PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

ter 3 we have studied the effects of the shape of trapping potentials to the propertiesof a Bose-Einstein condensate of an ideal gas. A rotation gives rise to the appearanceof a deconfining harmonic potential. Particularly, in the interesting fast-rotatingregime, when the rotation frequency approaches the confining frequency of the harmonictrap, the gas becomes unconfined and the condensate would disperse. Oneway to mitigate the deconfining effect is to use an additional, quartic term in thepotential. As a result, the total external potential in the co-rotating frame acquiresdifferent shapes, depending on the rotation frequency. It is important to understandhow this affects the properties of a BEC for the interpretation of data obtained inexperiments with fast rotating BECs. By employing the method of exact diagonalizationof the time-evolution operator from Chapter 2, we have obtained numericallyexact energy spectra of the harmonic plus quartic trapping potential for differentvalues of the rotation frequency. Using this, we have calculated the condensationtemperature and found that it decreases with an increase of the rotation frequency.We have also presented density profiles of the condensate and thermal cloud at differenttemperatures and have simulated the time-of-flight imaging procedure for thissetup. Interesting expansion dynamics has been found for the external potential inthe shape of a Mexican hat, in the over-critical rotation regime. In the initial stage ofthe expansion the gas expands inwards, into the previously unoccupied inner space,and only after that the common free expansion starts. This leads to an increase inthe typical time scales for the expansion of about one order of magnitude.Chapter 4 is dedicated to the review of the mean-field description of a weakly interactingBEC. We have presented several widely used approximation techniques. Inthe zero temperature limit, we have introduced nonlinear mean-field Gross-Pitaevskiiequation. In order to study BEC at finite temperature and to explore the BEC phasediagram, we have used Hartree-Fock framework in the form in which higher, thermallyexcited states are treated within the semiclassical approximation. Within thismean-field picture, a two-component model of a BEC naturally arises. In an approximativeway, the condensate and thermal component are introduced enabling us tokeep the intuition built on a noninteracting model. Depending on whether the interactionwithin each component is taken into account and how their mutual interactionis considered, several approximation schemes come into play. We have comparedtheir properties, and emphasized the identified drawbacks. The interaction-inducedshift of the condensation temperature has been re-derived. With these results, wehave shown how the noninteracting BEC picture is modified in the presence of weak135
short-range interactions.The excitation of collective oscillation modes is a direct way to probe the propertiesof a system, since frequencies of collective modes can be measured very accurately,with errors of below 1%. In Chapter 5 we have studied properties of collectivemodes subject to the harmonic modulation of the s-wave scattering length, i.e. contactinteraction strength. We have used time-dependent Gross-Pitaevskii equationand the variational approach based on a Gaussian Ansatz to describe condensatedynamics numerically. In the non-resonant regime, when the driving frequency doesnot match any of the eigenfrequencies of the condensate, we have found small amplitudeoscillations that correspond to the quadrupole and breathing mode, as expectedfor an axially-symmetric condensate. As the resonant regime is approached, nonlineardynamical features emerge in the excitation spectra: nonlinear mode coupling,higher-harmonics generation and shifts in the frequencies of excited modes. Wehave developed a perturbative approach in the modulation amplitude, based on thePoincaré-Lindstedt method, and have obtained analytical results for the mentionedeffects.At the end of each chapter, we have indicated a possible future research directionto further extend the study of the topics presented in this thesis.136
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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
Page 105 and 106: Chapter 4Mean-field description of
Page 107 and 108: 4. Mean-field description of an int
Page 125 and 126: Chapter 5Nonlinear BEC dynamics by
Page 127 and 128: 5. BEC excitation by modulation of
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Page 155: Chapter 6SummarySince the first exp
Page 159 and 160: of the ground state. Imaginary-time
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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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PhD thesis in English

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