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$Edge excitations of paired fractional quantum Hall states$

4. Mean-field description of an interacting BEC4 π n(r) r 2 /N [m -1 ]35000300002500020000150001000050000N = 4 × 10 7N = 3 × 10 7N = 2 × 10 7N = 1.46 × 10 7N = 1.09 × 10 70 50 100 150 200r [µm]ω = 125.664 HzT = 200 nKFigure 4.6: Plot of 4πn(r)r 2 /N versus the radial coordinate r within the GPSCmodel for the parameters T = 200 nK, ω = 125.664 Hz.versus the radial coordinate r within the GPSC model is shown in the graph. Fora very large number of atoms we see a single dominant peak corresponding to thecondensate. As the number of atoms decreases, the thermal component becomesvisible, and another broader peak emerges. Finally, below the critical value of thetotal number of atoms, the condensate component disappears completely.High accuracy of the GPSC model is confirmed by direct comparison with thenumerical results obtained from Monte Carlo simulations [76]. However, GPSCmodel fails precisely in describing the phase transition properly. It turns out thatthe system of Eqs. (4.19)-(4.21) does not admit a solution in the vicinity of thephase transition, when it is approached from the condensate side. For all physicalquantities to be well-behaved, it is crucial that the value of the chemical potentialis lower than the value of the classical energy of excited states, µ < H(⃗r, ⃗p). Unfortunately,for small values of n 0 (r), this condition is not satisfied. The inconsistencyarises since we treat the ground state quantum-mechanically and all the other statessemiclassically. For this reason, in the next subsection, we calculate thermodynamicproperties of interacting bosons within the TFSC model.4.2.5 Calculation of the condensation temperatureA representative phase diagram of ultra-cold gas of bosons in the harmonic trapis displayed in Fig. 4.7. Calculations are performed within the TFSC model. We97
4. Mean-field description of an interacting BECN 0 /N10.80.60.4N=10 6 , idealN=10 7 , idealN=10 6 , semi-idealN=10 7 , semi-idealN=10 6 , TFSCN=10 7 , TFSC0.2020 40 60 80 100 120 140 160 180 200T [nK]Figure 4.7: Condensate fraction N 0 /N versus the temperature T for ω = 100 Hz.0.50.4N=10 7 , idealN=10 7 , TFSCN=10 7 , semi-idealN 0 /N0.30.20.10120 125 130 135 140 145 150 155 160T [nK]Figure 4.8: Condensate fraction N 0 /N versus the temperature T for ω = 100 Hz.In the semi-ideal model the condensation temperature is the same as in the idealcase T 0 c = 154.77 nK, while the TFSC approximation yields a lower condensationtemperature T c = 149.5 nK, in agreement with the analytical calculation.have solved Eqs. (4.26) and (4.27) for the fixed number of particles in the giventemperature range and calculated the corresponding condensate fractions N 0 /N.From Fig. 4.8, we clearly see the decrease in the condensation temperature due tointeraction effects.We now focus on the analytical calculation of the shift in the condensation tem-98
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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-
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CONTENTS3.4.2 Time-of-flight graphs
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NomenclatureRoman Symbolsagk BLMNn(
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Chapter 1Introduction1.1 ForewordTh
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ently explored to illustrate the ve
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Summations in the last expression c
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Figure 1.1: The hallmark of the Bos
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we discuss in some detail the exper
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In the first papers [3, 4], the TOF
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where a BG is the off-resonant scat
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system given by( ) ǫ Bog ⃗k =
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Having the efficient numerical meth
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Chapter 2Properties of quantum syst
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2. Diagonalization of Transition Am
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||2. Diagonalization of Transition
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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
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Page 101 and 102: 3. Rotating ideal BECpancy numbers
Page 103 and 104: 3. Rotating ideal BEC6·10 4 0 0.02
Page 105 and 106: Chapter 4Mean-field description of
Page 107 and 108: 4. Mean-field description of an int
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Page 125 and 126: Chapter 5Nonlinear BEC dynamics by
Page 127 and 128: 5. BEC excitation by modulation of
Page 149 and 150: where n = 1, 2, 3, . . . is an inte
Page 155 and 156: Chapter 6SummarySince the first exp
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Page 163 and 164: Appendix B Time-dependent variation
Page 165 and 166: List of papers by Ivana VidanovićT
Page 167 and 168: 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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