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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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 26 27 28 29 30 30 33 . . . . 33 . . . . 34 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 39 . . . . . . 39 40 40 41 41 42 45 example 47 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 71 Paper I 71 Paper II 71 Paper III 71 Paper IV 71 Paper V 71 Paper VI 71 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 5 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 11 12 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 18 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. 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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. 71