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$Edge excitations of paired fractional quantum Hall states$

highly accurate information on energy levels of quantum systems. The precise informationon energy spectra is necessary for the characterization of the phase diagramof a Bose-Einstein condensate. Method that we have used is based on the exactdiagonalization of the time-evolution operator [9]. In order to optimally apply it,we have carefully analyzed numerical errors which arise for two reasons: numericalerrors which stem from the spatial discretization, as well as the errors due tothe approximative calculation of matrix elements. One of our main results is highlysuperior behavior of the dicretization error of the discretized evolution operator comparedto the common discretization error of the discretized Hamiltonian. Based onthe analytical and numerical considerations, we have shown that the diagonalizationof a time-evolution operator exhibits a non-perturbatively small discretization error,which vanishes exponentially with 1/∆ 2 , where ∆ is the discretization spacing,while standard discretization introduces errors polynomial in ∆. This is the mainreason that makes the diagonalization of the time-evolution operator the preferredmethod. The main difficulty in the application of the method - precise calculationof the matrix elements of time-evolution operator (transition amplitudes) can be directlyresolved using previously developed effective action approach [10, 11], whichyields transition amplitudes as high-order expansions in the short time of propagation.The efficiency of this method has been demonstrated on several one- andtwo-dimensional models.In Chapter 3 we have used the described numerical method to explore the phasediagram of rotating bosons in a non-standard external potential. Widely used confinementsare harmonic traps and many details of Bose-Einstein condensation insuch traps are already well understood. Rotation of a quantum gas is one way toreach strongly correlated phases [7] and is therefore highly relevant. One of the consequencesof rotation is the appearance of the deconfining centrifugal component inthe potential, which in the case of a very fast rotation frequency (exceeding the trappingfrequency) leads to the deconfinement. In order to avoid this, recent experiment[12] introduced an additional quartic potential to enhance trapping. Depending onthe value of the rotation frequency, the potential changes its shape from a simpleconvex one to the Mexican-hat-shaped potential. Using exact diagonalization of thetime-evolution operator, we have studied how the modification of the external trapinfluences properties of a Bose-Einstein condensate, such as condensation temperature,equilibrium density distribution of atoms and the expansion time of the cloudafter it is released from the trap.
The effects of weak interactions on the phenomenon of Bose-Einstein condensationare subject of Chapter 4. This chapter reviews the Hartree-Fock descriptionof a bosonic system. Hartree-Fock is one of the mean-field approximations andwithin this framework we have described ultracold bosons at zero temperature, aswell as in the vicinity of the Bose-Einstein phase transition. At zero temperaturewe have rederived famous Gross-Pitaevskii equation [13, 14] - nonlinear partial differentialequation, which governs the dynamics of the macroscopic wave functionof the condensate. For finite temperatures, we have scrutinized several widely usedimplementations of the mean-field approximation, with the emphasis on their benefitsand their physical drawbacks. Approximations of this type are very often usedin the interpretation of the experimental data, which makes them highly relevant.A very recent experimental progress in the observation of beyond-mean-field effectsdemands further improvements of these standard tools.One of the basic methods to characterize phases of matter, both experimentallyand theoretically, are properties of their excitation spectra, with collective modesbeing of special interest. In ultracold gases, collective modes are usually excited usingmodulation of the parameters of the external trap. In the recent experiment [15],however collective modes were excited using an alternative method - harmonic modulationof interaction. In its essence the experimental method relies on a Feshbachresonancetechnique, which enables modulation of interaction via a modulation ofthe external magnetic field. As an outcome of this dynamical protocol, oscillationsof the condensate size were induced and measured. Depending on the value of theexternal modulation frequency, either a linear response or resonant large-amplitudeoscillations were obtained. Since the main underlying equation is nonlinear, in thecase of large-amplitude oscillations strong nonlinear effects are expected. In Chapter5 we have performed numerical simulations of the system dynamics and identifiedmain nonlinear features of the obtained excitation spectra: beside the basic modes,higher harmonics appear together with their linear combinations. Most prominentnonlinear effects are nonlinearity-induced shifts in the frequencies of excited modes.In order to describe these results in an analytic way, we have developed perturbativeapproach where the small parameter is given by the modulation amplitude.The developed perturbative approach is based on the Poincaré-Lindstedt method,and gives analytical estimates for the shifts of the eigenfrequencies in the vicinity ofresonances.
Page 1 and 2: UNIVERSITY OF BELGRADEFACULTY OF PH
Page 3 and 4: Thesis advisor, Committee member:Dr
Page 6 and 7: lence for Computer Modeling of Comp
Page 8 and 9: dobijanje kondenzata odabrani su at
Page 10 and 11: Uticaj slabih interakcija na fenome
Page 12 and 13: Abstract of the doctoral dissertati
Page 16 and 17: Keywords: cold quantum gases, Bose-
Page 18 and 19: CONTENTS3.4.2 Time-of-flight graphs
Page 20 and 21: NomenclatureRoman Symbolsagk BLMNn(
Page 22 and 23: Chapter 1Introduction1.1 ForewordTh
Page 24 and 25: ently explored to illustrate the ve
Page 26 and 27: Summations in the last expression c
Page 28 and 29: Figure 1.1: The hallmark of the Bos
Page 30 and 31: we discuss in some detail the exper
Page 32 and 33: In the first papers [3, 4], the TOF
Page 34 and 35: where a BG is the off-resonant scat
Page 36 and 37: system given by( ) ǫ Bog ⃗k =
Page 38 and 39: Having the efficient numerical meth
Page 40 and 41: Chapter 2Properties of quantum syst
Page 42 and 43: 2. Diagonalization of Transition Am
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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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Chapter 5Nonlinear BEC dynamics by
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5. BEC excitation by modulation of
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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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PhD thesis in English

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