PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

2. Diagonalization of Transition AmplitudesTable 2.3: Top: Low-lying energy levels of the modified Pöschl-Teller potential,obtained by diagonalization using level p = 21 effective action. The parametersused: χ = 0.5, λ = 15.5, L = 5, ∆ = 0.02, t = 0.1. Bottom: Symmetric tableof scalar products 〈ψ k |ψlexact 〉 of numerically calculated and analytic eigenstates fork, l = 0, 1, 2, 3, 4.k E k E exactk|E k − E exactk | δE k0 −26.28125000000000000000000(174) −26.28125 2 × 10 −24 7 × 10 −261 −22.781250000000000000000(28812) −22.78125 3 × 10 −22 2 × 10 −232 −19.53124999999999999999(736443) −19.53125 3 × 10 −21 2 × 10 −223 −16.5312499999999999999(6571136) −16.53125 4 × 10 −20 2 × 10 −214 −13.7812499999999999(8195897101) −13.78125 2 × 10 −17 2 × 10 −185 −11.28124999999999(398393103608) −11.28125 6 × 10 −15 6 × 10 −166 −9.03124999999(8602255352218206) −9.03125 2 × 10 −12 2 × 10 −137 −7.031249999(773547728177905754) −7.03125 3 × 10 −10 4 × 10 −118 −5.2812499(74811672590174261082) −5.28125 3 × 10 −8 5 × 10 −90 1 2 3 40 1 − 4 · 10 −12 1.2 · 10 −13 2.8 · 10 −6 7.4 · 10 −14 3.0 · 10 −71 1.2 · 10 −13 1 − 1 · 10 −11 2.1 · 10 −13 4.5 · 10 −6 4.6 · 10 −142 2.8 · 10 −6 2.1 · 10 −13 1 − 2 · 10 −11 1.3 · 10 −13 5.9 · 10 −63 7.4 · 10 −14 4.5 · 10 −6 1.3 · 10 −13 1 − 4 · 10 −11 3.1 · 10 −134 3.0 · 10 −7 4.6 · 10 −14 5.9 · 10 −6 3.1 · 10 −13 1 − 5 · 10 −112.5 Numerical results for two-dimensional modelsIn this section we illustrate the application of the numerical method based on thediagonalization of transition amplitudes on two models in d = 2 spatial dimensions.The first model is the anharmonic oscillatorV (x, y) = k 22 (x2 + y 2 ) + k 424 (x2 + y 2 ) 2 , (2.36)which is used for a description of the trapping potential used in a recent experimentwith fast-rotating Bose-Einstein condensate of 87 Rb atoms [12, 64, 65].The graphs in Fig. 2.13 are calculated for k 2 = 0, when the potential is reducedto a pure quartic interaction. The analysis of errors is very similar as in the onedimensionalcases we studied in the previous section. The dependence of groundenergy errors on the space cutoff L is shown in Fig. 2.13(top), and we see the47
2. Diagonalization of Transition Amplitudes10 2 0 0.5 1 1.5 2 2.5 3|| E 0(p) (∆, L, t) - E 0exact10 -2 110 -410 -610 -810 -1010 -12p = 1p = 2p = 4p = 6p = 8p = 10p = 12p = 1410 -2 1L10 -4| E 0(p) (∆, L, t) - E 0exact|10 -610 -810 -1810 -1210 -1410 -1610 -180.020.04t0.060.08p = 1p = 2p = 4p = 6p = 8p = 100.10.12Figure 2.13: Deviations from the ground energy |E (p)0 (∆, L, t) − E 0 | as a function ofthe space cutoff L (top) and as a function of the time t (bottom) for the potential(2.36) for k 2 = 0 and k 4 = 24. The discretization parameters are ∆ = 0.2, t = 0.1on the top graph, and L = 3 on the bottom graph. Deviations are calculated usingthe ground energy E 0 = 1.47714975357799(4) obtained with p = 21 effective action.The dashed line in the bottom graph corresponds to the discretization error (2.19).usual saturation of errors for sufficiently large values of L. The saturated valuerapidly decreases (by several orders of magnitude) as we increase the level p of theeffective action used to calculate space-discretized matrix of the evolution operator.Fig. 2.13(bottom) shows the time dependence of ground energy errors, which are48
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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
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Abstract of the doctoral dissertati
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highly accurate information on ener
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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
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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
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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
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Page 79 and 80: magnetic field ⃗ B = 2M ⃗ Ω.3.
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Page 85 and 86: 3. Rotating ideal BEC350300SC appro
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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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