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

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4. Continuously monitored Bose condensates: quasiprobability distributionsFigure 4.2: Contour plots of the atomic Q function in polar coordinates, with the initial state∣ j, 0 ,forastrongmeasurement(Γ = 1600) and no atom collisions (Θ =0.00). The times of〉xsuccessive plots are separated approximately by the tunnelling period: t n ≃ 2πn/Ω, andareprintedin units of t 0 =1/Ω.1t=0.21t=6.41t=12.6θ/π0.5θ/π0.5θ/π0.50−1 0 1t=18.8 φ/π10−1 0 1t=25 φ/π10−1 0 1t=31.2 φ/π1θ/π0.5θ/π0.5θ/π0.50−1 0 1t=37.4 φ/π10−1 0 1φ/π0−1 0 1φ/πθ/π0.50−1 0 1φ/π85
4. Continuously monitored Bose condensates: quasiprobability distributionscentre, but for Θ = 0.50 (Fig. 4.4), the distribution shears drastically. Near the bottom ofthe Bloch sphere (θ = 0), the rotation around the Ĵz axis is at the tunnelling frequency Ω,but near the top, it is greatly retarded. This is consistent with the semiclassical dynamicalanalysis in Sec. 2.2.1, which showed that the tunnelling motion of trajectories near thesouth of the Bloch sphere was more robust in the presence of collisions than trajectoriesnear the top of the sphere. The resultant distribution lies almost parallel to the equatoras it rotates around the Ĵz axis, its elongation smearing out any definite phase in thetunnelling oscillations. The zig-zag motion is due to the nonlinear twisting effect near theĴ x poles.When the atom collisions are above the critical strength, as in Fig. 4.5, no rotationaround the vertical axis is evident. The band stays in place, but deforms and also expandsnear the base of the Bloch sphere. This is consistent with the suppression of tunnellingseen in the evolution of 〈 Ĵ x〉c .To obtain a clearer insight into how the measurement feedback affects the Q distribution,we can evolve the system under the measurement part of the Schrödinger equationwithout the tunnelling or collisional terms present. Our previous initial state ∣ ∣ j, 0〉x isstationary in this case, since it is an eigenstate of Ĵ x and Ĵ x 2 . So instead, we can beginwith the same state but have it rotated π/2 around the sphere, i.e. ∣ j, 0 . This state is〉ydistributed around a great circle with J y as its axis, as Fig. 4.6(a) shows. Under the influenceof a weak measurement, the band on one side grows at the expense of the other, as inFig. 4.6(b), until the Q function is located entirely on one side of the sphere. Which sidethis is varies from one run to another. After a longer time, or for a stronger measurementstrength, the Q function separates into two compact distributions, equally spaced aboveand below the equator (Fig. 4.6(c)). These distributions then form into crescent shapesbefore eventually joining to form a circular band around the Ĵx axis (Fig. 4.6(d)).This behaviour is an intuitive consequence of the conditional and nonlinear nature ofthemeasuredsystem. Thepresenceof 〈 Ĵ x in Eq. (3.25), which conditions the evolution〉con past measurement results, induces the localisation evident in Figs. 4.6(b&c); an imperfectmeasurement partially collapses the wave function around the measurement value,thereby narrowing the distribution. The diffusive spread around the Ĵx axis seen in Fig.4.6(d) betrays the presence of Ĵ 2 x in the stochastic Schrödinger equation, which generatesthis nonlinear precession.86
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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 35 and 36: 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
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6. Quantum effects in optical fibre
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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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