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$Edge excitations of paired fractional quantum Hall states$

5. BEC excitation by modulation of scattering lengththis is only an ad-hoc approximation. We stress that this issue concerning the trueresonant behavior can not be settled either by relying on a numerical calculationdue to inherent numerical artifacts related to finite numerical precision and finitecomputational time. To resolve it, one should rely on an analytical considerationalong the lines of Ref. [118] or use some analytical tool applicable at resonances,such as the resonant Bogoliubov-Mitropolsky method [124].10 3 1.94 1.96 1.98 2 2.02 2.04 2.06 2.0810 210 110 010 -110 -210 -3GP numerics 1GP numerics 2GP numerics 3Gaussian app.ΩFrequencyωFigure 5.12: Part of the Fourier spectrum of the time-dependent condensate widthfor P = 0.4, Q = 0.2, Ω = 2. For numerical solution of GP equation we use severaldiscretization schemes: GP numerics 1 (time step ε = 10 −3 , spacing h = 4 × 10 −2 ),GP numerics 2 (ε = 5 × 10 −4 , h = 2 × 10 −2 ), GP numerics 3 (ε = 5 × 10 −5 ,h = 5 × 10 −3 ). Details of the used numerical algorithms are given in AppendixA. For comparison we also show the corresponding spectrum obtained from theGaussian approximation (dotted-dashed line) and analytical result (5.21) for theposition of breathing mode (solid vertical line).In addition to comparison of our analytical results with numerical solutions basedon the Gaussian variational approximation, we present a comparison with the fullnumerical solution of the GP equation. In order to be able to perform Fourieranalysis with sufficient resolution, it is necessary to obtain an accurate solutionfor long evolution times. We do this by using the split-step method in combinationwith the semi-implicit Crank-Nicolson method [87]. As we refine the GP numerics byusing finer space and time discretization parameters, our numerical results becomestable as shown in Fig. 5.12. From the same figure, we observe quantitatively goodagreement between GP numerics and Gaussian approximation, reflected in close125
5. BEC excitation by modulation of scattering lengthvalues obtained for the breathing mode frequency. In addition, numerical values forthe breathing mode approach closely the analytical result of Eq. (5.21), shown by asolid vertical line in Fig. 5.12.It is well known that for a corresponding two-dimensional axially-symmetricsystem with a constant interaction and trapping frequency, the breathing modeoscillations can be described by an exact linear equation [131, 132]. However, inthe case of a time-dependent trapping frequency, the exact equation of motion isnonlinear [118]. To the best of our knowledge, for a time-dependent interactionstrength the corresponding exact equation does not exist in the literature, but onecan reasonably expect that nonlinear effects will remain in such systems, due to theinherent time dependence of the interaction.5.5 Axially-symmetric BECTo obtain results relevant for a comparison with the experiment reported in Ref. [15],we now study an axially-symmetric BEC. An illustration of the condensate dynamicsis shown in Fig. 5.13 for P = 1, Q = 0.2, λ z = 0.3. We plot numerical solutions ofEqs. (5.11) and (5.12) obtained for the equilibrium initial conditions u ρ (0) = u ρ0 ,˙u ρ (0) = 0, u z (0) = u z0 , and ˙u z (0) = 0. For the specified parameters, the equilibriumwidths are found to be u ρ0 = 1.09073, u z0 = 2.40754, and from the linear stabilityanalysis we find both the quadrupole mode frequency ω Q0 = 0.538735 and thebreathing mode frequency ω B0 = 2.00238. For a driving frequency Ω close to ω Q0 , weobserve large amplitude oscillations in the axial direction. An example of excitationspectra is shown in Fig. 5.14. Here, we have the three basic modes ω Q , ω B , Ω, andmany higher-order harmonics.5.5.1 Poincaré-Lindstedt methodIn order to extract information on the frequencies of the collective modes beyondthe linear stability analysis, we apply the perturbative expansion in the modulationamplitude Q:u ρ (t) = u ρ0 + Q u ρ1 (t) + Q 2 u ρ2 (t) + Q 3 u ρ3 (t) + . . ., (5.23)u z (t) = u z0 + Q u z1 (t) + Q 2 u z2 (t) + Q 3 u z3 (t) + . . .. (5.24)126
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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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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 145: 5. BEC excitation by modulation of
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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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