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

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A. Stochastic calculus in outlinewhile the Stratonovich calculus is equivalent to evaluating the increment at the centre ofthe time interval, which corresponds to an implicit algorithm.The Stratonovich calculus has the simplicity of conforming to the rules of standardcalculus, and the noise terms of Stratonovich equations can be regarded as the ideal limitof physical processes. However there is a mathematical convenience to the Ito calculus becauseat any particular time t, the dependent variable x(t) is independent of the stochasticincrement dW (t); standard calculus rules can be adapted by keeping terms that are secondorder in the differential dx and by using the results of Eq. (A.4).171
Appendix BQuasiprobability distributions usingsqueezed-state expansionsAs the numerical results from Ch. 7 indicate, the use of the coherent-state positive-Prepresentation in evaporative cooling simulations leads to phase-space excursions when theatoms are strongly interacting. Large phase-space excursions indicate that the positive-Pfunction is diffusing rapidly and that it may be developing finite boundary terms, leadingto the breakdown of the method.These phase-space excursions arise because the system is evolving in a state whichis not compactly described by a coherent-state expansion. Moments such as the Fourieratom density n(k) =a 1 (k)a ∗ 2 (k) becoming negative in single trajectories indicates that thestate description requires superpositions of coherent states.As a possible alternative basis set for phase-space representations, we now consider theset of squeezed states.B.1 Squeezed statesA general squeezed state for a singe mode is defined as:〉 ∣ 〉∣ ξ,α = D(α)S(ξ) ∣0 ,(B.1)in which the vacuum state is squeezed by S(ξ) and then displaced by D(α). The squeezingand displacement operators are defined respectively asS(ξ) = e 1 2 (ξ∗ â 2 −ξâ †2 )(B.2a)D(α) = e αâ† −α ∗â, (B.2b)172
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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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3. Homodyne measurements on a Bose-
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Chapter 4Continuously monitored con
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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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Page 123 and 124: Part IIQuantum evolution of a Bose
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Page 171: A. Stochastic calculus in outlineco
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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
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