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

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3. Homodyne measurements on a Bose-Einstein condensateFor N = 100 atoms, χ>10 −3 s −1 should give detectable phase shifts (0.1 radians), andshould be experimentally feasible. For example, in terms of the incident intensity I on theatom, the atomic linewidth γ a , the total photon number N P , and the saturation intensityI s , the dipole coupling constant is[52]√Ig d = γ a . (3.5)N P I sFor a cavity of length l irradiated with light of frequency ω, the photon number is N P =Ilw 2 /(ωc). So the atom-light coupling strength becomesµ = ωcγ2 a4lw 2 . (3.6)∆I sIn the simulations given later in this chapter, the maximum measurement strength isχ ≈ 10 3 s −1 , which can be achieved by using the parameter values listed in Table 3.1, basedon the 5S 1 −→ 5P 3 transition in Rubidium[52]. The narrow width of the intracavity field22means that the power required to achieve the necessary coupling strengths is extremelysmall.The cavity is driven by a strong coherent field of strength ɛ and strongly damped atthe rate γ, and thus the cavity field is close to the coherent state α 0 =2iɛ/γ. Themasterequation for the whole system (cavity field plus condensate) is]]2 ˙ˆρ tot = −iΩ[Ĵz , ˆρ tot − i2κ[Ĵ x , ˆρ tot− iɛ]+ iχ[â † âĴx, ˆρ tot[â † +â, ˆρ tot]+ γ 2− i(δ − Nχ2 ) [â † â, ˆρ tot](2âˆρ tot â † − â † âˆρ tot − ˆρ tot â † â), (3.7)where the initial cavity detuning δ = Nχ/2ischosentoremovetheN-dependent lineardispersion.The optical field may now be adiabatically eliminated from the master equation [126,181], under the assumption that the driving and damping terms dominate the couplingterm. To do this, we first transform the master equation to a displacement picture usingthe displacement operator D(α 0 )=exp(αâ † − α ∗ â) which creates a coherent state ∣ 〉α0from the vacuum state:˙˜ρ = D † (α 0 ) ˙ˆρ tot D(α 0 )=[(S ˜ρ + iχ â † â + α ∗ 0â + α 0â † + |α 0 | 2) ]Ĵ x , ˜ρ + γ ()2â˜ρâ † − â † â˜ρ − ˜ρâ † â , (3.8)2where S incorporates the condensate dynamics. For large damping γ, then〈〉∣ S ∣∣∣∣ ∣ γ ∣ ∼ χ|α 0 | 〈〉Ĵ x γ ∣ = ɛ 0 ≪ 1, (3.9)63
3. Homodyne measurements on a Bose-Einstein condensateTable 3.1: Fundamental and derived parameters of the intracavity system.Quantity Value Descriptionr 0 10µm length scalingw 60µm beam widthl 10cm cavity lengthI s 17W/m 2 saturation intensityω/2π 3.8 × 10 14 Hz optical frequencyγ a /2π 6MHz atomic linewidth∆/2π 300MHz optical detuningP 5.7 × 10 −9 W incident powerI 3.2W/m 2 intracavity intensityN P 15 photon numberg d 4.2 × 10 6 s −1 dipole coupling constantγ 1.5 × 10 9 s −1 cavity damping rateɛ 4.1 × 10 9 s −1 cavity drivingχ 1.2 × 10 3 s −1 coupling strength64
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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
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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Page 79 and 80: Chapter 4Continuously monitored con
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Page 107 and 108: 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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