PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

2. Diagonalization of Transition AmplitudesFor specific calculations one can have additional constraints. For example, if oneis interested only in energy eigenvalues, then the optimal parameters are obtainedby minimizing all errors and minimizing the ratio N cut = L/∆, which correspondsto the size of the transition operator matrix S = 2N cut that needs to be numericallydiagonalized. The minimization of N cut is performed in order to minimizecomputation time needed for the diagonalization, which roughly scales as Ncut 3 .On the other hand, if one is interested in details of energy eigenfunctions, then itmight be necessary to have a fixed small value for the discretization step ∆, whichwill allow all features of eigenstates to be visible. This is especially important forstudies of higher energy eigenfunctions which e.g. have many nodes, and in orderto study them it is necessary to have sufficient spatial resolution. In such case, thevalue of ∆ is fixed and other parameters are chosen so as to minimize the errors toa desired value. For example, with the discretization step of the order ∆ = 10 −3 wehave been able to accurately calculate several hundreds energy eigenfunction of thequartic anharmonic oscillator.Table 2.2 gives eigenvalues of the double-well potential, obtained from the quarticanharmonic potential (2.29) by setting the constant k 2 to some negative value. Ascan be seen, numerically obtained energy eigenvalues have the precision similar to theprevious case of the quartic potential without symmetry breaking. The double wellbehavior of the potential does not present any obstacle in its numerical treatmentby this method.Another situation in which one might be interested to keep the ratio N cut = L/∆,i.e. the size of the space-discretized evolution operator matrix as large as possibleis when a large number of energy eigenlevels is needed. The number of energyeigenvalues that can be calculated by the diagonalization is limited by the size ofthe matrix S = 2N cut . Usually the highest energy levels cannot be used due tothe accumulation of numerical errors, and therefore one needs to have a matrix ofsufficient size in order to study energy spectra. In such cases it is necessary to usehighly optimized libraries for numeric diagonalization. We have implemented theeffective actions as a C programming language code [60] and used LAPACK [62]library for numeric diagonalization to calculate large number of energy eigenvaluesand eigenfunctions.Even when one uses such a sophisticated tool, the highest eigenvalues cannotbe used due to accumulation of numerical errors. In order to assess the quality ofthe obtained results for higher energy eigenstates, it is necessary to compare the41
2. Diagonalization of Transition AmplitudesTable 2.2: Low-lying energy levels of the double-well potential, obtained by diagonalizationusing level p = 18 effective action. The parameters used: k 2 = −1,k 4 = 12, L = 16, ∆ = 0.1, t = 0.05. The errors are estimated by comparisonwith the diagonalization results obtained from higher-order effective actions, finerdiscretizations, larger space cutoffs, and lower values of the propagation time t.k E k |∆E k | δE k0 0.328826502590357561530(2) 7 × 10 −22 2 × 10 −211 1.41726810105965210733(23) 5 × 10 −21 4 × 10 −212 3.0819506284815341204(849) 3 × 10 −20 1 × 10 −203 5.019323060355788021(7990) 2 × 10 −19 4 × 10 −204 7.186203252338934478(3958) 5 × 10 −19 8 × 10 −205 9.54285734251209386(72421) 2 × 10 −18 2 × 10 −196 12.06403774639116375(04211) 4 × 10 −18 4 × 10 −197 14.7314279571006902(462590) 1 × 10 −17 7 × 10 −198 17.5310745155383834(413592) 3 × 10 −17 2 × 10 −189 20.4519281359123716(968554) 5 × 10 −17 3 × 10 −18numerical results with some known properties of the physical system. One suchproperty is density of states, defined formally asρ(E) =∞∑δ(E − E k ) , (2.30)k=0assuming that the system has a discrete spectrum. This highly relevant physicalquantity can be directly calculated using the numerically obtained spectra. On theother hand, it can be also analytically calculated using semiclassical approximation.This approximation is valid at least in the high-energy region, and we can use itto assess the quality of our numerical results. In semiclassical approximation, thedensity of states in d spatial dimensions is calculated asρ sc (E) =∫ d⃗xd⃗pδ(E − H(⃗x, ⃗p)) . (2.31)(2π)dreplacing the discrete spectrum with a continuous distribution of energy defined bythe classical Hamilton function H(⃗x, ⃗p). After integration over momenta, we obtain42
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UNIVERSITY OF BELGRADEFACULTY OF PH
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Thesis advisor, Committee member:Dr
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lence for Computer Modeling of Comp
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dobijanje kondenzata odabrani su at
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Uticaj slabih interakcija na fenome
Page 12 and 13: Abstract of the doctoral dissertati
Page 14 and 15: highly accurate information on ener
Page 16 and 17: Keywords: cold quantum gases, Bose-
Page 18 and 19: CONTENTS3.4.2 Time-of-flight graphs
Page 20 and 21: NomenclatureRoman Symbolsagk BLMNn(
Page 22 and 23: Chapter 1Introduction1.1 ForewordTh
Page 24 and 25: ently explored to illustrate the ve
Page 26 and 27: Summations in the last expression c
Page 28 and 29: Figure 1.1: The hallmark of the Bos
Page 30 and 31: we discuss in some detail the exper
Page 32 and 33: In the first papers [3, 4], the TOF
Page 34 and 35: where a BG is the off-resonant scat
Page 36 and 37: system given by( ) ǫ Bog ⃗k =
Page 38 and 39: Having the efficient numerical meth
Page 40 and 41: Chapter 2Properties of quantum syst
Page 42 and 43: 2. Diagonalization of Transition Am
Page 66 and 67: ||2. Diagonalization of Transition
Page 74: 2. Diagonalization of Transition Am
Page 79 and 80: magnetic field ⃗ B = 2M ⃗ Ω.3.
Page 81 and 82: 3. Rotating ideal BECof the rotatio
Page 83 and 84: 3. Rotating ideal BEC3.1 Numerical
Page 85 and 86: 3. Rotating ideal BEC350300SC appro
Page 87 and 88: 3. Rotating ideal BEC3.2 Finite num
Page 89 and 90: 3. Rotating ideal BECN - N 03.0·10
Page 91 and 92: 3. Rotating ideal BEC3.3 Global pro
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Page 105 and 106: Chapter 4Mean-field description of
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4. Mean-field description of an int
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Chapter 5Nonlinear BEC dynamics by
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5. BEC excitation by modulation of
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where n = 1, 2, 3, . . . is an inte
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Chapter 6SummarySince the first exp
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short-range interactions.The excita
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of the ground state. Imaginary-time
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(A.13). With another boundary condi
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Appendix B Time-dependent variation
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List of papers by Ivana VidanovićT
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References[1] S. N. Bose, Plancks g
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REFERENCES[21] W. Ketterle, D. S. D
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REFERENCES[45] A. Bogojević, A. Ba
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REFERENCES[70] M. R. Matthews, B. P
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REFERENCES[94] M.-O. Mewes, M. R. A
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REFERENCES[116] K. Staliunas, S. Lo
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CURRICULUM VITAE - Ivana Vidanović
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