenon initially predicted by Bose [10] and Einstein
[30, 31]. The first experimental realization of such a system was made with
4.3. QUANTUM MANY-PARTICLE PHENOMENA: VORTICES,
WIGNER CRYSTALS AND THE TONKS-GIRARDEAU GAS
35
Figure 4.1: A rotating Bose-Einstein condensate, consisting of a dilute gas
cloud of ∼ 106 87 Rb atoms [67]. The rotational frequency increases from
left to right, resulting in additional vortices in the condensate. Before the
pictures were taken, the size of the clouds were of the order of µm. The
angular momentum vector is perpendicular to the figure. Image courtesy of
c
J Dalibard, reprinted with permission, from reference [67], 2000
by the
American Physical Society.
liquid helium (see references [63, 64], and references therein), but it has now
also been achieved with a dilute gas of atoms [3, 22, 28, 51]. Apart from
these two examples, the concept of Bose-Einstein condensation is relevant
for a wide variety of physical systems [82].
Often, Bose-Einstein condensates have peculiar and counter-intuitive
properties, e.g. superfluidity and the occurence of quantized vortices [33].
In the so-called Gross-Pitaevskii equation, the full many-particle wavefunction is approximated with a so-called order parameter – the single orbital
that all particles are assumed to occupy [25]. If a single orbital should have
a vortex in it, the phase shift around the vortex would have to be an integer
of 2π, otherwise the wavefunction could not be continuous. For a single particle, this means that it can only have certain quantized values of angular
momentum. Since this single orbital actually describes all particles in the
system, it implies that the entire system, which may consist of millions of
particles, can only have certain values of total angular momentum.
For a harmonically trapped Bose-Einstein condensate, if the system is to
have more angular momentum than one unit per particle, the energetically
stable state is characterized by an array of vortices – instead of a single
vortex of multiple quantization [14, 33, 67]. See figure 4.1. For a sufficiently
anharmonic potential the situation may be different [33], with e.g. the
occurence of multiply quantized vortices [66], but the harmonic oscillator is
commonly considered as a good approximation to the trapping potential in
typical experiments [33, 82].
As they are rotating quantum fluids, the bosonic gases considered here
have many similarities to e.g. quantum Hall systems, in particular in the
36
CHAPTER 4. COLD ATOMIC QUANTUM GASES AND
NANOSTRUCTURES
regime of rapid rotation, where the number of vortices becomes comparable
to the number of particles (see e.g. the reviews [21, 106] and references
therein).
An interesting feature of rotating superfluids is the possibility of persistent currents [61] – metastable states that, once set rotating, keep rotating
without any external driving. A harmonically trapped condensate does not
support such a state: a potential maximum at the trap center is required
[59], or alternatively a toroidal trap [7, 33, 103], as demonstrated in the
experiment in reference [87].
One may also consider so-called spinor condensates, where the bosons
may belong to different components, distinguished by some internal degree
of freedom. For example, the atoms may be in different hyperfine states. For
the bosonic rubidium atoms used in e.g. the experiment in reference [91],
this internal degree of freedom does not significantly change the scattering
properties of the atoms [82]. But it results in different symmetry constraints
on the many-body wavefunction, which may change the system’s response
to rotation. Vortices may form in either component of the system, but
are typically filled by particles belonging to another component – so-called
coreless vortices. This is discussed in section 5.1, but see also e.g. references
in Papers I and II.
4.3.2
Wigner localization
If the interaction between the particles is strong and repulsive, and has a
sufficiently long range, it may be energetically favorable for the particles
to localize at individual positions in space. This phenomenon was first predicted by Wigner for the uniform electron gas [111], but the same arguments
can be applied for other systems with long-range repulsive interactions.
In very large systems, such a phase is often called a Wigner crystal,
referring to the resulting regular lattice of localized particles. For finite-size
systems such as quantum dots, concepts such as crystallization and phase
transitions may not be applicable, but the underlying physics remain –
the particles form individual wave packets because of the strong long-range
repulsion. States with emerging Wigner localization, but with considerable
overlap of the particles, are sometimes referred to as Wigner molecules. For
detailed discussions, see e.g. the reviews [85, 112].
With long-range interactions, particle localization may also be induced
if the system is rotated, see e.g. references [68, 85, 112]. The regime of
4.3. QUANTUM MANY-PARTICLE PHENOMENA: VORTICES,
WIGNER CRYSTALS AND THE TONKS-GIRARDEAU GAS
37
Figure 4.2: Wigner localized 40 Ca+ ions [77]. The ions are trapped in a
quasi-one-dimensional harmonic oscillator, and the strong Coulomb repulsion between them causes them to form individual localized wavepackets.
Since the ions absorb and emitt light they can be photographed, allowing direct observation of their positions. The distance between two ions is
roughly 10 µm. Image from reference [77], courtesy of H C Nägerl, with
kind permission of Springer Science+Business Media.
rapid rotation is also strongly related to quantum Hall physics (see e.g.
reference [69]).
Wigner localization has been observed in a two-dimensional electron gas
on a liquid-helium surface [43], and in different solid-state systems, see e.g.
references [29, 83]. A very clear demonstration is that with trapped ions,
see figure 4.2, and e.g. references [77, 110].
4.3.3
The Tonks-Girardeau gas
In a one-dimensional system, bosons with a repulsive short-range interaction
may form a strongly correlated state which has many similarities to that of
non-interacting fermions. If the interaction is infinitely strong, two bosons
will never be found at the same positions, thus mimicking the Pauli exclusion
principle for fermions [39]. However, while the density distribution becomes
identical to that of fermions, the momentum distribution is fundamentally
different – it actually features a large fraction of the bosons in the same
momentum orbital [40]. This state has been observed experimentally, with
cold atoms [79]. The state is named after Girardeau, who discovered that
the bosonic wavefunction in this regime corresponds to the absolute value
of the fermionic one [39], and Tonks, who studied properties of a classical
gas of hard elastic spheres confined in low-dimensional geometries [104].
Chapter 5
Summary of results
5.1
Papers I and II – Rotating two-component
Bose-Einstein condensates
We theoretically studied harmonically trapped rotating bosonic twocomponent gases, in particular vortices and wavefunction structures. See
also section 4.3.1.
In paper I we showed that similar to the case of a one-component system
[5], in the regime of low angular momentum, also for two components it
turns out that the system’s energy as a function of angular momentum is
(to numerical precision) described by simple analytical formulas. For a onecomponent system the full solution of the many-body Schrödinger equation
in this regime is known [92], and our results indicate that there may exist
similar solutions for the two-component case.
In paper II we saw that given the appropriate ratio between the populations of the two components, one of them may have a multiply quantized
vortex, in which the phase of the order parameter changes by a multiple 2π
around the vortex core. This does not happen for a rotated one-component
condensate in a harmonic trap [14, 33] – the presence of a smaller cloud
of non-rotating particles appears to be needed to support the larger vortex. Also, our results showed that the considered system could not support
persistent currents.
39
40
5.2
CHAPTER 5. SUMMARY OF RESULTS
Paper III – The Lee-Suzuki method for trapped
bosons
In Paper III we demonstrated that the Lee-Suzuki method [102] can be
useful to describe particle-particle correlations for trapped bosons with a
repulsive short-range interaction. Apart from Paper III, we have not yet
exploited this method for any of our published results, but plan to do so in
future studies on trapped cold gases.
In nuclear structure theory, the Lee-Suzuki approximation is typically
employed to describe short-ranged correlations, induced by the strong shortrange repulsion in nucleon-nucleon interactions [102]. In agreement with the
general conception within that field, in Paper III we experienced that convergence of the calculations was not significantly improved for a long-range
interaction. Because of this, we would not expect a signficant convergence
improvement for electronic systems, e.g. quantum dots – an application
which we initially hoped for. Our conception maintained after a few test
calculations with a few charged particles in a two-dimensional harmonic oscillator. However, references [54, 57, 81] do discuss increased convergence for
similar calculations. At present, we have not investigated this discrepancy
further, but note that the seemingly contrasting conclusions can possibly be
explained by the consideration of different system parameters, such as the
interaction strengths. On the other hand, a possible interpretation is that
the Lee-Suzuki approximation should always be employed, even when significant convergence improvement is not expected, if it requires a relatively
small additional computational effort.
5.3
Paper IV – Electrons in a nanowire quantum dot
We made an attempt to see if Wigner localization could be detected in a
few-electron nanowire quantum dot. First, we solved the Schrödinger equation for a few electrons in an isolated dot system, using the configuration
interaction method. We assumed the dot to be a quasi-1D system, and
thus the length of the dot was the parameter which determined the electron
density. This allowed us to determine which lengths were needed for the
electrons to show the effect of crystallization. See figure 1 of Paper IV,
where panels b, c, and d show the probability density for two electrons, for
different dot lengths.
Using quantum transport theory, the calculated many-particle energies
5.4. PAPER V – DIPOLAR BOSONS IN ONE DIMENSION
41
and corresponding wavefunctions were used to compute the current as a
function of bias and gate voltages. Finally, this was compared to experimental measurements, showing good agreement. However, due to experimental
difficulties, measurements for the largest length of the nanowires were never
made. Because of this the actual Wigner state was not observed, but just
a precursor of it.
5.4
Paper V – Dipolar bosons in one dimension
Bosonic dipolar atoms in a quasi-one-dimensional trap may form significantly different structures depending on the parameters of the interaction.
It is here assumed that the dipole moments of the particles are aligned by
an external field, perpendicular to the trap such that the resulting interaction is always repulsive. The effective strength and shape of the repulsive
interaction may then be tuned by the system parameters; the transverse
confinement strength, the magnitude of the dipole moments, and the particle density. The interaction can then, conceptually, be classified by a
short-range part and a long-range part with different strengths. This results in qualitatively different regimes for the structure of the many-particle
wavefunction. As long as the long-range part is weak, the bosons may be
either weakly interacting if the short-range part is also weak, or, for large
short-range repulsion, they may form a Tonks-Girardeau gas. But if the
long-range part is sufficiently strong, the bosons may instead Wigner localize.
5.5
Paper VI – Wigner states with dipolar atoms or
molecules
As discussed also in section 5.4 above, a system where it could be possible
to see Wigner localized particles is ultra-cold gases with dipolar interactions. The long-range repulsive parts of the interaction potential may be
sufficient to keep the particles well separated. However, dipole fields are
anisotropic. Possibly, this could allow for Wigner states of different kinds
than for electrons for which the Coulomb interaction is isotropic. We numerically investigated this for a few particles in a quasi-two-dimensional
confinement, where the interaction anisotropy may be tuned with an externally applied field. See figure 1 in Paper VI for an illustration of the system
setup.
42
CHAPTER 5. SUMMARY OF RESULTS
If one considers a system with trapped dipolar classical particles, the
ground-state configuration should depend on the anisotropy of the interaction. In the limit of very strong repulsion the quantum mechanical particles
should behave as classical ones, which is also what we found. Despite this,
the resulting behavior involves some complexity due to the quantum mechanical nature of the system. For a three-particle system, we found that,
somewhat surprisingly, the transitions between different ground-state geometries depends on whether the particles are bosons or fermions, even if
the individual particles are Wigner localized.
5.6
Additional results – Bilayer excitons
In a semiconductor, an exciton is a composite quasi-particle consisting of
an electron and a hole, bound by the attractive electrostatic interaction resulting from their different charges [49]. As an exciton is a boson, it may be
possible for several of them to form a Bose-Einstein condensate, as discussed
in e.g. references [6, 15, 45, 73]. Several possible realizations have been
considered, and in recent years bilayer systems have seemed most promising [13, 107]. In such systems, the electrons and holes are kept spatially
separated in two different layers, which suppresses the recombination rate
[13]. Many interesting experimental studies of such exciton gases have been
made, but still no undisputable evidence of condensation has been obtained
[13, 93, 107]. We were interested to see if any signatures of condensation
could be seen in a few-particle system of this kind, such as a quantum dot.
In a simple model, the particles are confined to two different strictly twodimensional layers, separated by a distance z0 . See the inset of figure 5.1.
For simplicity, we consider only spin-polarized electrons and holes. We also
assume the effective masses of electrons and holes to be equal (m∗ ). The particles are confined within their xy-planes by harmonic traps, parametrized
by the oscillator length l0 , same for both electrons and holes. The electrostatic repulsion between particles of same charge is the usual Coulomb
interaction:
Vsame charge (r, r0 ) =
e2
1
4π0 r |r − r0 |
(5.1)
In the term for the attractive interaction between electrons and holes the
distance between the two layers enters:
5.6. ADDITIONAL RESULTS – BILAYER EXCITONS
43
e2
1
p
(5.2)
4π0 r |r − r0 |2 + z02
The configuration interaction method was then used to find the manyparticle eigenstates, using a basis of two-dimensional harmonic oscillator
orbitals. Unfortunately, for small z0 , the attractive interaction results in
strong correlations that were difficult to describe, resulting in poor convergence of the calculated energies. The basis was truncated with a cutoff in the
single-particle space, such that the orbitals from up to ten oscillator shells
were used. This resulted in many-particle bases consisting of ∼ 107 states.
In an attempt to improve the convergence, a different oscillator length than
that of the confinement potential was used for the single-particle basis.
This helped, but the accuracy remained too poor to allow any quantitative
conclusions about the energy spectrum. Still, the obtained solutions may
provide information about the system, such as approximate density distributions. The data presented here did not vary qualitatively with the cutoff
energy.
Figure 5.1 shows pair-correlated density distributions for different values
of the layer separation z0 . The lengths, in the figure and below, are given in
effective Bohr radii, a∗0 = ~2 4π0 r /(m∗ e2 ). The oscillator length was set to
l0 = 10, which effectively makes the interaction terms dominate the singleparticle parts of the Hamiltonian. For z0 = 10 = l0 , the particles localize at
individual positions, as would be expected from a model with classical point
charges. But for z0 = 1, it appears as if strongly bound electron-hole pairs
are formed. For three non-interacting (polarized) fermions in a harmonic
trap, the density distribution has a local minimum at the trap center, due
to shell structure. Here, the overall density distribution instead resembles
a Gaussian function, as it would for condensed bosons in a harmonic trap.
These qualitative results agree very well with those of reference [34],
where an identical system was studied using a quantum Monte Carlo method.
In reference [58], density functional theory was instead used. The phase with
localized particles was seen, but the regime of possible condensation was not
considered.
A possible continuation of this project would have been to consider
hard-wall cylindrical confinements, modelling an experimental setup with
nanowire quantum dots. Such a system could possibly have been probed
experimentally using e.g. spectroscopy. However, the numerical issues mentioned above disallowed more quantitative predictions about the excitation
spectrum.
Vopposite charge (r, r0 ) = −
44
CHAPTER 5. SUMMARY OF RESULTS
Figure 5.1: Pair-correlated density distributions for the bilayer quantum
dot system discussed in the text, and shown schematically in the inset.
The white circle denotes the reference position of one particle (cf equation
3.2). Depending on the interlayer distance, weak or strong coupling between
electrons and holes can be seen. Lengths are given in a∗0 , effective Bohr radii.
See discussion in the text for more information.
Chapter 6
Outlook
Quantum few-particle systems can give rise to several non-trivial effects
and complex dynamics. Signatures of physical phenomena typically considered for very large systems, such as Bose-Einstein condensation or Wigner
localization, may be present also for very small systems. Naturally occurring few-particle systems are most notably atoms and nuclei, and also small
molecules. But other experimental few-particle systems with tunable parameters are becoming increasingly available, such as electronic quantum
dots and trapped cold atomic or molecular gases. Similar to the historical
development of quantum dots, where initially only large systems with at
least ∼ 103 particles were considered [85], experiments on trapped ultracold
gases are now also reaching the few-particle regime, in contrast to the first
experiments with 106 atoms or so [82]. This is a natural development, as
smaller systems require more detailed experimental control. Several recent
experiments, utilizing optical lattices, have prepared and analyzed systems
with single or very few cold atoms (see e.g. references [8, 20, 26, 36]).
A recent thesis [78] reports about on-going experimental work to realize
fermionic few-particle systems in a so-called microtrap, with possibilities
to detect single atoms, where the present accomplishment is roughly 120
trapped 6 Li atoms [78].
The research presented in this thesis and the included papers may provide insight into the behavior of small quantum many-particle systems, but
in most cases it is not discussed how to verify the predictions experimentally.
The exception is Paper IV, where a direct comparisons between theoretical
modeling and experiments were made. While measurements of electrical
currents through small systems is a well-established technique to probe the
45
46
CHAPTER 6. OUTLOOK
internal structure, it is not so for cold gases. But the field is developing –
e.g. references [20, 36] report about particle transfer through coupled wells.
Regarding the Lee-Suzuki approximation for cold gases, we have not
yet taken advantage of it for any of our published results, apart from the
method-oriented Paper III. During the work leading to Paper VI, we also
noted that it gave improved convergence for the considered dipolar forces,
although the results that were eventually published were produced without
the approximation. The problem is that while the approximation can produce accurate eigenenergies of a system, we are often also interested in the
expectation values of various operators. Since the obtained eigenstates are
expanded in an insufficient basis space, they are, in principle, wrong. They
may be used as approximations to the true states, but a more sophisticated
approach would be to transform the relevant operator, similar to what was
done with the interaction. Reference [75] suggests how one may obtain a
two-particle operator correction to an arbitrary one-particle operator.
A possible refinement of the Lee-Suzuki method could be to use analytical two-particle solutions, rather than numerical approximations. For
two particles with a zero-range interaction in a harmonic trap, analytical
solutions are presented in reference [12]. Direct diagonalization of Hamiltonians with zero-range interactions may not be mathematically well-defined
[48], but is still often considered as a useful model. By creating an effective
interaction for a given many-particle space, based on an analytical solution where the zero-range scattering is accounted for properly, could the
mathematical problems possibly be circumvented?
The method could be suitable to describe strongly correlated trapped
Fermi gases. The attractive short-range interaction often present in such
systems can result in strongly or loosely bound pairs of fermions. This
can lead to a Bose-Einstein condensed phase of bosonic molecules, or a
Bardeen-Cooper-Schrieffer (BCS) phase of loosely bound pairs [9, 38]. For
an unbalanced system, with only a single or a few particles of one spin
polarization placed in a sea of particles with opposite spin, there may be
so-called polarons [90]. As the standard configuration interaction approach
does not handle strong short-range interactions well, possibly the Lee-Suzuki
approximation could be useful to study few-particle variants of such systems.
Appendix A
An example
Section A.1 contains an explicit example of how the configuration interaction
method can be used. The calculation is then repeated in section A.2, using
the Lee-Suzuki approximation. Section A.3 summarizes some elements of
the second quantization formalism, which are needed for the calculations
below.
A.1
The configuration interaction method
For an example of the configuration interaction method, let us consider a
few identical bosons in a one-dimensional infinite well. This could serve as
a simple model for an ultra-cold gas of bosonic atoms trapped in a quasione-dimensional waveguide. We assume that the atoms interact only via
short-ranged van der Waals forces – here approximated with a Gaussian
function, for which the range is parametrized by σ. In dimensionless units,
the Hamiltonian is
N
1 d2
gX
1
(xi − xj )2
√
Ĥ =
−
+
exp −
2 dx2i
2
2σ 2
2πσ 2
i=1
N
X
(A.1)
j6=i
where g is a constant representing the strength of the interaction, which is
repulsive if g is positive. The particles are confined to the interval 0 < x < 1,
and the wavefunction is zero outside. In this example, the parameters are set
to g = 5 and σ = 0.01, which results in a fairly strong repulsive interaction
with short range.
47
48
APPENDIX A. AN EXAMPLE
The single-particle states are characterized by the quantum number n
(with n ≥ 1), and the corresponding wavefunctions are
φn (x) =
√
2 sin(nπx)
(A.2)
(1)
with energies En = π 2 n2 /2. Approximately, the energies of the first two
(1)
(1)
orbitals are E1 = 4.935 and E2 = 19.739.
For now, the many-particle space P (3) will be truncated by an energy
cutoff, E < 50.0. This yields three possible Fock states (the second quantization operators â†i are discussed in section A.3):
1
â†1 â†1 â†1 |000 . . . i = |30000 . . . i
3·2
1
(3)
|Φ2 i = √ â†2 â†1 â†1 |000 . . . i = |21000 . . . i
2
1 † † †
(3)
|Φ3 i = √ â2 â2 â1 |000 . . . i = |12000 . . . i
2
(3)
|Φ1 i = √
(A.3)
As the chosen single-particle orbitals are eigenfunctions of the one-particle
part of the Hamiltonian, the correponding matrix representation is diagonal
and can be constructed by just summing up the energies of the individual
particles:

(3)
(3)
(3)
(3)
(3)
(3)
hΦ1 |Ĥ (1) |Φ1 i hΦ1 |Ĥ (1) |Φ2 i hΦ1 |Ĥ (1) |Φ3 i

(3)
(3) 
(3)
(3)
(3)
(1)
=  hΦ(3)
|Φ1 i hΦ2 |Ĥ (1) |Φ2 i hΦ2 |Ĥ (1) |Φ3 i  =
2 |Ĥ
(3)
(3)
(3)
(3)
(3)
(3)
hΦ3 |Ĥ (1) |Φ1 i hΦ3 |Ĥ (1) |Φ2 i hΦ3 |Ĥ (1) |Φ3 i


14.804
0
0

 (A.4)
0
29.609
0
=
0
0
44.413

[Ĥ (1) ]P (3)
The two-particle part is more complicated. Because the interaction conserves the mirror symmetry (parity) of the potential, some matrix elements
of the interaction are zero. But in general the integrals must be calculated
numerically. The following ones are needed for the chosen subspace, P (3) :
A.1. THE CONFIGURATION INTERACTION METHOD
h1, 1|Ĥ (2) |1, 1i
=
h1, 1|Ĥ
(2)
|1, 2i = h1, 2|Ĥ
h1, 2|Ĥ
(2)
|1, 2i
h1, 1|Ĥ
(2)
h1, 2|Ĥ
(2)
h2, 2|Ĥ
(2)
|2, 2i
(2)
7.495
|1, 1i =
=
0
9.988
|2, 2i = h2, 2|Ĥ
(2)
|1, 1i =
|2, 2i = h2, 2|Ĥ
(2)
|1, 2i = 0
=
49
4.988
7.480
(A.5)
These can then be used in the matrix representation of the interaction.
(3)
(3)
Below, the matrix element hΦ1 |Ĥ (2) |Φ3 i is calculated as an example. It
turns out that only a single term in the sum will give a non-zero contribution
to this matrix element – only a single term in the two-particle operator
actually couples h300 . . . | to |120 . . . i.
(3)
(3)
hΦ1 |Ĥ (2) |Φ3 i =
X hi, j|Ĥ (2) |k, li
√
h300 . . . |â†i â†j âl âk |120 . . . i =
δij +δkl
(
2)
i≤j,k≤l
h1, 1|Ĥ (2) |2, 2i
√ √
h300 . . . |â†1 â†1 â2 â2 |120 . . . i =
2· 2
√
√
h1, 1|Ĥ (2) |2, 2i
√ √
2|110 . . . i =
=
h200 . . . | 3 â†1 â2
2· 2
√ √
√ √
h1, 1|Ĥ (2) |2, 2i
√ √
h100 . . . | 2 3
=
2 1|100 . . . i =
2· 2
√
= 3h1, 1|Ĥ (2) |2, 2i = 8.639
=
The other matrix elements can be calculated similarly, and we get


22.485
0
8.639

0
27.470
0
[Ĥ (2) ]P (3) = 
8.639
0
27.456
(A.6)
(A.7)
If the total matrix [Ĥ]P (3) = [Ĥ (1) ]P (3) +[Ĥ (2) ]P (3) is diagonalized, one finds
the lowest eigenvalue to be
(3) E1 = 35.251
(A.8)
P (3)
50
APPENDIX A. AN EXAMPLE
However, this is not necessarily a good approximation of the ground state energy. One must redo the calculations also for other cutoff energies, yielding
other truncations of the basis space, and examine the energy as a function
of increasing cutoff. Ideally, the energy becomes independent of the cutoff
and convergence is achieved. In table A.1, ground state energies for different
cutoffs are presented.
A.2
The Lee-Suzuki approximation
As can be seen in table A.1, the size of the matrix that needs to be diagonalized easily becomes very large. In particular for a short-range interaction,
one typically needs to use very many single-particle orbitals in the basis to
be able to describe a wavefunction with sufficient spatial resolution. The
Lee-Suzuki approximation [102] may be useful here, as discussed in section 2.2. Most published calculations made with the Lee-Suzuki method
have used harmonic oscillator orbitals as basis functions (see e.g. references
[57, 54, 75, 74, 102]), but this is no fundamental restriction.
Let us take the same example as above, with the cutoff E < 50.0. The
calculations will now be done again, with the only difference that another
interaction operator will be used. By examining the corresponding manyparticle basis P (3) (equation A.3), one can see which two-particle matrix
elements for the interaction that are needed (cf equation A.5). The relevant
two-particle states are
1
(2)
|Φ1 i = √ â†1 â†1 |000 . . . i = |20000 . . . i = |1, 1i
2
|Φ2 i = â†2 â†1 |000 . . . i = |11000 . . . i = |1, 2i
1
(2)
|Φ3 i = √ â†2 â†2 |000 . . . i = |02000 . . . i = |2, 2i
2
(2)
(A.9)
These states will constitute the Lee-Suzuki two-particle model space, P (2) .
We now need to solve the Schrödinger equation for only two interacting
particles. In this example this is done just like the calculation above for
three particles, with the configuration interaction method. By using the
first 100 orbitals of the quantum well, and all the two-particle states thereby
possible, good convergence is achieved for the energies. As the model space
P (2) consists of three states, we need the first three two-particle solutions.
(2)
(2)
(2)
Their energies are E1 = 15.334, E2 = 32.482 and E3 = 45.557. The
A.2. THE LEE-SUZUKI APPROXIMATION
(2)
(2)
51
(2)
corresponding eigenstates |Ψ1 i, |Ψ2 i and |Ψ3 i are expanded in several
basis states, but the parts involving P (2) , assembled in matrix form, are


(2)
(2)
(2)
(2)
(2)
(2)
hΦ1 |Ψ1 i hΦ1 |Ψ2 i hΦ1 |Ψ3 i

(2)
(2)
(2)
(2)
(2) 
A =  hΦ(2)
=
2 |Ψ1 i hΦ2 |Ψ2 i hΦ2 |Ψ3 i 
(2)
(2)
(2)
(2)
(2)
(2)
hΦ3 |Ψ1 i hΦ3 |Ψ2 i hΦ3 |Ψ3 i


−0.984
0
0.143

0
0.988
0
=
−0.149
0
−0.960
(A.10)
Using the singular-value decomposition A = U ΣV † where U and V are
unitary matrices, and Σ a diagonal matrix with the singular values of A
[41], the effective interaction within the P (2) -space is (cf equations 2.4 and
3.5; and equation A.5 for the original interaction):

(2)
E1
0
0


(2)
(2)
†
(1)
[Ĥeff ]P (2) = U V †  0
E2
0  V U − [Ĥ ]P (2) =
(2)
0
0
E3


(2)
(2)
(2)
h1, 1|Ĥeff |1, 1i h1, 1|Ĥeff |1, 2i h1, 1|Ĥeff |2, 2i


(2)
(2)
(2)
=  h1, 2|Ĥeff
|1, 1i h1, 2|Ĥeff |1, 2i h1, 2|Ĥeff |2, 2i  =
(2)
(2)
(2)
h2, 2|Ĥeff |1, 1i h2, 2|Ĥeff |1, 2i h2, 2|Ĥeff |2, 2i


6.130
0
4.434
7.808
0  (A.11)
= 0
4.434
0
5.413

The matrix representation of the effective interaction in the many-particle
space becomes (cf equation A.7)


18.390
0
7.680
(2)

0
21.746
0
[Ĥeff ]P (3) = 
(A.12)
7.680
0
21.029
(2)
Diagonalization of the matrix [Ĥ]P (3) = [Ĥ (1) ]P (3) + [Ĥeff ]P (3) gives the
ground state energy as
(3) E1 = 31.458
(A.13)
P (3) ,Lee-Suzuki
52
APPENDIX A. AN EXAMPLE
Cutoff energy
20.0
50.0
100.0
500.0
1000.0
5000.0
10000.0
50000.0
Basis size
1
3
9
97
266
2892
8124
89817
(3)
Estandard
37.290
35.251
32.893
31.677
31.460
31.191
31.152
31.135
(3)
ELee-Suzuki
31.198
31.458
31.034
31.138
31.135
31.135
31.135
31.135
Table A.1: Ground state energy for a system with N = 3 particles as
described in the beginning of section A.1, for different basis space truncations, either using the standard configuration interaction method or the
modified one with the Lee-Suzuki approximation. The Lee-Suzuki approximation here gives faster convergence. While Estandard always decreases
monotonically, ELee-Suzuki in this case initially seems to oscillate around the
asymptotic value.
As can be seen in table A.1, the energy now converges faster when the LeeSuzuki approximation is used. In particular, much smaller matrices need to
be considered to achieve similar convergence.
A.3
Second quantization formalism
This appendix briefly summarizes the second quantization formalism, as it
is described in many textbooks, such as e.g. reference [44]. The standard
definitions are repeated here to make the example comprehensive.
With a complete set of single-particle orbitals, {φi (x)}, one can construct
a complete set of basis functions for the many-particle space [44]. The index
i may be a tuple of indices, consisting of all quantum numbers needed to
uniquely specify the orbital, and likewise the argument x may be a vector
variable. The many-particle basis states must be either symmetric or antisymmetric with respect to particle exchange, depending on whether the
particles are bosons or fermions. For fermions, such basis states are called
Slater determinants, while for bosons they are sometimes called permanents.
A common name is Fock states. They are often represented by arrays of
integers, denoting the number of particles in each single-particle orbital, e.g.
|301500i. For bosons, any number of particles may occupy the same orbital,
A.3. SECOND QUANTIZATION FORMALISM
53
while for fermions there can only be zero or one particle in each.
The creation operator â†i acts on a state and creates an additional particle in the orbital i, while the annihilation operator âi removes a particle.
A complete definition is given in reference [44]. Apart from the creation or
annihilation of a particle, the operators also give a numerical factor, which
may be zero. For bosons:
â†i |n1 n2 . . . ni . . . i =
âi |n1 n2 . . . ni . . . i =
√
√
ni + 1|n1 n2 . . . (ni + 1) . . . i
ni |n1 n2 . . . (ni − 1) . . . i
(A.14)
And for fermions:
â†i |n1 n2 . . . 0i . . . i = (−1)
P
â†i |n1 n2
j<i
nj
|n1 n2 . . . 1i . . . i
. . . 1i . . . i = 0
(A.15)
âi |n1 n2 . . . 0i . . . i = 0
P
âi |n1 n2 . . . 1i . . . i = (−1)
j<i
nj
|n1 n2 . . . 0i . . . i
For bosons, the operators obey the following commutation rules:
âi , â†j = âi â†j − â†j âi = δij
âi , âj = â†i , â†j = 0
For fermions, they instead obey anti-commutation rules:
âi , â†j = âi â†j + â†j âi = δij
âi , âj = â†i , â†j = 0
(A.16)
(A.17)
A typical Hamiltonian of an isolated quantum few-particle system can be
written as
Ĥ = Ĥ (1) + Ĥ (2)
(A.18)
where Ĥ (1) is a one-particle operator, for example the kinetic energy operator plus an external trapping potential, and Ĥ (2) is a two-particle operator,
the interaction between the particles. In second quantization, the Hamilton
operator can be written as
Ĥ =
X
i,j
hij â†i âj +
(1)
1 X (2) † †
hijkl âi âj âl âk
2
i,j,k,l
(A.19)
54
APPENDIX A. AN EXAMPLE
where the sums run over all possible indices for the single-particle orbitals,
and
ˆ
(1)
hij = φ∗i (x)Ĥ (1) φj (x)dx
(A.20)
and
ˆ ˆ
(2)
φ∗i (x)φ∗j (x0 )Ĥ (2) φk (x)φl (x0 )dxdx0
hijkl =
(A.21)
as in reference [44]. By grouping terms in the two-particle operator, and using properly (anti-)symmetrized
and normalized two-particle states |i, ji =
√ δ
† †
ij
âj âi |000 . . . i/( 2) one may rewrite equation A.19 as
Ĥ =
X
hij â†i âj +
(1)
i,j
X hi, j|Ĥ (2) |k, li † †
√
âi âj âl âk
( 2)δij +δkl
i≤j,k≤l
(A.22)
√
where the 2 in the denominator is needed to keep everything properly
normalized when two bosons are in the same orbital. The matrix elements
for fermions are then
hi, j|Ĥ (2) |k, li =
1 (2)
(2)
(2)
(2)
(h
− hijlk + hjilk − hjikl )
2 ijkl
(A.23)
while for bosons they are
(2)
(2)
(2)
(2)
hi, j|Ĥ (2) |k, li = 12 (hijkl + hijlk + hjilk + hjikl )
(2)
(2)
√1 (h
iikl + hiilk )
2
(2)
(2)
hi, j|Ĥ (2) |k, ki = √12 (hijkk + hjikk )
(2)
hi, i|Ĥ (2) |k, ki = hiikk
hi, i|Ĥ (2) |k, li =
(A.24)
(2)
In this text there is a need to distinguish between the two quantitites hijkl
and hi, j|Ĥ (2) |k, li. The notation used here to denote them is not universal,
e.g. it differs from that in reference [44].
Appendix B
Quantum harmonic oscillator
In this appendix, I give some formulas and equations that I have found useful
for studies of a few particles trapped in harmonic oscillator confinements.
Much of it is standard textbook material that may be found elsewhere, but
some things are not. The notation used in this chapter does not necessarily
conform to that elsewhere in this thesis. Also, the step operators here should
not be confused with those of the second quantization formalism.
B.1
B.1.1
One dimension
One particle
For a quantum mechanical particle in a one-dimensional harmonic oscillator
potential the eigenstates are |nx i = √n1 ! (b̂†x )nx |0i, where the step operator
x
b̂†x and its complex conjugate b̂x are defined in any standard textbook on
quantum mechanics, for example Messiah’s [70]. The commutator between
them
is [b̂x , b̂†x ] = 1. The wavefunctions of the eigenstates are φnx (x) =
q
2
1 √
H (x)e−x /2 , where Hnx is a Hermite polynomial, defined as
nx !2nx π nx
Hnx (x) = (−1)nx ex
B.1.2
2
dnx
dxnx
2
(e−x ).
Two particles
For two particles in a one-dimensional harmonic oscillator the eigenstates
in absolute particle coordinates are
55
56
APPENDIX B. QUANTUM HARMONIC OSCILLATOR
(b̂†x,1 )nx,1 (b̂†x,2 )nx,2
p
|nx,1 nx,2 i =
|00i
nx,1 !nx,2 !
(B.1)
and their wavefunctions are products as φnx,1 (x1 )φnx,2 (x2 ). One may define
new coordinates for the two-particle system:
(
2
xcm = x1√+x
2
(B.2)
2
xrel = x1√−x
2
xcm is here called the center-of-mass coordinate and xrel the relative coordinate. One can have step operators for the center-of-mass motion and the
relative motion, b̂†x,cm = √12 (b̂†x,1 + b̂†x,2 ) and b̂†x,rel = √12 (b̂†x,1 − b̂†x,2 ). Using
these operators one can create the basis states
|nx,cm nx,rel i =
(b̂†x,cm )nx,cm (b̂†x,rel )nx,rel
p
|00i
nx,cm !nx,rel !
(B.3)
with wavefunctions φnx,cm (xcm )φnx,rel (xrel ). The two-particle states in the
different coordinate systems can be expanded in terms of each other. The
overlap integrals are (the x-subscripts are omitted here for clarity):
♥(ncm , nrel ; n1 , n2 ) = hncm nrel |n1 n2 i =
1
1
= h00| √
(b̂cm )ncm (b̂rel )nrel √
(b̂† )n1 (b̂†2 )n2 |00i =
ncm !nrel !
n1 !n2 ! 1
n n
1
b̂1 + b̂2 cm b̂1 − b̂2 rel (b̂†1 )n1 (b̂†2 )n2
√
√
√
= h00| √
|00i =
ncm !nrel !
n1 !n2 !
2
2
r
ncm +nrel
1
n1 !n2 !
√
(−1)n2 ×
= δncm +nrel ,n1 +n2
ncm !nrel !
2
min(ncm ,n2 )
X
ncm
nrel
×
(−1)k
(B.4)
k
n2 − k
k=max(0,n2 −nrel )
The final formula can be obtained by using the binomial identity to expand
the powers in ncm and nrel , and finally rewriting the resulting summations.
The Kronecker delta can be interpreted as conservation of energy. Equation
B.4 is the 1D equivalent to Moshinsky’s transformation brackets for two
B.2. TWO DIMENSIONS
57
particles in a 3D harmonic oscillator [72]. Another version of the explicit
formula is given by Felline [32], who employs a more general coordinate
transformation than equation B.2.
One may derive various symmetry relations, for example:
♥(ncm , nrel ; n1 , n2 ) = ♥(n1 , n2 ; ncm , nrel )
♥(ncm , nrel ; n2 , n1 ) = (−1)nrel ♥(ncm , nrel ; n1 , n2 )
(B.5)
Also, various recursion relations for the overlap integrals may be derived.
For example, by applying one of the âcm -operators on the ket instead of the
bra one obtains
1
♥(ncm , nrel ; n1 , n2 ) = √
2
r
n1
♥(ncm − 1, nrel ; n1 − 1, n2 )+
ncm
r
n2
♥(ncm − 1, nrel ; n1 , n2 − 1)
+
ncm
(B.6)
For the special case of ncm = 0, the explicit formula in equation B.4 simplifies to
♥(0, nrel ; n1 , n2 ) = δnrel ,n1 +n2
1
√
2
nrel
r
(−1)
n2
nrel !
n1 !n2 !
(B.7)
The explicit expression in equation B.4 is not well suited for numerical
evaluation, but equations B.6 and B.7 provide a stable way to calculate the
overlap bracket.
B.2
B.2.1
Two dimensions
One particle
In Cartesian coordinates the one-particle states in an isotropic twodimensional harmonic oscillator are
|nx ny i =
(b̂†x )nx (b̂†y )ny
p
|00i
nx !ny !
(B.8)
with wavefunctions φnx (x)φny (y). However, it is often convenient to use the
polar coordinates r and ϕ instead. Following Messiah [70], one can define
58
APPENDIX B. QUANTUM HARMONIC OSCILLATOR
b̂†+ =
√1 (b̂†
2 x
+ ib̂†y ) and b̂†− =
√1 (b̂†
2 x
− ib̂†y ) which are step operators in polar
b̂†+
coordinates. The operator
increases the energy by one unit and increases
the angular momentum by one unit, while the operator b̂†− also increases
the energy by one unit but decreases the angular momentum by one unit.
The Hamiltonian can be written as b̂†+ b̂+ + b̂†− b̂− + 1, and the one-particle
eigenstates in polar coordinates are
|n+ n− i =
(b̂†+ )n+ (b̂†− )n−
p
|00i
n+ !n− !
(B.9)
with energy En+ n− = n+ + n− + 1 and angular momentum n+ − n− , where
the quantum numbers must fulfill n+ ≥ 0 and n− ≥ 0. An alternative set
of quantum numbers is nr and mϕ , where nr is the number of radial nodes
in the wavefunction (nr ≥ 0), and mϕ is the angular momentum (mϕ can
be any integer). The two notations are equivalent in the sense that they
represent exactly the same eigenstates. The relation between them is
(
n+
n−
= (2nr + |mϕ | + mϕ )/2
⇔
= (2nr + |mϕ | − mϕ )/2
(
= (n+ + n− − |n+ − n− |)/2
= n+ − n−
(B.10)
The eigenstates in polar coordinates can be expressed in terms of the eigenstates in Cartesian coordinates. By using the definitions of the polar step
operators in terms of the cartesian ones, one can derive the following relation, where i is the imaginary unit:
nr
mϕ
hnx ny |n+ n− i = iny ♥(nx , ny ; n+ , n− )
(B.11)
The wavefunctions, often referred to as Fock-Darwin orbitals [27, 35], are
imϕ ϕ
φnr mϕ (r, ϕ) = Rnr mϕ (r) e√2π with the radial parts
Rnr mϕ (r) = (−1)
|m |
nr
√
s
2
nr !
2 −r 2 /2
ϕ|
r|mϕ | L|m
nr (r )e
(nr + |mϕ |)!
(B.12)
where Lnr ϕ is a generalized Laguerre polynomial, as defined in reference
[1]. The phase factor (−1)nr is perhaps unusual, but is essential for the
consistency of this text, and the validity of the overlap formulas presented
here. Equation B.12, except for the phase factor, is given in many places
B.2. TWO DIMENSIONS
59
in the literature. A detailed derivation is given by Kristinsdóttir [53], although the states were originally presented by Fock [35] and Darwin [27],
and more recently in e.g. references [19] and [50]. However, an alternative
way to formally get the full two-dimensional wavefunction is to expand it
in Cartesian orbitals, using equation B.11. By evaluating this expansion at
x → +∞, keeping y = 0, and noting that for large enough x a single term
in the resulting sums will dominate, one can see that the radial wavefunction will always be positive at r → ∞. Because the sign of the Laguerre
polynomials is (−1)nr as r → ∞ [1], the phase factor (−1)nr is required in
equation B.12.
B.2.2
Two particles
In polar coordinates, the two-particle states in absolute particle coordinates
are
(b̂†+,1 )n+,1 (b̂†−,1 )n−,1 (b̂†+,2 )n+,2 (b̂†−,2 )n−,2
p
|0000i
n+,1 !n−,1 !n+,2 !n−,2 !
(B.13)
For polar center-of-mass and relative coordinates one has the step operators b̂†±,cm = √12 (b̂†±,1 + b̂†±,2 ) and b̂†±,rel = √12 (b̂†±,1 − b̂†±,2 ), so that the
corresponding states are
|n+,1 n−,1 n+,2 n−,2 i =
|n+,cm n−,cm n+,rel n−,rel i =
=
(b̂†+,cm )n+,cm (b̂†−,cm )n−,cm (b̂†+,rel )n+,rel (b̂†−,rel )n−,rel
p
|0000i (B.14)
n+,cm !n−,cm !n+,rel !n−,rel !
Because of the decoupling of “positive” and “negative” motions (the associated step operators commute, which can be shown by expanding them in the
Cartesian ones), it turns out that the overlap integral between two-particle
states in the different coordinate systems can be written as
hn+,cm n−,cm n+,rel n−,rel |n+,1 n−,1 n+,2 n−,2 i =
= ♥(n+,cm , n+,rel ; n+,1 , n+,2 ) · ♥(n−,cm , n−,rel ; n−,1 , n−,2 ) (B.15)
The Kronecker delta present in the ♥-function gives that the overlap is zero
unless n+,cm + n+,rel = n+,1 + n+,2 and n−,cm + n−,rel = n−,1 + n−,2 .
60
APPENDIX B. QUANTUM HARMONIC OSCILLATOR
In (nr , mϕ )-notation this implies 2nr,cm + |mϕ,cm | + 2nr,rel + |mϕ,rel | =
2nr,1 +|mϕ,1 |+2nr,2 +|mϕ,2 | (conservation of energy) and mϕ,cm +mϕ,rel =
mϕ,1 + mϕ,2 (conservation of angular momentum). Equation B.15 is the 2D
equivalent to Moshinsky’s transformation brackets for two particles in a 3D
harmonic oscillator [72]. Please note that for equation B.15 to be valid, the
phase convention in equation B.12 must be used. An alternative, explicit,
expression for the overlap is given in reference [18]; however, there appears
to be either a typographical error present or an unclear phase convention of
the orbitals used.
Bibliography
[1]
M Abramowitz, I A Stegun (1965), Handbook of Mathematical Functions, Dover Publications
[2]
E Anderson, Z Bai, C Bischof, S Blackford, J Demmel, J Dongarra,
J Du Croz, A Greenbaum, S Hammarling, A McKenney, D Sorensen,
LAPACK Users’ Guide (3rd Ed.), SIAM, ISBN 0-89871-447-8; see
also www.netlib.org/lapack
[3]
M H Anderson, J R Ensher, M R Matthews, C E Wieman, E A Cornell (1995), Observation of Bose-Einstein Condensation in a Dilute
Atomic Vapor, Science 269, 198
[4]
M A Baranov (2008), Theoretical progress in many-body physics with
ultracold dipolar gases, Physics Reports 464, 71
[5]
G F Bertsch, T Papenbrock (1999), Yrast Line for Weakly Interacting
Trapped Bosons, Physical Review Letters 83, 5412
[6]
J M Blatt, K W Böer, W Brandt (1962), Bose-Einstein Condensation
of Excitons, Physical Review 126, 1691
[7]
F Bloch (1973), Superfluidity in a Ring, Physical Review A 7, 2187
[8]
I Bloch (2005), Ultracold quantum gases in optical lattices, Nature
Physics 1, 23
[9]
I Bloch, J Dalibard, W Zwerger (2008), Many-body physics with ultracold gases, Reviews of Modern Physics 80, 885
[10]
Bose (1924), Plancks Gesetz und Lichtquantenhypothese (Planck’s law
and light quantum hypothesis), Zeitschrift für Physik 26, 178
61
62
[11]
BIBLIOGRAPHY
T J Bukowski, J H Simmons (2002), Quantum Dot Research: Current State and Future Prospects, Critical Reviews in Solid State and
Material Sciences 27, 119
[12] T Busch, B-G Englert, K Rza̧żewski, M Wilkens (1998), Two Cold
Atoms in a Harmonic Trap, Foundations of Physics 28, 549
[13]
L V Butov (2007), Cold exciton gases in coupled quantum well structures, Journal of Physics: Condensed Matter 19, 295202
[14]
D A Butts, D S Rokshar (1999), Predicted signatures of rotating BoseEinstein condensates, Nature 397, 327
[15]
R C Casella (1963), A Criterion for Exciton Binding in Dense
Electron-Hole Systems – Application to Line Narrowing Observed in
GaAs, Journal of Applied Physics 34, 1703
[16]
E Caurier, F Nowacki (1999), Present status of shell model
techniques, Acta Physica Polonica B 30, 705; see also
sbgat194.in2p3.fr/~theory/antoine/menu.html
[17]
L S Cederbaum, O E Alon, A I Streltsov (2006), Coupled-cluster theory for systems of bosons in external traps, Physical Review A 73,
043609
[18]
E Chacón (1973), Basis functions for the nuclear three-body problem,
Revista Mexicana de Física 22, 97
[19]
T Chakraborty (1999), Quantum Dots: A survey of the properties of
artificial atoms, North-Holland, ISBN 0-444-50258-0
[20]
P Cheinet, S Trotzky, M Feld, U Schnorrberger, M Moreno-Cardoner,
S Fölling, I Bloch (2008), Counting Atoms Using Interaction Blockade
in an Optical Superlattice, Physical Review Letters 101, 090404
[21]
N R Cooper (2008), Rapidly rotating atomic gases, Advances in
Physics 57, 539
[22]
E A Cornell, C E Wieman (2002), Nobel Lecture: Bose-Einstein condensation in a dilute gas, the first 70 years and some recent experiments, Reviews of Modern Physics 74, 875
[23]
C J Cramer (2004), Essentials of Computational Chemistry – Theories
and Models (2nd Ed.), John Wiley & Sons, ISBN 0-470-09181-9
63
[24]
T D Crawford, C D Sherrill, E F Valeev, J T Fermann, R A King,
M L Leininger, S T Brown, C L Janssen, E T Seidl, J P Kenny,
W D Allen (2007), PSI3: An Open-Source Ab Initio Electronic Structure Package, Journal of Computational Chemistry 28, 1610; see also
www.psicode.org
[25]
F Dalfovo, S Giorgini, L P Pitaevskii, S Stringari (1999), Theory
of Bose-Einstein condensation in trapped gases, Reviews of Modern
Physics 71, 463
[26]
J G Danzl, M J Mark, E Haller, M Gustavsson, R Hart, J Aldegunde,
J M Hutson, H-C Nägerl (2010), An ultracold high-density sample of
rovibronic ground-state molecules in an optical lattice, Nature Physics
6, 265
[27]
C G Darwin (1930), The Diamagnetism of the Free Electron, Proceedings of the Cambridge Philosophical Society 27, 86
[28]
K B Davis, M-O Mewes, M A Joffe, M R Andrews, W Ketterle (1995),
Evaporative Cooling of Sodium Atoms, Physical Review Letters 74,
5202
[29]
V V Deshpande, M Bockrath, L I Glazman, A Yacoby (2010), Electron
liquids and solids in one dimension, Nature 464, 209
[30]
A Einstein (1924), Quantentheorie des einatomigen idealen Gases
(Quantum theory of monatomic ideal gases), Sitzungsberichte
der Preussichen Akademie der Wissenschaften – PhysikalischMatematische Klasse, p 261
[31]
A Einstein (1925), Quantentheorie des einatomigen idealen Gases,
Zweite Abhandlung (Quantum theory of monatomic ideal gases,
Part 2), Sitzungsberichte der Preussichen Akademie der Wissenschaften – Physikalisch-Matematische Klasse, p 3
[32]
C Felline (2004), Applications of effective field theories to the manybody nuclear problem and frustrated spin chains, PhD thesis, Florida
State University
[33]
A L Fetter (2009), Rotating trapped Bose-Einstein condensates, Reviews of Modern Physics 81, 647
64
BIBLIOGRAPHY
[34]
A V Filinov, M Bonitz, Y E Lozovik (2003), Excitonic clusters in coupled quantum dots, Journal of Physics A: Mathematical and General
36, 5899
[35]
V Fock (1928), Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld (Remark on quantization of the harmonic oscillator in a magnetic field), Zeitschrift für Physik 47, 446
[36]
S Fölling, S Trotzky, P Cheinet, M Feld, R Saers, A Widera, T Müller,
I Bloch (2007), Direct observation of second order atom tunneling,
Nature 448, 1029
[37]
M Galassi, J Davies, J Theiler, B Gough, G Jungman, P Alken,
M Booth, F Rossi (2010), GNU Scientific Library Reference Manual
(Ed. 1.14); see also www.gnu.org/software/gsl
[38]
S Giorgini, L P Pitaevskii, S Stringari (2008), Theory of ultracold
atomic Fermi gases, Reviews of Modern Physics 80, 1215
[39]
M Girardeau (1960), Relationship between Systems of Impenetrable
Bosons and Fermions in One Dimension, Journal of Mathematical
Physics 1, 516
[40]
M D Girardeau, E M Wright, J M Triscari (2001), Ground-state properties of a one-dimensional system of hard-core bosons in a harmonic
trap, Physical Review A 63, 033601
[41]
G H Golub, C F van Loan (1996), Matrix Computations (3rd Ed.),
The John Hopkins University Press, ISBN 0-8018-5414-8
[42] M Greiner, C A Regal, D S Jin (2003), Emergence of a molecular
Bose-Einstein condensate from a Fermi gas, Nature 426, 537
[43]
C C Grimes, G Adams (1979), Evidence for a Liquid-to-Crystal Phase
Transition in a Classical, Two-Dimensional Sheet of Electrons, Physical Review Letters 42, 795
[44]
E K U Gross, E Runge, O Heinonen (1991), Many-Particle Theory,
IOP Publishing, ISBN 0-7503-0155-4
[45]
E Hanamura, H Haug (1977), Condensation effects of excitons,
Physics Reports 33, 209
65
[46]
R Hanson, L P Kouwenhoven, J R Petta, S Tarucha, L M K Vandersypen (2007), Spins in few-electron quantum dots, Reviews of Modern
Physics 79, 1217
[47] A Harju (2005), Variational Monte Carlo for Interacting Electrons in
Quantum Dots, Journal of Low Temperature Physics 140, 181
[48]
K Huang (1963), Statistical Mechanics, John Wiley & Sons
[49]
H Ibach, H Lüth (2003), Solid-State Physics (3rd Ed.), SpringerVerlag, ISBN 3-540-43870-X
[50]
L Jacak, P Hawrylak, A Wójs (1998), Quantum Dots, Springer-Verlag,
ISBN 3-540-63653-6
[51]
W Ketterle (2002), Nobel lecture: When atoms behave as waves: BoseEinstein condensation and the atom laser, Reviews of Modern Physics
74, 4431
[52]
D E Knuth (1998), The Art of Computer Programming Vol. 3
(2nd Ed.), Addison-Wesley, ISBN 0-201-89685-0
[53]
L H Kristinsdóttir (2009), The Role of Inter-Species Interaction
Strength in a Two-Component Rotating Boson Gas, Master thesis,
Lund University
[54]
S Kvaal, M Hjorth-Jensen, H Møll Nilsen (2007), Effective interactions, large-scale diagonalization, and one-dimensional quantum dots,
Physical Review B 76, 085421
[55]
S Kvaal (2008), Geometry of effective Hamiltonians, Physical Review
C 78, 044330
[56]
S Kvaal (2008), Open source FCI code for quantum dots and effective
interactions, arXiv: 0810.2644v1
[57]
S Kvaal (2009), Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions, Physical Review B 80, 045321
[58]
K Kärkkäinen, M Koskinen, M Manninen, S M Reimann (2004),
Electron-hole bilayer quantum dots: phase diagram and exciton localization, Solid State Communications 130, 187
66
BIBLIOGRAPHY
[59]
K Kärrkäinen, J Christensson, G Reinisch, G M Kavoulakis,
S M Reimann (2007), Metastability of persistent currents in trapped
gases of atoms, Physical Review A 76, 043627
[60]
T Lahaye, C Menotti, L Santos, M Lewenstein, T Pfau (2009), The
physics of dipolar bosonic quantum gases, Reports on Progress in
Physics 72, 126401
[61]
A J Leggett (1999), Superfluidity, Reviews of Modern Physics 71,
S318
[62]
A J Leggett (2001), Bose-Einstein condensation in the alkali gases:
Some fundamental concepts, Reviews of Modern Physics 73, 307
[63]
F London (1938), The λ-Phenomenon of Liquid Helium and the BoseEinstein Degeneracy, Nature 141, 643
[64]
F London (1938), On the Bose-Einstein Condensation, Physical Review 54, 947
[65]
R B Lehoucq, D C Sorensen, C Yang (1998),
Users’ Guide,
SIAM, ISBN 978-0-898714-07-4;
www.caam.rice.edu/software/ARPACK
[66]
E Lundh (2002), Multiply quantized vortices in trapped Bose-Einstein
condensates, Physical Review A 65, 043604
[67]
K W Madison, F Chevy, W Wohlleben, J Dalibard (2000), Vortex
Formation in a Stirred Bose-Einstein Condensate, Physical Review
Letters 84, 806
[68]
P A Maksym (1996), Eckardt frame theory of interacting electrons in
quantum dots, Physical Review B 53, 10871
[69]
S S Mandal, M R Peterson, J K Jain (2003), Two-Dimensional
Electron System in High Magnetic Fields: Wigner Crystal versus
Composite-Fermion Liquid, Physical Review Letters 90, 106403
[70]
A Messiah (1965), Quantum Mechanics I, North-Holland
[71]
O Morsch, M Oberthaler (2006), Dynamics of Bose-Einstein condensates in optical lattices, Reviews of Modern Physics 78, 179
ARPACK
see also
67
[72]
M Moshinsky, Y F Smirnov (1996), The Harmonic Oscillator in Modern Physics, Harwood Academic Publishers GmbH, ISBN 3-71860621-6
[73]
S A Moskalenko (1962), Reversible optico-hydrodynamic phenomena
in a nonideal exciton gas, Soviet Physics – Solid State 4, 199
[74]
P Navrátil, J P Vary, B R Barrett (2000), Properties of 12 C in the Ab
Initio Nuclear Shell Model, Physical Review Letters 84, 5728
[75]
P Navrátil, J P Vary, B R Barrett (2000), Large-basis ab initio no-core
shell model and its application to 12 C, Physical Review C 62, 054311
[76]
P Navrátil, S Quaglioni, I Stetcu, B R Barrett (2009), Recent developments in no-core shell-model calculations, Journal of Physics G:
Nuclear and Particle Physics 36, 083101
[77]
H C Nägerl, W Bechter, J Eschner, F Schmidt-Kaler, R Blatt (1998),
Ion strings for quantum gates, Applied Physics B 66, 603
[78]
T B Ottenstein (2010), Few-body physics in ultracold Fermi gases,
PhD thesis, Ruperto-Carola University of Heidelberg
[79]
B Paredes, A Widera, V Murg, O Mandel, S Fölling, I Cirac,
G V Shlyapnikov, T W Hänsch, I Bloch (2004), Tonks-Girardeau gas
of ultracold atoms in an optical lattice, Nature 429, 277
[80]
F Pederiva, C J Umrigar, E Lipparini (2000), Diffusion Monte Carlo
study of circular quantum dots, Physical Review B 62, 8120; (2003),
Erratum, Physical Review B 68, 089901
[81]
M Pedersen Lohne, G Hagen, M Hjorth-Jensen, S Kvaal, F Pederiva (2010), Ab initio computation of circular quantum dots, arXiv:
1009.4833v1
[82]
C J Pethick, H Smith (2008), Bose-Einstein Condensation in Dilute Gases (2nd Ed.), Cambridge University Press, ISBN 978-0-52184651-6
[83]
B A Piot, Z Jiang, C R Dean, L W Engel, G Gervais, L N Pfeiffer,
K W West (2008), Wigner crystallization in a quasi-three-dimensional
electronic system, Nature Physics 4, 936
68
BIBLIOGRAPHY
[84]
L P Pitaevskii, S Stringari (2003), Bose-Einstein Condensation,
Clarendon Press, ISBN 0-19-850719-4
[85]
S M Reimann, M Manninen (2002), Electronic structure of quantum
dots, Reviews of Modern Physics 74, 1283
[86]
M Rontani, C Cavazzoni, D Bellucci, G Goldoni (2006), Full configuration interaction approach to the few-electron problem in artificial atoms, Journal of Chemical Physics 124, 124102; see also
www.s3.infm.it/donrodrigo
[87]
C Ryu, M F Andersen, P Cladé, Vasant Natarajan, K Helmerson,
W D Phillips (2007), Observation of Persistent Flow of a BoseEinstein Condensate in a Toroidal trap, Physical Review Letters 99,
260401
[88]
H Saarikoski, S M Reimann, A Harju, M Manninen (2010), Vortices
in quantum droplets: Analogies between boson and fermion systems,
Reviews of Modern Physics 82, 2785
[89]
L Samuelson, M T Björk, K Deppert, M Larsson, B J Ohlsson,
N Panev, A I Persson, N Sköld, C Thelander, L R Wallenberg (2004),
Semiconductor nanowires for novel one-dimensional devices, Physica
E 21, 560
[90]
A Schirotzek, C-H Wu, A Sommer, M W Zwierlein (2009), Observation of Fermi Polarons in a Tunable Fermi Liquid of Ultracold Atoms,
Physical Review Letters 192, 230402
[91]
V Schweikhard, I Coddington, P Engels, S Tung, E A Cornell (2004),
Vortex-Lattice Dynamics in Rotating Spinor Bose-Einstein Condensates, Physical Review Letters 93, 210403
[92]
R A Smith, N K Wilkin (2000), Exact eigenstates for repulsive bosons
in two dimensions, Physical Review A 62, 061602(R)
[93]
D W Snoke (2003), When should we say we have observed Bose condensation of excitons?, Physica Status Solidi B 238, 389
[94]
B Stroustrup (2006), The C++ Programming Language (Special Ed.),
Addison-Wesley, ISBN 0-201-70073-5
69
[95]
K Suzuki, S Y Lee (1980), Convergent Theory for Effective Interaction
in Nuclei, Progress of Theoretical Physics 64, 2091
[96]
K Suzuki (1982), Construction of Hermition Effective Interaction in
Nuclei, Progress of Theoretical Physics 68, 246
[97]
K Suzuki (1982), Unitary-Model-Operator Approach to Nuclear Effective Interaction. I, Progress of Theoretical Physics 68, 1627
[98]
K Suzuki (1982), Unitary-Model-Operator Approach to Nuclear Effective Interaction. II, Progress of Theoretical Physics 68, 1999
[99]
K Suzuki, R Okamoto (1983), Degenerate Perturbation Theory in
Quantum Mechanics, Progress of Theoretical Physics 70, 439
[100] K Suzuki, R Okamoto (1986), Unitary-Model-Operator Approach to
Nuclear Many-Body Problem. I, Progress of Theoretical Physics 75,
1388
[101] K Suzuki, R Okamoto (1986), Unitary-Model-Operator Approach to
Nuclear Many-Body Problem. II, Progress of Theoretical Physics 76,
127
[102] K Suzuki, R Okamoto (1994), Effective Interaction Theory and
Unitary-Model-Operator Approach to Nuclear Saturation Problem,
Progress of Theoretical Physics 92, 1045
[103] J Tempere, J T Devreese, E R I Abraham (2001), Vortices in
Bose-Einstein condensates confined in a multiply connected LaguerreGaussian optical trap, Physical Review A 64, 023603
[104] L Tonks (1936), The Complete Equation of State of One, Two and
Three-Dimensional Gases of Hard Elastic Spheres, Physical Review
50, 955
[105] D Vautherin (1973), Hartree-Fock Calculations with Skyrme’s Interaction. II. Axially Deformed Nuclei, Physical Review C 7, 296
[106] S Viefers (2008), Quantum Hall physics in rotating Bose-Einstein condensates, Journal of Physics: Condensed Matter 20, 123202
[107] Z Vörös, D W Snoke (2008), Quantum well excitons at low density,
Modern Physics Letters B 22, 701
70
BIBLIOGRAPHY
[108] E Waltersson, E Lindroth (2010), Performance of the coupled cluster
singles and doubles method on two-dimensional quantum dots., arXiv:
1004.4081v1
[109] W G van der Wiel, S De Franceschi, J M Elzerman, T Fujisawa,
S Tarucha, L P Kouwenhoven (2003), Electron transport through double quantum dots, Reviews of Modern Physics 75, 1
[110] C E Wieman, D E Pritchard, D J Wineland (1999), Atom cooling,
trapping, and quantum manipulation, Reviews of Modern Physics 71,
S253
[111] E Wigner (1934), On the Interaction of Electrons in Metals, Physical
Review 46, 1002
[112] C Yannouleas, U Landman (2007), Symmetry breaking and quantum
correlations in finite systems: studies of quantum dots and ultracold
Bose gases and related nuclear and chemical methods, Reports on
Progress in Physics 70, 2067
[113] M W Zwierlein, C A Stan, C H Schunck, S M F Raupach, A J Kerman,
W Ketterle (2004), Condensation of Pairs of Fermionic Atoms near
a Feshbach Resonance, Physical Review Letters 92, 120403
Acknowledgments
First, I would like to thank my supervisor Stephanie Reimann. I have enjoyed the years of my PhD studies very much. And I want to thank everyone
else I have collaborated with, and learned from, within and outside of the
group – Andreas, Christian, Francesc, Frank, Georg, Georgios, Jeremy, Johan, Kimmo, Liney, Magnus, Matti, Peter, Sara, Sven, Yongle – and many
others. I also thank my office mates during the years, for many things other
than science-related. And thanks to everyone at the division, for the nice
lunch hours, friday cookies, exciting cs sessions, and many other things.
I also want to thank a number of persons who have simplified my work:
Lennart for his computer support, and Ewa, Katarina, Sigurd and Yvonne
for helping me with lots of practical things. And also the teams behind
some of the free software packages I have used, in particular the formidable
arpack and the excellent gsl.
And I thank Sofia, for her wonderful love and support, and Alva, for
teaching me new and fascinating things about life.
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