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$Edge excitations of paired fractional quantum Hall states$

5. BEC excitation by modulation of scattering length1000101001Amplitude1010.10.11.9622.042.080.01Q = 0.3Q = 0.050.0011.5 2 2.5 3 3.5 4 4.5 5Figure 5.7: Oscillation amplitude (u max − u min )/2 versus driving frequency Ω forP = 0.4. In the inset, we zoom to the first peak to emphasize that both the shapeand the value of a resonance depend on the modulation amplitude Q and thatresonances occur at a driving frequency Ω, which differs from ω 0 . The solid verticallines correspond to ω 0 and 2ω 0 .Ωu(t)2.521.51Ω = 1.032u(t)1.81.61.41.21Ω = 1.3750.50.80 100 200 300 400 5000 1000 2000 3000ttFigure 5.8: Exact numerical solution of Eq. (5.15) for the condensate width u(t)versus t for P = 0.4, Q = 0.3, corresponding to ω 0 = 2.06638. We observe largeamplitude oscillations for Ω ≈ ω 0 /2 in the left panel, while in the right panel the“resonant” behavior is present for Ω ≈ 2ω 0 /3.numerically, since it is necessary to perform a fine tuning of the external frequency.However, they clearly demonstrate nonlinear BEC properties and an experimentalobservation of these phenomena is certainly of high interest. We note that theobserved resonance pattern of the form Ω ≈ 2ω 0 /n (where n is an integer) arisesalso in the case of a parametrically driven system described by the Mathieu equation,119
5. BEC excitation by modulation of scattering length10 3-40 -30 -20 -10 0 10 20 30 4010 210 110 010 -110 -210 -310 -410 -510 -6Ω = 2Frequency10 210 110 0ω - Ω10 -110 -20.05 0.06 0.07 0.08Frequency10 2 Ω + ω10 0 2Ω 2ω10 -210 -44 4.05 4.1 4.15 4.2Frequency10 210 1 Ω ω10 010 -110 -21.9 2 2.1 2.2Frequency10 110 0 2Ω + ω Ω + 2ω10 -110 -2 3Ω3ω10 -36 6.05 6.1 6.15 6.2FrequencyFigure 5.9: Fourier transform of u(t) for P = 0.4, Q = 0.1, and Ω = 2. First plotpresents the complete spectrum on a semi-log scale, while the subsequent plots focuson regions of interest in the spectrum.for instance, in the context of the Paul trap [130].To examine such excited modes directly, we look at the Fourier transform of thecondensate width u(t). To this end, we numerically solve Eq. (5.15) and find theFourier transform of its solution using the MATHEMATICA software package [54].An example of such an excitation spectrum for P = 0.4, Q = 0.1, and Ω = 2 isgiven in Fig. 5.9. The spectrum contains two prominent modes: a breathing modeof frequency ω (close, but not equal to ω 0 ), and a mode that corresponds to thedriving frequency Ω, along with many higher-order harmonics of the general formmΩ + nω, where m and n are integers.In Fig. 5.10 we juxtapose two zoomed-in Fourier spectra for two different drivingfrequencies for P = 0.4 and Q = 0.2. On the left plot, we show a spectrum forΩ = 1. The vertical solid line corresponds to ω 0 and we find the peak in the120
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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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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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