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

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4. Continuously monitored Bose condensates: quasiprobability distributionsdistribution remains symmetric about the equator of the Bloch sphere, and its behaviouris similar to the previous strong measurement simulations with ∣ ∣ j, 0〉xas the initial state(Fig. 4.2): it periodically splits into two components which move apart, sometimes movingover the poles to come down along a different longitude, which corresponds to the phasejumps in the tunnelling oscillations.When subcritical atom-atom interactions are included (Θ < 1), the Q function evolutionis very similar to the noninteracting case: the distribution gradually moves off thesouth pole of the Bloch sphere, remaining compact as it precesses around the Ĵz axis. Forvery strong self-interactions (as in Fig. 4.11, where Θ = 2.0), the Q function develops therotating asymmetries, but it never (up to time t = 200t 0 ) fully moves off the bottom ofthe sphere, leading only to oscillations of small amplitude.4.4 Quasiprobability distributions using optical coherent statesThe technique of mapping operator equations onto stochastic phase-space equations iscommon in quantum optics.Different phase-space representations are used to deriveFokker-Planck equations from the master equation. This technique is the basis of the secondpart of this thesis, in which quantum features are fully simulated by way of quasiprobabilitydistributions. Here we apply the technique to the conditional master equation (Eq.(3.21)) to get a probabilistic semiclassical description.Exactly what structure in Hilbert space is mapped onto the phase-space equationsdepends on what overcomplete set of states is used to represent ˆρ c . The most commonlyused states are the coherent states ∣ ∣ α〉of the boson field.They have the advantageof producing correspondences between the annihilation (and creation) operators and thephase-space variables that are simple in form 1 . The (conditional) Glauber-Sudarshan Pfunction is defined by∫ˆρ c =P (α) ∣ ∣ α〉〈α∣ ∣d 2 α, (4.18)1 For an overview of the properties of coherent states and the phase-space representations constructedfrom them, see the fourth chapter of [62]93
4. Continuously monitored Bose condensates: quasiprobability distributionsFigure 4.9: Contour plots of the atomic Q function in polar coordinates, with the initial state∣ j, −j , for a medium strength measurement (Γ =16) and no atom collisions (Θ =0.00). The〉ztimes of successive plots are separated approximately by the tunnelling period: t n ≃ 2πn/Ω, andare printed in units of t 0 =1/Ω.1t=0.251t=6.51t=12.75θ/π0.5θ/π0.5θ/π0.50−1 0 1t=19 φ/π10−1 0 1t=25.25 φ/π10−1 0 1t=31.5 φ/π1θ/π0.5θ/π0.5θ/π0.50−1 0 1t=37.75 φ/π10−1 0 1t=44 φ/π10−1 0 1t=50 φ/π1θ/π0.5θ/π0.5θ/π0.50−1 0 1φ/π0−1 0 1φ/π0−1 0 1φ/π94
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Open Quantum Dynamics of Mesoscopic
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AcknowledgementsMy thanks must firs
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AbstractThe properties of an atomic
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CONTENTS4 Continuously monitored Bo
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List of Figures2.1 Two-mode approxi
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LIST OF FIGURES7.5 Angular momentum
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LIST OF ABBREVIATIONS AND SYMBOLSI{
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1. Condensation ‘without forces
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1. Condensation ‘without forces
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Chapter 2Properties of an atomic Bo
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2. Properties of an atomic Bose con
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Page 43 and 44: 2. Properties of an atomic Bose con
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Page 79 and 80: Chapter 4Continuously monitored con
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Page 107 and 108: Chapter 5Weak force detection using
Page 109 and 110: 5. Weak force detection using a dou
Page 123 and 124: Part IIQuantum evolution of a Bose
Page 125 and 126: 6. Quantum effects in optical fibre
Page 141 and 142: Chapter 7Quantum simulations of eva
Page 143 and 144: 7. Quantum simulations of evaporati
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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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