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$Edge excitations of paired fractional quantum Hall states$

5. BEC excitation by modulation of scattering lengthis given in Fig. 5.16. Our analytical perturbative result for the shifted quadrupolemode frequency contains poles at ω Q0 , 2ω Q0 , ω B0 −ω Q0 , ω Q0 +ω B0 and ω B0 . Similarly,for the shifted frequency of the breathing mode poles in the perturbative solutionare found at ω Q0 , ω B0 , 2ω B0 , ω B0 − ω Q0 and ω Q0 + ω B0 . In both figures we seeexcellent agreement of the perturbatively obtained results with the exact numerics.In the experiment from Ref. [15], excitations of a highly elongated and stronglyrepulsive BEC were considered with the system parameters given in Eq. (5.9). Forthat case, according to Eq. (5.10) we get ω Q0 ≪ ω B0 , and the driving frequencywas chosen in the range (0, 3ω Q0 ). Good agreement of real-time dynamics obtainedfrom the variational approximation with the exact solution of the time-dependentGP simulation occurs even for long propagation times, as can be seen in Fig. 5.5,which implies a good accuracy of the Gaussian approximation for calculating thefrequencies of the excited modes. From the real-time dynamics shown in Fig. 5.5, weobserve the excitation of the slow quadrupole mode as an out-of phase oscillation inthe axial and in the radial direction. In addition, in the radial direction we observefast breathing mode oscillations. This is typical for highly elongated condensates[133] and our analysis for the experimental parameters shows a strong excitationof the quadrupole mode, but also a significant excitation of the breathing modein the radial direction. Due to the large modulation amplitude Q, many higherorder harmonics are excited, and, most importantly, we find frequency shifts of thequadrupole mode of about 10% in Fig 5.17. From the same figure we notice that,due to the chosen frequency range for Ω, only resonances located at ω Q and 2ω Qare observed. The presence of nonlinear effects is already mentioned in Ref. [15].However, we conclude that frequency shifts calculated here have to be taken intoaccount for extracting the resonance curves from the underlying experimental data.To achieve more clear-cut experimental observation of the nonlinearity-inducedfrequency shifts calculated in this paper, we suggest a different trap geometry fromthe one used in Ref. [15]. Measurements of stable BEC modes can be performedfor about 1 s, and in order to extract precise values of the excited frequencies inthe Fourier analysis, several oscillation periods should be captured within this timeinterval. A higher frequency of the quadrupole mode, that can be realized by usinga larger trap aspect ratio λ z , in combination with a higher modulation frequencywould fulfill this condition. According to the results presented in Ref. [15], resonantdriving may lead to condensate fragmentation. However, our numerical resultsindicate frequency shifts of 10 % even outside the resonant regions according to131
5. BEC excitation by modulation of scattering length0.040.038Frequency ω Q0.0360.0340.0320.03ω Q0ω Q, (a)ω Q, (n)0 0.02 0.04 0.06 0.08 0.1 0.12ΩFigure 5.17: Frequency of the quadrupole mode ω Q versus driving frequency Ω forthe experimental parameters from Eq. (5.9). We plot linear response result ω Q0 ,second-order analytical result ω Q,(a) , and numerical values ω Q,(n) .Figs. 5.11 and 5.17, and this is where experimental measurements should be performed.Although an increase in λ z leads to a more pronounced nonlinear mixing ofquadrupole and breathing mode and may complicate condensate dynamics further,it may be possible to perform a Fourier analysis of experimental data, analogous toRef. [101], and to compare it with the excitation spectra presented here. To achievea complete matching of experimental data and our calculations, it may turn out thathigher-order corrections to Eq. (5.7), which arise due to the nonlinear dependenceof scattering length on the external magnetic field, have to be taken into account.5.6 Conclusions and outlookMotivated by recent experimental results, in this Chapter, we have studied nonlinearBEC dynamics induced by a harmonically modulated interaction at zero temperature.We have used a combination of an analytic perturbative approach, numericalanalysis based on Gaussian approximation, and numerical simulations of a full timedependentGross-Pitaevskii equation. We have presented numerically calculated relevantexcitation spectra and found prominent nonlinear features: mode coupling,higher harmonics generation, and significant shifts in the frequencies of collectivemodes. In addition, we have provided an analytical perturbative framework that132
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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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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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