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

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2. Properties of an atomic Bose condensate in a double-well potentialTable 2.1: Calculations of the tunnelling period T 0 /t 0 , based on different choices of z 0 ′ ,withcomments on the significance of each choice. The tunnelling frequency is calculated from Ω=38 ω 0z 02 ′ exp(− 1 2 z′ 2 0 ).z ′ 0 3 T 0 /t 0 =4πω 0 /Ω Comments4.00 6.24 × 10 3 Position of well minimum3.65 1.97 × 10 3 Maximum of local mode3.40 938 Minimises high frequency components3.21 562 Corresponds to observed tunnelling period3.11 439 Calculated from critical value of ΘGaussians. If the position of the initial distribution is reduced further to z =3.4 ,thenthe unwanted oscillations are reduced to a minimum (Figs. 2.7(c) and 2.8(c)), althoughthere are now several contributing frequencies.The question arises how to calculate the tunnelling frequency Ω, which depends onthe overlap of the local wave functions. If the local modes are no longer centred at theminima of the trapping potential, then the calculation of Ω based on z 0 would no longerbe accurate. Could the calculation be done assuming the same form for Ω but with z 0replaced by z 0 ′ , chosen to be the centre of the initial distribution that gives the cleanestJosephson oscillations? In fact the observed period corresponds to a value of z 0 ′ =3.21,which is smaller again, evidently because the local potentials are now asymmetric, andskewed towards the origin. Starting at this value does not give the cleanest oscillations, asshown in Figs. 2.7(d) and 2.8(d). The different results for the calculation of the tunnellingperiod are summarised in the Table 2.1.What remains of the self-trapping effect for strong atom interactions in these GPsimulations?Figures 2.9 and 2.10 show the evolution of atom density and the meanposition calculated from this, for different collisional strengths, for a z 0 =4.0 trappingpotential and an initial distribution centred at z =3.65. The tunnelling dynamics exhibitthe qualitative predictions of the two-mode analysis.There is a critical value of thenonlinearity (0.1015
2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.7: Evolution of the mean position of a condensate in a double-well potential with minimaat x = ±4r 0 (in (a) to (d)) and x = ±3r 0 (in (e) and (f)), calculated from the GP equation. Thehigh frequency components present in the tunnelling waveform for different initial conditions canbe seen. The initial distribution is centred at 4.00z 0 in (a), 3.65z 0 in (b), 3.40z 0 in (c), 3.21z 0 in(d), 3.00z 0 in (e), and 2.30z 0 in (f).43(a)43(b)2211x/r 00x/r 00−1−2−3−40 200 400 600 800 1000t/t 04321(c)−1−2−3−40 200 400 600 800 1000t/t 04321(d)x/r 00x/r 00−1−2−3−40 200 400 600 800 1000t/t 032(e)−1−2−3−40 200 400 600 800 1000t/t 032(f)11x/r 00x/r 00−1−1−2−2−30 20 40 60 80 100t/t 0−30 20 40 60 80 100t/t 036
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 35: 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
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4. Continuously monitored Bose cond
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Chapter 5Weak force detection using
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5. Weak force detection using a dou
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Part IIQuantum evolution of a Bose
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6. Quantum effects in optical fibre
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Chapter 7Quantum simulations of eva
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7. Quantum simulations of evaporati
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A. Stochastic calculus in outlineco
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Appendix BQuasiprobability distribu
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B. Quasiprobability distributions u
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Bibliography[1] Agrawal, G. P. Nonl
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BIBLIOGRAPHY[22] Carter,S.J.Quantum
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BIBLIOGRAPHY[45] Drummond, P. D., C
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BIBLIOGRAPHY[68] Gordon,D.,andSavag
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BIBLIOGRAPHY[92] Jaksch,D.,Gardiner
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BIBLIOGRAPHY[114] Marshall, R. J.,
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BIBLIOGRAPHY[135] Naraschewski, M.,
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BIBLIOGRAPHY[158] Shelby, R. M., Le
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BIBLIOGRAPHY[179] Wiseman, H. M. Qu
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. . . but this book is already too
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