PhD thesis in English

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$Edge excitations of paired fractional quantum Hall states$

2. Diagonalization of Transition Amplitudespresent the method and notation, and identify the sources of the errors present inreal-space discretization approaches. Then we analyze in detail the above questions1 and 2, and discuss the effects of discretization on the numerically calculated valuesof the observables for a given physical system. We analytically derive estimates forerrors stemming from space discretization coarseness, finite size effects, and choice ofthe evolution time parameter t. All the analytically derived results are numericallyverified to hold on several instructive models.Errors associated with the time of evolution parameter t (question 3 above) mustbe carefully taken into account and may substantially limit the precision of numericalcalculation in the diagonalization method. This problem was not addressed atall in Ref. [9], but has been addressed recently [38, 39, 40, 41, 42] using various approaches.We significantly improve the method by applying the recently introducedeffective action approach [43, 44, 10, 45, 46, 11] to completely resolve the problemformulated in question 3. We stress that use of higher-order effective actionsrepresents an efficient and numerically inexpensive way to calculate transition amplitudes,and leads to many orders of magnitude increase in precision of calculatedproperties of the system. We will demonstrate on several lower-dimensional modelshow use of higher-order effective actions significantly reduces numerical errors andsystematically improves the obtained energy eigenvalues and eigenstates.This Chapter gives a complete analysis of the method based on the diagonalizationof transition amplitudes, providing us with necessary analytical knowledge toestimate errors of all types associated with this method and to numerically very accuratelycalculate large numbers of energy eigenvalues and eigenstates. This invitesvarious applications of the method to the study of few-body quantum systems, someof which are discussed throughout the thesis.The expressions written throughout the second and third section are, for compactnessof notation, for one particle in one dimension. Extension to more particlesand dimensions is straightforward, just as with the above short-time transitionamplitude. Note that we are working in imaginary time, which is well suited fornumerical calculations and does not affect in any way calculated energy levels norother time-independent properties of the system. We have also set to unity in thisChapter.21
2. Diagonalization of Transition Amplitudes2.1 Space-discretized Schrödinger equationIn the coordinate representation the time-independent Schrödinger’s equation takesthe form∫dy 〈x|Ĥ|y〉〈y|ψ〉 = E 〈x|ψ〉 . (2.2)The standard way to numerically implement exact diagonalization is to go fromcontinuous coordinates x to ones living on a discrete space grid x n = n∆, where∆ is a given spacing and n ∈ Z. Integrations in the above equation are performedusing the simple rectangular quadrature rule, or some higher-order finite-differenceformula. This completes the transition to the space-discretized counterpart of thecontinuous theory, however, to represent this on a computer we still have to restrictthe integers n to a finite range. This is equivalent to introducing a space cutoff L, orputting the system in a infinitely high potential box. For example, the rectangularquadrature rule leads to the following space-discretized Schrödinger equationN∑cut−1m=−N cutH nm 〈m∆|ψ〉 = E(∆, L) 〈n∆|ψ〉 , (2.3)where H nm = ∆ · 〈n∆|Ĥ|m∆〉, N cut = [L/∆], and square brackets represent theinteger part of the argument. As a result, we have obtained a 2N cut × 2N cut matrixthat represents the Hamiltonian of the system. The eigenvalues of this matrix dependon the two parameters introduced in the above discretization process: cutoffL and discretization step ∆. Continuous physical quantities are recovered in thelimit L → ∞ and ∆ → 0. The outlined procedure is very useful in dealing withspatially localized physical problems, such as electronic structure calculations insemiconductor and polymer physics [47].The two approximations involved in the discretization procedure, characterizedby parameters ∆ and L, are common steps in solving eigenproblems of Hamiltoniansin e.g. electronic structure calculations [47], and as such have been extensivelyanalyzed. The imposed constraint on the values of spatial coordinates to the finiteinterval (−L, L) is a valid approach for capturing information on localized eigenstates.In this approximation the system is effectively surrounded by an infinitelyhigh wall, and as the cutoff L tends to infinity, we approach the exact energy levelsalways from above, which is a typical variational behavior. Therefore, we designateerrors associated with the cutoff L as variational. The effects of the discretization22
Page 1 and 2: UNIVERSITY OF BELGRADEFACULTY OF PH
Page 3 and 4: Thesis advisor, Committee member:Dr
Page 6 and 7: lence for Computer Modeling of Comp
Page 8 and 9: dobijanje kondenzata odabrani su at
Page 10 and 11: Uticaj slabih interakcija na fenome
Page 12 and 13: Abstract of the doctoral dissertati
Page 14 and 15: highly accurate information on ener
Page 16 and 17: Keywords: cold quantum gases, Bose-
Page 18 and 19: CONTENTS3.4.2 Time-of-flight graphs
Page 20 and 21: NomenclatureRoman Symbolsagk BLMNn(
Page 22 and 23: Chapter 1Introduction1.1 ForewordTh
Page 24 and 25: ently explored to illustrate the ve
Page 26 and 27: Summations in the last expression c
Page 28 and 29: Figure 1.1: The hallmark of the Bos
Page 30 and 31: we discuss in some detail the exper
Page 32 and 33: In the first papers [3, 4], the TOF
Page 34 and 35: where a BG is the off-resonant scat
Page 36 and 37: system given by( ) ǫ Bog ⃗k =
Page 38 and 39: Having the efficient numerical meth
Page 40 and 41: Chapter 2Properties of quantum syst
Page 44 and 45: 2. Diagonalization of Transition Am
Page 66 and 67: ||2. Diagonalization of Transition
Page 74: 2. Diagonalization of Transition Am
Page 79 and 80: magnetic field ⃗ B = 2M ⃗ Ω.3.
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Page 83 and 84: 3. Rotating ideal BEC3.1 Numerical
Page 85 and 86: 3. Rotating ideal BEC350300SC appro
Page 87 and 88: 3. Rotating ideal BEC3.2 Finite num
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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
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Chapter 4Mean-field description of
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4. Mean-field description of an int
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Chapter 5Nonlinear BEC dynamics by
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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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PhD thesis in English

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