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

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B. Quasiprobability distributions using squeezed-state expansionsξ m that would minimise the spread in phase space. But not only do we not know thisoptimum value, we also have no guarantee that it exists as a constant independent of thesystem dynamics. In fact, we would expect the optimum value ξ m to be correlated toα (for the single-mode case the phase of ξ m must be such as to minimise the amplitudefluctuations). So we need a phase-space representation in which ξ is a variable:∫ˆρ = dαdα ∗ dξdξ ∗ P (α, α ∗ ,ξ,ξ ∗ ) ∣ ∣α, ξ 〉〈 α, ξ ∣ (B.11)To obtain evolution equations for ξ, we need to rewrite the effect of the creation andannihilations on the squeezed state in terms of partial derivatives with respect to ξ, inthe manner of Eq. (B.7). To prevent any higher order derivative terms from appearingin the evolution equation for the P distribution, we need direct operator correspondencesfor â 2 and â †2 . One may consider doing this via the squeezing operator S(ξ,ξ ∗ ). Thesimplest way to differentiate this operator (because [ â 2 , â †2] does not commute with â 2 orâ †2 )appearstobeL(ξ,ξ ∗ )S(ξ,ξ ∗ ) =(= 1 2ξ ∗ ∂∂ξ ∗ + ξ ∂ )S(ξ,ξ ∗ )∂ξ(ξ ∗ â 2 − ξâ †2) S(ξ,ξ ∗ ),(B.12)which is not useful, since we cannot generate operator correspondences separately for â 2and â †2 . The exponent of S(ξ,ξ ∗ ) can be factored to find ∂ ∂ξ S andlike[156]∂∂ξ ∗ S using resultse ξ∗Â−ξÂ† = e −ξÂ† e − ln cosh r ˆBe ξ∗Â ,(B.13)where Â = 1 2â2 and ˆB =â † â + 1 2. But this does not appear to lead to anything moreuseful.Alternatively the phase of ξ can be locked into that of α to ensure an expansion inamplitude-squeezed states:ˆρ =∫dαdα ∗ drP (α, α ∗ ,r) ∣ ∣ α, r〉〈α, r∣ ∣.(B.14)This reduces the phase-space dimensionality, but has complications of its own: sinceS(ξ,ξ ∗ ) −→ S(α, α ∗ ,r), the correspondences for â and â † must be rederived.Thus there are many ways in which generalised phase-space representations couldbe generated. However, finding one which gives exact phase-space equations (regardlessof whether these give improved statistics) is not a trivial matter, without having thesimplicity of coherent states.175
Bibliography[1] Agrawal, G. P. Nonlinear Fiber Optics, second ed. Academic Press, 1995.[2] Anderson, B. P., and Kasevich, M. A. Macroscopic quantum interference fromatomic tunnel arrays. Science 282 (Nov. 1998), 1686–1689.[3] Anderson, M. H., Ensher, J. R., Matthews, M. R., Weiman, C. E., andCornell, E. A. Observations of Bose-Einstein condensation in a dilute atomicvapor. Science 269, 5221 (July 14 1995), 198–201.[4] Andrews, M. R., Townsend, C. G., and Ketterle, W. Observation of interferencebetween two Bose condensates. Science 275, 5300 (Jan. 31 1997), 637.[5] Anglin, J. R., and Zurek, W. H. Vortices in the wake of rapid Bose-Einsteincondensation. Physical Review Letters 83, 9 (Aug. 30 1999), 1707–1710.[6] Arecchi,F.T.,Courtens,E.,Gilmore,R.,andThomas,H.Atomic coherentstates in quantum optics. Physical Review A 6, 6 (Dec. 1972), 2211–2236.[7] Baboiu, D.-M., Mihalache, D., and Panoiu, N.-C. Combined influence ofamplifier noise and intrapulse Raman scattering on the bit-rate limit of optical fibercommunication systems. Optics Letters 20, 18 (Sept. 18 1995), 1865–1867.[8] Berg-Sørensen, K. Kinetics for evaporative cooling of a trapped gas. PhysicalReview A 55, 1 (Feb. 1997), 1281–1287.[9] Bergman, K., Haus, H. A., and Shirasaki, M. Applied Physics B 55, 3 (Sept.1 1992), 242.[10] Bernstein, L., Eilbeck, J. C., and Scott, A. C. The quantum theory of localmodes in a coupled system of nonlinear oscillators. Nonlinearity 3 (1990), 293–323.176
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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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Chapter 4Continuously monitored con
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Chapter 5Weak force detection using
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Part IIQuantum evolution of a Bose
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Page 179 and 180: BIBLIOGRAPHY[22] Carter,S.J.Quantum
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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
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