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

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2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.10: Tunnelling oscillations of a condensate in a double-well potential with minima at x =±4r 0 , calculated from the GP equation with different strengths of the nonlinear atom interactions,both above and below the critical value. In (a) K =0.0000, in(b)K =0.1000, in(c)K =0.1015,in (d) K =0.1020, in(e)K =0.1500, andin(f)K =0.5000. Note that (a), (e) and (f) extendfor only half the time as (b) to (d).43(a)43(b)2211x/r 00x/r 00−1−2−3−40 200 400t/t 0600 800 10004321(c)−1−2−3−40 500 1000t/t 01500 200043.532.5(d)x/r 00x/r 02−1−2−31.510.5x/r 0−40 500 1000 1500 2000t/t 03.83.63.43.232.8(e)x/r 000 500 1000t/t 01500 200043.83.63.43.2(f)2.62.42.220 200 400 600 800 1000t/t 032.82.60 200 400 600 800 1000t/t 039
2. Properties of an atomic Bose condensate in a double-well potentialsmall oscillations that decrease in period and amplitude as K increases further (Figs.2.9(d-f) and 2.10(d-f)). Both types of behaviour are consistent with the description interms of Jacobi elliptic functions predicted from the two-mode analysis.What is notpredicted correctly is just what this critical value is: the self-trapping effect occurs forlarger nonlinearities than that predicted from the two-mode analysis. If we assume thatat the critical value Ω = κN, then we get a smaller period T 0 (and hence z 0 ′ ) than theobserved value, and different from any other value discussed above. See Table 2.1 for thecomparison with other calculations of tunnelling period. This is yet another consequenceof the large overlap between the two local modes.In summary, the main features of the semiclassical two-mode description survive inthe full Gross-Pitaevskii treatment.An adjustment of the tunnelling frequency Ω andthe critical ratio Θ allows the two-mode treatment to correctly predict behaviour even inclosely spaced wells. These adjustments would be unnecessary for trapping potentials witha large separation, in which case the unwanted oscillations should not appear. But thetunnelling period increases exponentially with the well separation 2z 0 , which makes thenumerical integration quite demanding on computer time, especially when the nonlinearityin Eq. (2.46) is large.2.3 Quantum dynamicsBecause of the conservation of total particle number, the two-mode Hamiltonian (Eq.(2.14)) translates into an angular momentum model which allows us to represent thequantum dynamics on the surface of a Bloch sphere. To see this, we can remove the firstterm in Eq. (2.14) by transforming to an interaction picture, resulting in the HamiltonianĤ 2b = Ω 2 (ĉ 1ĉ † 2 +ĉ† 1ĉ2)+κ((ĉ † 1 )2 ĉ 2 1 +(ĉ † 2 )2 ĉ 2 2). (2.47)Introducing the angular momentum operators, which are similar to the transformationsin the semiclassical case (Eq. 2.31):Ĵ z = 1 2 (ĉ† 1ĉ2 +ĉ † 2ĉ1), (2.48a)Ĵ x = 1 2 (ĉ† 2ĉ2 − ĉ † 1ĉ1), (2.48b)Ĵ y = i 2 (ĉ† 2ĉ1 − ĉ † 1ĉ2), (2.48c)40
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 39: 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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