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$Edge excitations of paired fractional quantum Hall states$

separate treatment.5. BEC excitation by modulation of scattering lengthIn Ref. [116] it was predicted that a harmonic modulation of the scattering lengthleads to the creation of Faraday patterns in BEC, i.e. density waves. Up to now,Faraday patterns have been experimentally induced by harmonic modulation of thetransverse confinement strength [111], which is studied analytically and numericallyin Ref. [129]. Here we focus only on the nonlinear properties of low-lying collectivemodes and do not consider possible excitations of Faraday patterns.In order to obtain analytical insight into the condensate dynamics induced bythe harmonic modulation of the s-wave scattering length described by Eq. (5.7), weuse the Gaussian variational approximation approximation. We consider an axiallysymmetric BEC, (λ x = λ y = 1, u x = u y ≡ u ρ ), excited by modulation of theinteraction strength, which preserves the axial symmetry of the condensate duringits time evolution. For this reason, we use a simplified axially-symmetric form ofEqs. (5.3):ü ρ (t) + u ρ (t) − 1u ρ (t) 3 −P(t)u ρ (t) 3 u z (t)= 0 , (5.11)ü z (t) + λ 2 zu z (t) − 1u z (t) − P(t)= 0 , (5.12)3 u ρ (t) 2 u z (t) 2which we will refer to as a Gaussian approximation.To estimate the accuracy of the Gaussian approximation for describing the dynamicsinduced by the harmonic modulation of the interaction strength, we compareits solution with an exact numerical solution of the GP equation. In Fig. 5.5, weplot the resulting time-dependent axial and radial condensate widths ρ rms (t) andz rms (t), calculated as root mean square valuesρ rms (t) =z rms (t) =√√2π2π∫ ∞−∞∫ ∞−∞dzdz∫ ∞0∫ ∞0ρ dρ |ψ(ρ, z, t)| 2 ρ 2 , (5.13)ρ dρ |ψ(ρ, z, t)| 2 z 2 , (5.14)of the solution of the GP equation, as well as numerical solutions of Eqs. (5.11)and (5.12). We assume that initially the condensate is in the ground state. In thevariational description, this translates into initial conditions u ρ (0) = u ρ0 , ˙u ρ (0) = 0,u z (0) = u z0 , ˙u z (0) = 0, where u ρ0 and u z0 are the time-independent solutions115
5. BEC excitation by modulation of scattering lengthAxial condensate widthRadial condensate width35302520151.61.51.41.31.21.11variationalGP numerics0 200 400 600 800 1000 1200variationalGP numerics1.21.1t1220 225 230 235 2400 50 100 150 200 250Figure 5.5: Time-dependent axial and radial condensate widths calculated as rootmean square averages. Comparison of the numerical solution of the time-dependentGP equation with a solution obtained by using the Gaussian approximation for theactual experimental parameters in Eq. (5.9) and Ω = 0.05.tof Eqs. (5.11) and (5.12), while in GP simulations we reach the ground state byperforming an imaginary-time propagation until convergence to the ground stateis achieved [87]. For solving the GP equation (4.15), we use the split-step Crank-Nicolson method [87] explained in detail in Appendix A. From the typical resultspresented in Fig. 5.5, which correspond to the actual experimental parameters, itis evident that we have a good qualitative agreement between the two approaches,even for long times of the dynamical evolution.116
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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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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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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
Page 169 and 170: REFERENCES[21] W. Ketterle, D. S. D
Page 171 and 172: REFERENCES[45] A. Bogojević, A. Ba
Page 173 and 174: REFERENCES[70] M. R. Matthews, B. P
Page 175 and 176: REFERENCES[94] M.-O. Mewes, M. R. A
Page 177 and 178: REFERENCES[116] K. Staliunas, S. Lo
Page 179 and 180: CURRICULUM VITAE - Ivana Vidanović
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