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

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2. Properties of an atomic Bose condensate in a double-well potentialIn terms of these operators, the total atom number operator is∫ˆN = d 3 r ˆψ † (t, r), ˆψ(t, r). (2.3)The trapping potential is a double-well potential in which the interwell coupling occursalong the x-axis:V (r) =b ( x 2 − q02 ) 2 1 +2 mω2 t (y 2 + z 2 ), (2.4)where ω t is the trap frequency in the y − z plane and where b gives the strength of theconfinement in the x direction. This potential has elliptic fixed points at r 1 =+q 0 x,r 2 = −q 0 x, at which the linearised motion is harmonic with frequency ω 0 = q 0 (8b/m) 1/2 .For simplicity, we set ω t = ω 0 and scale the length by r 0 = √ /2mω 0 ,whichistheposition uncertainty in a harmonic oscillator ground state.The barrier height is thengiven by B =(ω 0 /8)(q 0 /r 0 ) 2 .A full analysis of the quantum dynamics resulting from the many-body HamiltonianEq. (2.1) is not tractable, however considerable insight can be gained within a two-modeapproximation or with a semiclassical picture. For a suitable choice of B, thetwolowestsingle-particle energy eigenstates will lie beneath the barrier, with the third eigenstateseparated by a relatively large energy gap. We will assume that the atoms interact sufficientlyweakly such that, near zero temperature, they have condensed into the two lowestsingle-particle states. This enables a treatment of the many-body problem with a twomodeapproximation, as Fig. 2.1 illustrates. The resulting model is sufficiently simple toenable an analytic solution to be found for the semiclassical equations, and to permit atractable numerical comparison of the semiclassical description with the full quantum dynamics.We will first derive the two-mode Hamiltonian before discussing the semiclassicaldynamics in Sec. 2.2.We can expand the potential around each minimum into local harmonic potentials:V (r) =Ṽ (2) (r − r j )+ ... , (2.5)where j =1, 2andṼ (2) (r) =4bq 2 0|r| 2 . (2.6)The local-mode solutions of the individual wells u j (r) arethenu j (r) = −(−1)j(2πr0 2) e − 1 3 4 (x−q 0) 2 +y 2 +z 2 )/r0 2 (2.7)421
2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.1: Two-mode approximation for a condensate in a double-well potential.c c+c c1 12 2+hΩ2 q 0u (x)1u (x)2with energy E 0 . They are approximately orthogonal with a correction to the orthogonalitygiven by the overlap between the modes of opposite wells:∫d 3 ru ∗ j(r)u k (r) = δ j,k +(1− δ j,k )e − 1 2 q2 0 /r2 0= δ j,k +(1− δ j,k )ɛ, j, k =1, 2. (2.8)The energy eigenstates of the global double-well potential may then be approximated aslinear combinationsu ± (r) ≈ 1 √2[u 1 (r) ± u 2 (r)] , (2.9)with corresponding eigenvalues E ± = E 0 ±R,and∫R = d 3 ru ∗ 1 (r)[V (r) − Ṽ (2) (r − r 1 )]u 2 (r). (2.10)The matrix element R, which is of order ɛ 1 , describes the coupling between the localmodes. The tunnelling frequency, Ω, between the two minima is then given by the energylevel splitting of these two lowest states:Ω= 2R = 3 8 ω 022q02r02e 1 2 q2 0 /r2 0 . (2.11)
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 and 8: CONTENTS4 Continuously monitored Bo
Page 9 and 10: List of Figures2.1 Two-mode approxi
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: Chapter 2Properties of an atomic Bo
Page 25 and 26: 2. Properties of an atomic Bose con
Page 62 and 63: 3. Homodyne measurements on a Bose-
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3. Homodyne measurements on a Bose-
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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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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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