PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

Euler-Lagrange equations of motion for variational parameters q ∈ {ϕ σ , σ 0 , φ σ , u σ }have the formand for the above Lagrangian read:d ∂Ldt ∂ ˙q − ∂L∂q = 0 , ˙φ σ u σ − 22Mϕ σ − M ˙σ 0 + 2φ σ σ 0 = 0, (B.1) ˙ϕ σ + 2σ 0 ˙φσ + 22M ϕ σφ σ + 42M φ2 σσ 0 + Mω 2 σσ 0 = 0 (B.2)1u 3 σφ σ − M 2˙u σu σ= 0, (B.3)+ 22M φ2 σu σ + Mω2 σ2 u σ − g(t)2(2π) 3/2 1u x u y u z u σ= 0. (B.4)Eqs. (B.1) and (B.3) give the explicit relation between ϕ σ and σ 0 , and between φ σand u σ . By inserting Eqs. (B.1) and (B.3) into Eqs. (B.2) and (B.4), we obtainvariational equations which we will refer to as a Gaussian approximation:¨σ 0 (t) + λ 2 σ σ 0 = 0 , (B.5)ü σ (t) + λ 2 σ u σ(t) − 1u σ (t) − P(t)3 u σ (t)u x (t)u y (t)u z (t)= 0 . (B.6)We also note that solutions of Eqs. (B.5) and (B.6) for σ 0 (t) and u σ (t) can be afterwardsinserted into Eqs. (B.1) and (B.3) to obtain time-evolution of the phaseparameters ϕ σ and φ σ . The last two equations are given in the dimensionless form:we choose a convenient frequency scale ω (for example, the external trap frequencyin one of the spatial directions) and express all lengths in the units of the characteristicharmonic oscillator length l = √ /Mω, time in units of ω −1 , and externalfrequencies in units of ω: λ σ = ω σ /ω, σ ∈ {x, y, z}. The dimensionless interactionparameter P(t) is given byP(t) =√g(t) 2(2π) 3/2 ωl = 3 π N a(t) .l143
List of papers by Ivana VidanovićThis thesis is based on the following publications:1. I. Vidanović, A. Balaž, H. Al-Jibbouri, and A. Pelster, Nonlinear Bose-EinsteincondensateDynamics Induced by a Harmonic Modulation of the s-wave ScatteringLength, Phys. Rev. A 84, 013618 (2011), [Chapter 5]2. A. Balaž, I. Vidanović, A. Bogojević, and A. Pelster, Ultra-fast convergingpath-integral approach for rotating ideal Bose-Einstein condensates, Phys.Lett. A 374, 1539 (2010), [Chapter 3]3. I. Vidanović, A. Bogojević, A. Balaž, and A. Belić, Properties of QuantumSystems Via Diagonalization of Transition Amplitudes. II. Systematic Improvementsof Short-time Propagation, Phys. Rev. E 80, 066706 (2009),[Chapter 2]4. I. Vidanović, A. Bogojević, and A. Belić, Properties of Quantum SystemsVia Diagonalization of Transition Amplitudes. I. Discretization Effects, Phys.Rev. E 80, 066705 (2009), [Chapter 2]Publications not included in the thesis:1. A. Balaž, I. Vidanović, D. Stojiljković, D. Vudragović, A. Belić, and A. Bogojević,SPEEDUP Code for Calculation of Transition Amplitudes Via theEffective Action Approach, Commun. Comput. Phys. 11, 739 (2012).2. I. Vidanović, S. Arsenijević, and S. Elezović-Hadzić, Force-induced Desorptionof Self-avoiding Walks on Sierpinski Gasket Fractals, Eur. Phys. J. B. 81,291 (2011).3. A. Balaž, I. Vidanović, A. Bogojević, A. Belić, and A. Pelster, Fast ConvergingPath Integrals for Time-Dependent Potentials: II. Generalization to Many-Body Systems and Real-Time Formalism, J. Stat. Mech. P03005 (2011).144
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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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magnetic field ⃗ B = 2M ⃗ Ω.3.
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3. Rotating ideal BECof the rotatio
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3. Rotating ideal BEC3.1 Numerical
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3. Rotating ideal BEC350300SC appro
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3. Rotating ideal BEC3.2 Finite num
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3. Rotating ideal BECN - N 03.0·10
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3. Rotating ideal BEC3.3 Global pro
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3. Rotating ideal BECever, it does
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3. Rotating ideal BECT c [nK]120110
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3. Rotating ideal BEC3.533210 3 κ
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3. Rotating ideal BECn(x, y)10·10
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3. Rotating ideal BECpancy numbers
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3. Rotating ideal BEC6·10 4 0 0.02
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Chapter 4Mean-field description of
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4. Mean-field description of an int
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Page 155 and 156: Chapter 6SummarySince the first exp
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Page 167 and 168: References[1] S. N. Bose, Plancks g
Page 169 and 170: REFERENCES[21] W. Ketterle, D. S. D
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Page 173 and 174: REFERENCES[70] M. R. Matthews, B. P
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Page 177 and 178: REFERENCES[116] K. Staliunas, S. Lo
Page 179 and 180: CURRICULUM VITAE - Ivana Vidanović
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