PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

5. BEC excitation by modulation of scattering lengththe first-order correction ω 1 vanishes, leading to a frequency shift which is quadraticin Q:ω = ω 0 +Q212u 200 ω3 0where the polynomial P(Ω) is given byP(Ω)(Ω 2 − ω 2 0 )2 (Ω 2 − 4ω 2 0 ) + . . . , (5.21)P(Ω)=Ω 4 [ −240Pu 5 0 + 36u6 0 (−4 + 3u4 0 ω2 0 )]+Ω 2 [ −1100P 2 − 30Pu 0 (44 − 65u 4 0 ω2 0 ) + 9u2 0 (−44 + 127u4 0 ω2 0 − 44u8 0 ω4 0 )]+5600P 2 ω 2 0 + 840Pu 0 ω 2 0(8 − 3u 4 0ω 2 0) + 36u 2 0ω 2 0(56 − 39u 4 0ω 2 0 + 8u 8 0ω 4 0). (5.22)MATHEMATICA notebook which implements this analytical calculation is availableat our web site [60].5.4.2 Results and discussionThe result given by Eq. (5.21) is the main achievement of our analytical analysis inthe previous section. It is obtained within a second-order perturbative approach inQ and it describes the breathing mode frequency dependence on the driving Ω andthe modulation amplitude Q as a result of nonlinear effects. Due to the underlyingperturbative expansion, we do not expect Eq. (5.21) to be meaningful at the preciseposition of the resonances. However, by comparison with numerical results basedon the variational equation, we find that Eq. (5.21) represents quite a reasonableapproximation even close to the resonant region.To illustrate this, in Fig. 5.11 we show two such comparisons. In the upperpanel we consider the parameter set P = 0.4 and Q = 0.1, and observe significantfrequency shifts only in the narrow resonant regions. We notice an excellent agreementof numerical values with the analytical result given by Eq. (5.21). In the lowerpanel we consider the parameter set P = 1 and Q = 0.8, with much stronger modulationamplitude. In this case we observe significant frequency shifts for the broaderrange of modulation frequencies Ω. In spite of a strong modulation, we still seea qualitatively good agreement of numerical results with the analytical predictiongiven by Eq. (5.21). In principle, better agreement can be achieved using higherorderperturbative approximation. The dashed line on both figures represents Ω/2,given as a guide to the eye. It also serves as a crude description of what we observenumerically in the range Ω ≈ 2ω 0 .123
5. BEC excitation by modulation of scattering length2.12.08ω 0analytical ωnumerical ωFrequency2.062.042.02Frequency22.32.252.22.152.12.0521.951.91 1.5 2 2.5 3 3.5 4 4.5 5ω 0analytical ωnumerical ωΩ1 2 3 4 5 6ΩFigure 5.11: Frequency of the breathing mode versus the driving frequency Ω forP = 0.4 and Q = 0.1 (top), and P = 1 and Q = 0.8 (bottom). The dashed linerepresents Ω/2 and is given to guide the eye.The presence of two poles at Ω = ω 0 and Ω = 2ω 0 in Eq. (5.21) implies thepossible existence of real resonances in the BEC with a harmonically modulatedinteraction. A perturbative expansion to higher orders would introduce some additionalpoles, responsible for higher-order “resonant” behavior observed at Ω ≈ 2ω 0 /n(n ≥ 3). Still, the poles seem to be only an artifact of our approximative perturbativescheme, not present in the exact description. For example, a simple resummationperformed using the second-order perturbative result removes these effects, although124
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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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Page 125 and 126: Chapter 5Nonlinear BEC dynamics by
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Page 143: 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
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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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