PhD thesis in English

More documents

Recommendations

Info

$Edge excitations of paired fractional quantum Hall states$

5. BEC excitation by modulation of scattering lengthnext step, we use these solutions and solve for x 2 (t) and y 2 (t) and so on. At eachlevel n ≥ 1 we impose the initial conditions u ρn (0) = 0, ˙u ρn (0) = 0, u zn (0) = 0,and ˙u zn (0) = 0. At the first level of perturbation theory, equations for x and yare decoupled, i.e. x 1 (t) and y 1 (t) are normal modes: x 1 (t) describes quadrupoleoscillations, while y 1 (t) describes breathing oscillations. However, at the secondorder of perturbation theory y 1 (t) enters the equation for x 2 (t) and also x 1 (t) appearsin equation for y 2 (t), i.e. we have a nonlinear mode coupling.We have performed the explicit calculation to the second order by using thesoftware package MATHEMATICA [54]. We have obtained an excellent agreementof second-order analytical results and numerical results, as can be seen in Fig. 5.13for a moderate value of a modulation amplitude Q. The first secular terms appearat the level n = 3. The expressions are cumbersome, but the relevant behavior isobtained from the following terms in the equation for x 3 (t):ẍ 3 (t) + ωQ0x 2 3 (t) + C Q cos ω Q0 t + . . . = 0, (5.31)which leads tox 3 (t) = − C Q2ω Q0t sin ω Q0 t + . . . (5.32)The last term can be absorbed into the first-order solutionu ρ (t) = A Q cos ω Q0 t − C QQ 2t sin ω Q0 t + . . .2ω Q0≈ A Q cos [(ω Q0 + ∆ω Q0 )t], (5.33)and can be interpreted as a frequency shift of the quadrupole mode, quadratic in Q:ω Q = ω Q0 + ∆ω Q0 = ω Q0 + Q 2 + . . . (5.34)2ω Q0 A QThe coefficients A Q and C Q are calculated using the MATHEMATICA code availableat our site [60], and their explicit form is too long to be presented here. Along thesame lines we have also calculated the frequency shift of the breathing mode.C Q5.5.2 Results and discussionThe main results of our calculation in this section are shown in Figs. 5.15 and5.16. In Fig. 5.15 we plot the analytically obtained frequency of the quadrupole129
5. BEC excitation by modulation of scattering length0.55Frequency ω Q0.5450.540.5350.530 1 2 3 4 5Ωω Q0ω Q, (a)ω Q, (n)Figure 5.15: Frequency of the quadrupole mode ω Q versus driving frequency Ω forP = 1, Q = 0.2, and λ z = 0.3. We plot linear response result ω Q0 , second-orderanalytical result ω Q,(a) and numerical values ω Q,(n) .Frequency ω B2.012.0082.0062.0042.00221.9981.9961.9941.9921.990 1 2 3 4 5 6Ωω B0ω B, (a)ω B, (n)Figure 5.16: Frequency of the breathing mode ω B versus driving frequency Ω forP = 1, Q = 0.2, and λ z = 0.3. We plot linear response result ω B0 , second-orderanalytical result ω B,(a) and numerical values ω B,(n) .mode versus the driving Ω, using the second order perturbation theory togetherwith the corresponding numerical result based on the Fourier analysis of solutionsof Eqs. (5.11) and (5.12). An analogous plot for the frequency of the breathing mode130
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 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
Page 64 and 65:
Page 66 and 67:
||2. Diagonalization of Transition
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74:
Page 77 and 78:
Page 79 and 80:
magnetic field ⃗ B = 2M ⃗ Ω.3.
Page 81 and 82:
3. Rotating ideal BECof the rotatio
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
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 149: where n = 1, 2, 3, . . . is an inte
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?