PhD thesis in English

More documents

Recommendations

Info

$Edge excitations of paired fractional quantum Hall states$

3. Rotating ideal BECBy rearranging the summation order in the previous equation, we finally obtain:N = B 0 (µ, T) += B 0 (µ, T) +∞∑ ∑ ∞e jβµ e −jβEn ,j=1n=1∞∑e ( jβµ Z 1 (jβ) − e ) −jβE 0, (3.7)j=1where the summation on the right-hand side of the previous equation corresponds tothe cumulant expansion. In order to avoid any double-counting, we have subtractedthe contribution of the ground state within the single-particle partition functionbecause a possible macroscopic occupation of the ground state is separately takeninto account.The BEC phase transition is achieved only in the thermodynamic limit of aninfinite number of atoms, thus making numerical studies of the condensation increasinglydifficult. Usually, the problem is solved by fixing the chemical potential µat the low temperatures of the condensate phase to the ground-state energy, i.e. bysetting µ = E 0 . This requires that the ground state is treated separately by associatinga macroscopic value N 0 to the ground-state occupation number B 0 (µ, T), forT < T c . Thus, from Eq. (3.7), the total number of particles in the condensate phasefollows to be:∞∑ (N = N 0 + ejβE 0Z 1 (jβ) − 1 ) . (3.8)j=1The equation (3.8) yields the temperature dependence of N 0 . Within the gas phase,where the macroscopic occupation of the ground state vanishes, i.e. we have N 0 =0, Eq. (3.7) determines the temperature dependence of the chemical potential µ.Therefore, the value of β c = 1/k B T c , which characterizes the boundary betweenboth phases, follows from Eq. (3.8) by setting N 0 = 0 and µ = E 0 :N =∞∑ [ejβ cE 0Z 1 (jβ c ) − 1 ] . (3.9)j=1We conclude that, for a given number N of ideal bosons, the condensation temperaturecan be exactly calculated only if both the single-particle ground-state energyE 0 and the full temperature dependence of the one-particle partition function (3.5)are known.61
3. Rotating ideal BEC3.1 Numerical calculation of energy eigenvalues and eigenstatesA very efficient method for calculating properties of few-body quantum systems,that we use, is the direct diagonalization of the space-discretized propagator inimaginary time. The approach is able to give very accurate energy eigenvalues evenfor moderate values of the propagation time t of the order 0.1, as shown in detail inChapter 2. Note that throughout this Chapter we use dimensionless units, in whichall energies are expressed in terms of ω, while the length unit is the correspondingharmonic oscillator length √ /Mω.Table 3.1 presents the first several energy eigenvalues for the two-dimensional(x − y) part of the BEC potential (3.4) for the non-rotating case (η = 0), as well asfor the critically-rotating condensate (η = 1). The table on the left gives the energyspectrum of the potential with the anharmonicity κ = κ BEC used in the experiment[12], while the right table shows the spectrum for the much larger anharmonicityκ = 10 3 κ BEC . The degeneracies of numerically obtained eigenstates in all casesTable 3.1: Lowest energy levels of the xy-part of the BEC potential (3.4) for nonrotating(η = 0) and critically rotating (η = 1) condensate with the quartic anharmonicityκ = κ BEC (left) and κ = 10 3 κ BEC (right). They are obtained by using levelp = 21 effective action with the discretization parameters of Table 3.3. The spacing∆ was always chosen so that L/∆ = 100, and the propagation time was t = 0.2 forκ = κ BEC and t = 0.05 for κ = 10 3 κ BEC . Errors are given by the precision of thelast digit, typically 10 −12 to 10 −13 , and are estimated by comparing the numericalresults obtained with different discretization parameters.E n /ω, κ = κ BECn η = 0 η = 10 1.0009731351803 0.11626671641341 2.0029165834022 0.26746899689052 2.0029165834022 0.26746899689053 3.0058275442161 0.44269273752694 3.0058275442161 0.44269273752705 3.0067964582067 0.47252757249416 4.0097032385903 0.63681788049837 4.0097032385903 0.63681788049848 4.0116368851078 0.68481424703569 4.0116368851078 0.6848142470357E n /ω, κ = 10 3 κ BECn η = 0 η = 10 1.468486725893 1.1626671641341 3.213056378201 2.6746899689052 3.213056378201 2.6746899689053 5.163819069871 4.4269273752694 5.163819069871 4.4269273752705 5.406908088225 4.7252757249416 7.282930987460 6.3681788049827 7.282930987460 6.3681788049828 7.690584058915 6.8481424703579 7.690584058915 6.84814247035762
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 42 and 43: 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.
Page 81: 3. Rotating ideal BECof the rotatio
Page 85 and 86: 3. Rotating ideal BEC350300SC appro
Page 87 and 88: 3. Rotating ideal BEC3.2 Finite num
Page 89 and 90: 3. Rotating ideal BECN - N 03.0·10
Page 91 and 92: 3. Rotating ideal BEC3.3 Global pro
Page 93 and 94: 3. Rotating ideal BECever, it does
Page 95 and 96: 3. Rotating ideal BECT c [nK]120110
Page 97 and 98: 3. Rotating ideal BEC3.533210 3 κ
Page 99 and 100: 3. Rotating ideal BECn(x, y)10·10
Page 101 and 102: 3. Rotating ideal BECpancy numbers
Page 103 and 104: 3. Rotating ideal BEC6·10 4 0 0.02
Page 105 and 106: Chapter 4Mean-field description of
Page 107 and 108: 4. Mean-field description of an int
Page 125 and 126: Chapter 5Nonlinear BEC dynamics by
Page 127 and 128: 5. BEC excitation by modulation of
Page 133 and 134:
5. BEC excitation by modulation of
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Page 143 and 144:
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
where n = 1, 2, 3, . . . is an inte
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Chapter 6SummarySince the first exp
Page 157 and 158:
short-range interactions.The excita
Page 159 and 160:
of the ground state. Imaginary-time
Page 161 and 162:
(A.13). With another boundary condi
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
show all

PhD thesis in English

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?