PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

4. Mean-field description of an interacting BEC0.450.40.35g = 0g = 125.484g = 627.40.3ψ(r/l)0.250.20.150.10.0500 1 2 3 4 5 6r/lFigure 4.1: Wave functions of the condensate for different interaction strengths. Weassume a spherically symmetric harmonic trap V (⃗r) = Mω 2 r 2 /2. Radial coordinateis expressed in units of l = √ /Mω, while the interaction strength is given in thedimensionless units g ≡ 4πaN/l.that the repulsive interaction leads to the broadening of the density profile comparedto the noninteracting Gaussian. This is illustrated in Fig. 4.1, where we show wavefunctions of the condensate obtained by numerically solving the GP equation fordifferent interaction strengths. To find the condensate ground state, we perform theimaginary-time propagation of the GP equation [87]. More details on the numericalalgorithms are given in Appendix A. Due to its simplicity, TF approximation iswidely used for the interpretation of experimental data obtained for systems with alarge number of atoms.In a similar manner, starting from the real-time formalism and neglecting thecondensate depletion, in the mean-field framework we obtain the time-dependentGP equation:]∂ψ(⃗r, t)i =[− 2∂t 2M ∆ + V (⃗r) + g|ψ(⃗r, t))|2 ψ(⃗r, t), (4.15)which describes the condensate dynamics at T = 0. This equation also allows thestudy of the condensate excitation spectra, which is essential information for probingsystem’s properties.The time-dependent GP equation can be also derived variationally, from the89
4. Mean-field description of an interacting BECaction principle [19] based on the Lagrangian∫L GP = d⃗r L GP (⃗r) , (4.16)where the Lagrangian density is given byL GP (⃗r) = i 2) (ψ ∂ψ∗∂t − ψ∗∂ψ + 2∂t 2M |∇ψ|2 + V (⃗r)|ψ| 2 + g 2 |ψ|4 . (4.17)By performing the extremization of the action with respect to ψ(⃗r, t),δ ∫ t 2t 1dtL GPδψ ∗ (⃗r, t)= 0, (4.18)we again arrive at the GP Eq. (4.15). This is an important observation, since itrepresents the basis of the time-dependent variational analysis. We exploit thisapproach later in Chapter 5.4.2 Finite-temperature properties of a BECAfter briefly exploring the properties of a weakly interacting BEC in the zerotemperaturelimit, we now continue the study of its finite-temperature propertiesin the HF approximation, Eqs. (4.9)-(4.11). To facilitate an almost analytical approach,we abolish Eq. (4.11) and treat excited states within a semiclassical approximation,while keeping the generalized GP equation (4.9) for the condensate phase.To apply the semiclassical approximation, similarly to derivation of Eq. (1.13), weuse the classical Hamiltonian H(⃗r, ⃗p) = ⃗p 2 /2M + 2g(n 0 (⃗r) + n th (⃗r)), in accordancewith Eq. (4.11), and replace a summation over the discrete eigenstates in Eq. (4.10)by an integration over momentum ⃗p. We assume a spherically symmetric trap,i.e. V (⃗r) = V (r), and search for the equilibrium configuration (∂ψ(⃗r, τ)/∂τ = 0),when ψ(⃗r, τ) ≡ ψ(r). Eqs. (4.10) and (4.11) are now reduced to a single equation,and, together with the equation for a total number of particles, we have the following90
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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 105 and 106: Chapter 4Mean-field description of
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Page 125 and 126: Chapter 5Nonlinear BEC dynamics by
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Page 155 and 156: Chapter 6SummarySince the first exp
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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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