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$Edge excitations of paired fractional quantum Hall states$

3. Rotating ideal BECtrapping potential is switched off compared to the undercritical case. Further studyshould incorporate the effects of weak interactions on the Bose-Einstein condensationin the external potential (3.4), either in the mean-field framework or by the fullnumerical treatment. Finally, we note that approach presented here can also beused for numerical studies of properties of rotating ultra-cold Fermi gases [75].83
Chapter 4Mean-field description of an interacting BECIn the previous Chapter we have considered BEC of an ideal Bose gas, and,despite neglecting the interactions, we had to apply sophisticated numerical methodsfor characterization of the BEC phase transition and properties of the condensate.Now we extend the description of a Bose gas to include interactions. For an exactmicroscopic description of an interacting Bose gas, we need to consider a full manybodyHamiltonian. For systems with a dominant two-body contact interaction, theHamiltonian has the form (1.24):∫Ĥ =d⃗r(− ˆψ † (⃗r) 22M ∇2 ˆψ(⃗r) + V (⃗r) ˆψ† (⃗r) ˆψ(⃗r) + g )2 ˆψ † (⃗r) ˆψ † (⃗r) ˆψ(⃗r) ˆψ(⃗r) . (4.1)In the case of ultra-cold atomic gases, the bosons are alkali atoms, which are notelementary particles. However, due to very low temperatures, the atoms can beconsidered to be always in their ground states and their internal structure can beneglected. The exact treatment of the system described by the Hamiltonian (4.1)is possible only by numerical simulations based on Monte Carlo approach [76, 77].Such simulations are ubiquitously time-consuming and significantly limit the scopeof problems that can be addressed. For this reason, a number of approximativemethods have been developed. Simplified approximative approaches are much easierfor the numerical implementation and sometimes even provide analytical insightsinto the studied problem. Furthermore, very often, information on the parametersof the system, such as the temperature or condensate fraction, are extracted fromexperimental data indirectly, by fitting to expressions derived within one of simplifiedschemes.In this Chapter we review the mean-field Hartree-Fock (HF) approximation fora many-body Hamiltonian (4.1). The HF approximation is the most-widely usedapproach to study finite-temperature properties of weakly interacting BECs [78, 5].We will show within this mean-field scheme how the presence of the interactioninfluences properties of the condensate, both in the zero-temperature limit, and in84
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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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Page 54 and 55: 2. Diagonalization of Transition Am
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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
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Page 101 and 102: 3. Rotating ideal BECpancy numbers
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Page 107 and 108: 4. Mean-field description of an int
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
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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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