Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

More documents

Recommendations

Info

2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.16: The ground-state spatial density distribution ρ = 〈 ˆψ† (x) ˆψ(x) 〉 ,scaledbythepeakvalue. The continuous line is for κ ≪ 1 and the dashed line is for κ ≫ 1.Normalised density: ρ/ρ 010.80.60.40.2Ω >> κ NΩ
2. Properties of an atomic Bose condensate in a double-well potentialThis is just the joint probability distribution to detect an atom at x 1 and x 2 which hasbeen considered by [95]. For the double-well system, this becomesG 2 (x 1 ,x 2 ,t) =++++〈 N2〉4 − Ĵ x2 u 2 1(x 1 )u 2 2(x 2 )〈〉(N − 1)Ĵz − ĴxĴz − ĴxĴz u 2 1(x 1 )u 1 (x 2 )u 2 (x 2 )〈〉+ (N − 1)Ĵz + ĴxĴz + ĴxĴz u 1 (x 1 )u 2 (x 1 )u 2 2 (x 2)〈 ( )〉N N2 2 − 1 + Ĵx(Ĵx +1− N) u 2 1 (x 1)u 2 1 (x 2)〈 ( )〉N N+2 2 − 1 + Ĵx(Ĵx − 1+N) u 2 2(x 1 )u 2 2(x 2 )〈〉+ 4Ĵ z 2 − N u 1 (x 1 )u 1 (x 2 )u 2 (x 1 )u 2 (x 2 )〈〉(N − 1)Ĵz − ĴxĴz − ĴxĴz u 1 (x 1 )u 2 1(x 2 )u 2 (x 1 )〈〉+ (N − 1)Ĵz + ĴxĴz + ĴxĴz u 1 (x 2 )u 2 2 (x 1)u 2 (x 2 )〈 N2〉4 − Ĵ x2 u 2 1(x 2 )u 2 2(x 1 ). (2.75)We will look at the case where the atoms considered come from different wells, i.e. x 1 < 0and x 2 > 0, which means that u i (x 3−i ) ≪ u j (x j ), for i, j =1, 2. The lines in Eq. (2.75)have been arranged in order of size, from zeroth order terms to fourth order terms. Fora condensate in the ground state for either of the limiting cases (κ ≪ 1orκ ≫ 1),〈Ĵx〉=〈Ĵx Ĵ z〉=〈Ĵz Ĵ x〉= 0, and so Eq. (2.75) becomesG 2 (x 1 ,x 2 ,t) =++++〈 N2〉4 − Ĵ x2 u 2 1(x 1 )u 2 2(x 2 )〈〉 (u2(N − 1)Ĵz 1 (x 1 )u 1 (x 2 )u 2 (x 2 )+u 1 (x 1 )u 2 (x 1 )u 2 2(x 2 ) )〈 ( ) 〉 N N (u2 2 − 1 + Ĵ x2 21 (x 1 )u 2 1(x 2 )+u 2 2(x 1 )u 2 2(x 2 ) )〉+〈4Ĵ z 2 − N u 1 (x 1 )u 1 (x 2 )u 2 (x 1 )u 2 (x 2 )〈〉 (u1(N − 1)Ĵz (x 1 )u 2 1 (x 2)u 2 (x 1 )+u 1 (x 2 )u 2 2 (x 1)u 2 (x 2 ) )〈 N2〉4 − Ĵ x2 u 2 1(x 2 )u 2 2(x 1 ). (2.76)The ‘first order’ term on the second line of Eq. (2.76) indicates the size of the interferencefringes and scales as 〈〉Ĵ z /N relative to the peak value, the same as with the densityfunction. (See Fig. 2.17 for how this term varies with N and κ.) This term disappearswhen κ ≫ 1 and approaches a maximum when κ ≪ 1. In the latter case, the correlation52
Page 1 and 2: Open Quantum Dynamics of Mesoscopic
Page 3 and 4: AcknowledgementsMy thanks must firs
Page 5 and 6: AbstractThe properties of an atomic
Page 7 and 8: CONTENTS4 Continuously monitored Bo
Page 9 and 10: List of Figures2.1 Two-mode approxi
Page 11 and 12: LIST OF FIGURES7.5 Angular momentum
Page 14 and 15: LIST OF ABBREVIATIONS AND SYMBOLSI{
Page 17 and 18: 1. Condensation ‘without forces
Page 19 and 20: 1. Condensation ‘without forces
Page 21 and 22: Chapter 2Properties of an atomic Bo
Page 23 and 24: 2. Properties of an atomic Bose con
Page 51: 2. Properties of an atomic Bose con
Page 62 and 63: 3. Homodyne measurements on a Bose-
Page 70: 3. Homodyne measurements on a Bose-
Page 79 and 80: Chapter 4Continuously monitored con
Page 81 and 82: 4. Continuously monitored Bose cond
Page 103 and 104:
4. Continuously monitored Bose cond
Page 105 and 106:
4. Continuously monitored Bose cond
Page 107 and 108:
Chapter 5Weak force detection using
Page 109 and 110:
5. Weak force detection using a dou
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Part IIQuantum evolution of a Bose
Page 125 and 126:
6. Quantum effects in optical fibre
Page 127 and 128:
Page 129 and 130:
Page 131 and 132:
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
Page 139 and 140:
Page 141 and 142:
Chapter 7Quantum simulations of eva
Page 143 and 144:
7. Quantum simulations of evaporati
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
A. Stochastic calculus in outlineco
Page 173 and 174:
Appendix BQuasiprobability distribu
Page 175 and 176:
B. Quasiprobability distributions u
Page 177 and 178:
Bibliography[1] Agrawal, G. P. Nonl
Page 179 and 180:
BIBLIOGRAPHY[22] Carter,S.J.Quantum
Page 181 and 182:
BIBLIOGRAPHY[45] Drummond, P. D., C
Page 183 and 184:
BIBLIOGRAPHY[68] Gordon,D.,andSavag
Page 185 and 186:
BIBLIOGRAPHY[92] Jaksch,D.,Gardiner
Page 187 and 188:
BIBLIOGRAPHY[114] Marshall, R. J.,
Page 189 and 190:
BIBLIOGRAPHY[135] Naraschewski, M.,
Page 191 and 192:
BIBLIOGRAPHY[158] Shelby, R. M., Le
Page 193 and 194:
BIBLIOGRAPHY[179] Wiseman, H. M. Qu
Page 195:
. . . but this book is already too
show all

Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

Create successful ePaper yourself

Delete template?

Save as template?