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

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7. Quantum simulations of evaporatively cooled Bose condensatesto rubidium, is m =1.5 × 10 −25 kg, with the other parameters as in Table 7.1. Theseparameters correspond to relatively weakly interacting atoms, in order to reduce the samplingerror, discussed above, which increases rapidly with longer times and larger couplingconstants. All results are plotted in normalised units, with space scaled by x 0 =0.76µm,and time scaled by t 0 =0.79ms.7.5.1 Evolution of momentum distributionFigure 7.2 shows individual trajectories for the evolution of the stochastic fields using thesinusoidal profiles for the potential (V 1 ) and absorption (Γ 1 ). The trajectories correspondto (the real part of) the product n(k) = a 1 (k)a ∗ 2 (k), which when averaged gives theobserved momentum distribution. Figure 7.2(a) shows the result of including the nonlinearterm, but without the noise terms, which corresponds to a solution of the GP equation.The final momentum distribution is small, with no evidence of condensation, and is almostidentical to the case where there are no atom interactions at all. This contrasts with thecase shown in Figs. 7.2(b&c), which contains all noise terms and shows a final atomdensity that is quite narrow and tall, with a width corresponding to a temperature belowthe critical temperature for condensation.The peak final momentum state populationis much greater than one and also greater than that of the initial conditions.Thus itappears that spontaneous emission into what will be the condensate mode is much moreimportant, for these parameters and over this timescale, than the stimulated effect due tothe nonlinear term.This condensation effect is more pronounced in the three-dimensional case (Fig. 7.2(b&c))than in two dimensions (Fig. 7.2(d)), showing that the evaporative cooling process is moreefficient with the extra degree of freedom and the greater number of atoms that are present.As is usual in quantum mechanics, only the ensemble averages of the simulations havean operational meaning.Thus, while individual stochastic realisations have a definitecoherent phase, these phases are different for distinct stochastic realisations; the ensembleaverage has no absolute phase information. The average evolution of a two-dimensionalcondensate is shown in Fig. 7.3(a). Since this is an ensemble average, it corresponds to themeasurable momentum distribution. In this case, the condensate is only weakly occupied.Figure 7.3(b) shows the average over several three-dimensional trajectories. The finalpeak momentum density is a lot lower than in individual paths. Since the condensate does155
7. Quantum simulations of evaporatively cooled Bose condensatesFigure 7.2: Single-trajectory simulations of a three-dimensional Bose condensate, showing theatom density n(k) Fourier space versus time. Time has been normalised by t 0 =0.79ms andmomentum by k 0 =1.32 × 10 6 m −1 . The evolution in time of n(k) along one dimension is shownin (a) (b) and (d), while (c) gives a two-dimensional cross section at time t = 100t 0 . In (a), thestochastic terms are absent to give a solution to the GP equation, while (b) and (c) show the resultsof the full +P equations. For comparison, (d) shows the result for a two-dimensional simulation.156
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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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Page 107 and 108: Chapter 5Weak force detection using
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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
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Page 171 and 172: A. Stochastic calculus in outlineco
Page 173 and 174: Appendix BQuasiprobability distribu
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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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Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

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