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

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CONTENTS7 Quantum simulations of evaporatively cooled Bose condensates 1407.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.1.1 Important issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.1.2 Evaporative cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1427.2 Quantum simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1437.3 Phase-space equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1457.3.1 Positive-P technique . . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.3.2 Wigner technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.4 Computational method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1507.4.1 Constraints and limitations . . . . . . . . . . . . . . . . . . . . . . . 1527.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.5.1 Evolution of momentum distribution . . . . . . . . . . . . . . . . . . 1557.5.2 Wigner results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.5.3 Angular momentum and vortices . . . . . . . . . . . . . . . . . . . . 1597.5.4 Other trap geometries and cooling schemes . . . . . . . . . . . . . . 1617.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1667.6.1 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167A Stochastic calculus in outline 169A.1 The Fokker-Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.2 The Wiener process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169A.3 Ito and Stratonovich stochastic calculi . . . . . . . . . . . . . . . . . . . . . 170B Quasiprobability distributions using squeezed-state expansions 172B.1 Squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172B.2 Squeezed-state P functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1747
List of Figures2.1 Two-mode approximation for a condensate in a double-well potential. . . . 222.2 Period and amplitude of the semiclassical solutions. . . . . . . . . . . . . . . 282.3 Semiclassical evolution of the mean position. . . . . . . . . . . . . . . . . . 302.4 Bifurcation diagrams for fixed points of the semiclassical system. . . . . . . 312.5 Phase portraits of semiclassical dynamics in the double-well system. . . . . 332.6 Mean-field ground state for a double-well potential. . . . . . . . . . . . . . . 342.7 Evolution of the mean position calculated from the GP equation. . . . . . . 362.8 Spectrum of the tunnelling oscillations calculated from the GP equation. . . 372.9 Evolution of the atom density calculated from the GP equation. . . . . . . . 382.10 Tunnelling oscillations calculated from the GP equation with different strengthsof the nonlinear atom interactions. . . . . . . . . . . . . . . . . . . . . . . . 392.11 Quantum collapses and revivals in the tunnelling oscillations. . . . . . . . . 422.12 Energy spectra for condensates with N = 100 atoms. . . . . . . . . . . . . . 442.13 Energy spectrum for a N = 400 condensate. . . . . . . . . . . . . . . . . . . 442.14 Condensate fraction versus temperature for N = 100. . . . . . . . . . . . . . 482.15 Condensate fraction versus temperature for N = 400. . . . . . . . . . . . . . 492.16 Ground-state spatial density distribution ρ = 〈 ˆψ† (x) ˆψ(x) 〉 . . . . . . . . . . 512.17 Plots of the moment 〈〉Ĵ z /N in the ground state. . . . . . . . . . . . . . . . 512.18 Spatial second-order correlation function G 2 (x). . . . . . . . . . . . . . . . . 532.19 Evolution in a noisy double-well potential, in the absence of collisions. . . . 562.20 Evolution in a noisy potential, with collisions included. . . . . . . . . . . . . 572.21 Long-time simulations of a condensate in a noisy double-well potential. . . . 583.1 The homodyne detection scheme. . . . . . . . . . . . . . . . . . . . . . . . . 618
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: CONTENTS4 Continuously monitored Bo
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
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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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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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