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$Edge excitations of paired fractional quantum Hall states$

Chapter 2Properties of quantum systems viadiagonalization of transition amplitudesDeep in the condensate phase, features of cold atoms are determined by low-lyingenergy levels: ground state and few excited states corresponding to the thermalcloud. On the other hand, thermodynamic properties and details of the BEC phasetransition are determined by the full energy spectrum. As we have seen in Chapter1, the exact calculation of the condensation temperature, even for an ideal bosonicgas, requires a summation over the whole energy spectrum of the system. Dueto its simplicity, the semiclassical approximation is widely used for this purpose.In this Chapter we work out details of a numerical method based on the exactdiagonalization of a time-evolution operator that allows us access to a very largenumber of numerically exact energy levels of few-body systems. Afterwards, inChapter 3, we use the method to find the condensation temperature of the fastrotatingideal gas in an anharmonic trap.In the standard operator formulation of quantum mechanics, the description ofa physical system is based on constructing the Hamilton operator Ĥ. Properties ofquantum systems are then obtained by solving the corresponding time-independentSchrödinger equation,Ĥ|ψ〉 = E|ψ〉 . (2.1)Exact solutions can be found only for a very limited set of simple models. A widevariety of analytical approximation techniques has been developed in the past fortreatment of such problems. In addition, the last two decades have seen a rapidgrowth in the application of different numerical methods for solving the Schrödingerequation. Approaches based on real-space discretization start from some given finitedifferenceprescription. Such methods have been extensively studied in the past, andthe main difficulties follow from the finite-difference representations of the kineticoperator.A numerical approach based on diagonalizing of the evolution operator, intro-19
2. Diagonalization of Transition Amplitudesduced in Ref. [9], does not suffer from problems with the representation of differentialoperators on real-space grids, and has substantial advantages in practical applicationsto few-body problems. Effectively, in this way the problem is transferred fromthat of representing the kinetic operator on a real-space grid to the calculating ofcorresponding transition amplitudes. Detailed analysis of the errors associated withthe implementation of this approach is the main objective of this Chapter. It providesfull understanding of the method and allows its optimal use, as well as furthersignificant improvements within a generalized calculation scheme.The advantages of the method discussed here [9, 35, 36, 37] follow from two keyproperties. First, the objects being diagonalized are transition amplitudes, which arewell defined irrespective of discretization scheme, i.e. the exponential of the Hamiltonianeffectively regularizes the kinetic operator, making possible representations ofthe evolution operator that do not depend on the space grid. Second, the successfuldiagonalization of the evolution operator exp(−tĤ) for any time of propagation timmediately gives the solution of the eigenproblem for the Hamiltonian. Thus, thetime of propagation in this approach is just an auxiliary parameter. Said anotherway, we use the time-dependent evolution operator to extract time-independentinformation regarding the quantum system. If one could calculate transition amplitudesexactly, then the obtained results for the energy eigenproblem would notdepend on the time of propagation. However, in practical applications one uses someapproximation scheme to calculate the amplitudes, and in this case the precision ofthe obtained results for energy eigenvalues and eigenstates does depend on time t.The general applicability of the outlined method follows from the fact that one canuse short-time propagation amplitudes to obtain highly accurate results.In order to complete this numerical method and make it generally applicable, itis necessary to address the following key questions:1. How to analytically estimate the effects of spatial discretization?2. How to optimize the choice of evolution time t, so as to minimize errors?3. How to accurately calculate transition amplitudes?The authors in Ref. [9] have only briefly commented on the first two questions,and numerically determined the values of parameters that can be used for precisecalculations of energy eigenvalues and eigenstates for several models. In this Chapterwe address the above questions, which have not been fully answered before. First we20
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 42 and 43: 2. Diagonalization of Transition Am
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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
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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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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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