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$Edge excitations of paired fractional quantum Hall states$

3. Rotating ideal BECTable 3.2: Lowest energy levels of the xy-part of the BEC potential (3.4) for overcriticallyrotating (η = 1.04) condensate with the quartic anharmonicity κ = κ BECand κ = 10 3 κ BEC according to the same numerical procedure as in Table 3.1.E n /ω, η = 1.04n κ BEC 10 3 κ BEC0 -0.6617041825660 1.1356938262061 -0.6465857464220 2.6281299037902 -0.6465857464220 2.6281299037903 -0.6032113415949 4.3638766339294 -0.6032113415948 4.3638766339295 -0.5349860004310 4.6676535829636 -0.5349860004309 6.2904447340077 -0.4451224795419 6.2904447340078 -0.4451224795419 6.7772107731699 -0.3362724309903 6.777210773169correspond to the expected structure of the spectrum, which can be deduced fromthe symmetry of the problem. In addition to this, the interesting case of criticalrotation (η = 1) allows a further verification of the numerical results. To this endwe recall that the energy eigenvalues of a pure quartic oscillator, to which V BECreduces in this case, are proportional to κ 1/3 due to a spatial rescaling in the underlyingSchrödinger equation. Therefore, we expect that the energy eigenvaluesfor κ = 10 3 κ BEC are precisely 10 times larger than the corresponding eigenvaluesfor κ = κ BEC . Comparing the rightmost columns in Table 3.1 we see exactly thisscaling. This demonstrates conclusively that the presented method can be successfullyapplied also in this deeply non-perturbative parameter regime. Furthermore,Table 3.2 gives the energy spectrum of an over-critically rotating (η = 1.04) condensate,illustrating that the same approach can be used in this delicate regime aswell.With these results single-particle partition functions Z 1 (β) can now be calculatedaccording to Eq. (3.5). This is especially suitable for the low-temperature regime,when higher energy levels give a negligible contribution. Although the above describedapproach is able to accurately give several thousands of energy eigenvalues,their number is always necessarily limited. This is easily seen from Fig. 3.2, wherewe compare the density of states for a critically rotating condensate with the corre-63
3. Rotating ideal BEC350300SC approx.L = 22.3, L/∆ = 100ρ(E)2502001501005000 20 40 60 80 100 120 140E / − hω ⊥Figure 3.2: Numerically calculated density of states for xy-part of the BEC potentialwith κ = κ BEC for a critically rotating condensate, obtained by using level p = 21effective action. The discretization parameters are L = 22.3, L/∆ = 100, andt = 0.2. The dashed line is the corresponding semiclassical approximation for thedensity of states.sponding semiclassical approximation for the density of states (2.38). This comparisonallows us to estimate the maximal reliable two-dimensional energy eigenvalueE max which can be obtained numerically for a given set of discretization parameters.For example, from Fig. 3.2 we can estimate E max ≈ 90 for η = 1 with the discretizationparameters L = 22.3, L/∆ = 100, and t = 0.2. Table 3.3 gives estimatesfor the maximal reliable energy eigenvalue for the anharmonicities κ = κ BEC andκ = 10 3 κ BEC for several values of rotation frequencies. These results are obtainedfrom numerical calculations using the SPEEDUP codes [60]. This table gives alsoan overview over those discretization parameters which were used for a numericaldiagonalization of the BEC potential (3.4) in order to calculate both global and localproperties of the condensate throughout this Chapter.In the low-temperature limit the finiteness of the number of known energy eigenstatesdoes not present a problem. In fact, a precise knowledge of a large number ofenergy eigenvalues makes this approach a preferred method for a numerically exacttreatment of low-temperature phenomena. On the other hand, the high-temperatureregime, where thermal contributions of higher energy states play a significant role,is not treatable in the same way. This regime is usually not relevant for studies of64
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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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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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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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