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$Edge excitations of paired fractional quantum Hall states$

2. Diagonalization of Transition Amplitudessame order if one uses sufficiently fine discretization (∆ = 0.05). Therefore, thesaturation of errors on the left-hand graph are caused by the errors due to the timeof propagation. However, if one uses discretization which is not sufficiently fine, thesaturation of errors can be also caused by the discretization effects. Such effects can|| E 0(p) (∆, L, t) - E 0exact10 -2 110 -410 -610 -810 -1010 -1210 -1410 -1610 -1810 -20p = 1p = 3p = 5p = 7p = 9p = 11p = 131 1.5 2 2.5 3 3.5L10 5 0.001 0.01 0.1|| E 0(p) (∆, L, t) - E 0exact10 -5 110 -1010 -1510 -20∆ = 0.05∆ = 0.10∆ = 0.2010 -25Figure 2.9: Deviations from the ground energy |E (p)0 (∆, L, t) − Eexact 0 | as a functionof the space cutoff L (top) and as a function of the time t (bottom). The groundenergy is obtained using different levels p = 1, 3, 5, 7, 9, 11, 13 (top to bottom) ofthe effective action for the quartic anharmonic potential, with parameters k 2 = 1,k 4 = 48, ∆ = 0.05, t = 0.02 on top graph, and L = 4 on bottom graph. Theexact ground energy E0 exact = 0.95156847272950001114693 . .. is taken from Ref. [61].Dashed lines on the graph (b) correspond to the discretization error (2.19).t39
2. Diagonalization of Transition AmplitudesTable 2.1: Low-lying energy levels of the anharmonic quartic potential, obtainedby diagonalization using level p = 13 effective action. The parameters are k 2 = 1,k 4 = 48, L = 5, ∆ = 0.05, t = 0.01. For higher energy eigenvalues, errors areestimated by comparison with the diagonalization results obtained from higherordereffective actions, finer discretizations, larger space cutoffs, and lower values ofthe propagation time t.k E k |∆E k | δE k0 0.9515684727295000111468(8) 5 × 10 −23 6 × 10 −231 3.292867821434465922691(67) 4 × 10 −22 2 × 10 −222 6.30388056744652609989(522) 2 × 10 −21 4 × 10 −223 9.72732317270370501553(448) 5 × 10 −21 5 × 10 −224 13.4812758360385893838(1489) 2 × 10 −20 2 × 10 −215 17.5141323992530709259(6206) 3 × 10 −20 2 × 10 −216 21.7909563917965158973(8744) 6 × 10 −20 3 × 10 −217 26.286125156056810490(92289) 2 × 10 −19 7 × 10 −218 30.979882837938369575(08213) 2 × 10 −19 8 × 10 −219 35.856438766665971146(24181) 3 × 10 −19 9 × 10 −21be also analytically estimated to be proportional to −2 exp(−2π 2 t/∆ 2 ) cosh(π 2 (k +1)/L∆)/t, as we have shown in the previous section.Table 2.1 gives low-lying energy eigenvalues of the anharmonic oscillator for aparticular choice of the parameters of the potential and discretization parameters.In principle, one can achieve arbitrary precision by the use of appropriately chosendiscretization parameters. Of course, for arbitrary precision calculations one has touse one of the software packages able to support such calculations. For example,we have used MATHEMATICA [54] in order to be able to achieve high-precisionresults presented on the above graphs. The important conclusion is that even for verymoderate values of discretization parameters, the use of higher-order effective actionsleads to very small errors, which may be practically implemented with minimalcomputing resources.The analysis of errors such as the one presented in Fig. 2.9 is sufficient to estimateoptimal values of discretization parameters. In general, for a desired numericalprecision of energy eigenvalues, the optimal values of parameters are chosen so thatall types of errors are approximately the same. The overall error is always dominatedby the largest of all errors, and therefore it is optimal to have all errors of the sameorder of magnitude.40
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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
Page 10 and 11: 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 95 and 96: 3. Rotating ideal BECT c [nK]120110
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Page 99 and 100: 3. Rotating ideal BECn(x, y)10·10
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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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