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

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2. Properties of an atomic Bose condensate in a double-well potentialand so the eigenstates of the Hamiltonian should be close the Ĵ 2 x eigenstates. The groundstate will then be close to the zero weight state ∣ j, 0 , with all other states being doubly〉xdegenerate. This ground state corresponds to an equal number of atoms in each well, andhence a minimum in the local atom density.Figure 2.12 shows the energy eigenvalues of the scaled HamiltonianĤ 2 = Ĥ2Ω = Ĵz +2 Θ N Ĵ x, 2 (2.52)for different values of the scaled interaction strength Θ. In Fig. 2.12(a), where Θ = 0.5, thelow-lying part of the spectrum is dominated by the eigenstates of Ĵz, with a characteristiclinear increase of the energy with the integer m labeling the sequence of states. In bothFigs. 2.12(b) and 2.12(c) (with Θ = 2 and Θ = 20 respectively), the upper part of thespectrum is modified by the presence of the Ĵ x 2 term, which produces the doublet structure.For a very large interaction strength, such as Θ = 100 as in Fig. 2.12(d), the eigenstatesapproach the doubly degenerate Ĵ 2 x eigenstates, with the energy increasing quadraticallywith the integer m labeling of eigenstates. The result for a larger condensate (400 atoms)with Θ = 2 is shown in Fig. 2.13. This shows a similar division of the spectrum into alinear lower half and an quadratic upper half.When Θ is neither in the limit of being very small or very large, approximate expressionsfor the ground state can be written down using perturbation theory. For example,when Θ > 1, the corrections to the ground state and its corresponding energy can befound as follows. The Hamiltonian is separated into unperturbing and perturbing parts:Ĥ 2 = Ĥ(0) + ˆV ,whereĤ(0) =2κĴ 2 x and ˆV =ΩĴz. Then the ground state and energy arewritten as expansions which go to first order in ˆV :∣ (1)〉 ∣E 0 = ∣E (0)〉 ∣0 +∆∣E (0)〉0(2.53a)E (1)0 = E (0)0 +∆E (0)0 , (2.53b)where ∣ ∣ E(0)0〉is the eigenvector of Ĥ (0) with eigenvalue E (0)0 . Substituting these expansionsin to the eigenvalue equation and equating terms of like order in ˆV gives(Ĥ − E(0) 0 )∆∣ ∣ (0)〉 (0)E 0 =(∆E 0 − ˆV ) ∣ (0)〉 E 0 . (2.54)Using the orthonormality of the ∣ (0)〉 ∣E m and assuming that ∆∣E (0)〉 ∣0 is orthogonal to ∣E (0)〉 0 ,we can substitute the expansion∆ ∣ (0)〉 ∑ ∣ 〉E 0 = B m ∣E (0)m(2.55)m≠043
2. Properties of an atomic Bose condensate in a double-well potentialFigure 2.12: Energy spectra, scaled by Ω, for N = 100 and different values of the atominteractionstrength. In (a) Θ=0.5, in(b)Θ=2,in(c)Θ=20,andin(d)Θ = 100.E6040200(a)E120(b)10080604020−20−40−600 20 40 60 80 100 120m12001000800600(c)0−20−40−600 20 40 60 80 100 120m6000500040003000(d)E4002000E200010000−2000 20 40 60 80 100 120m−10000 20 40 60 80 100 120mFigure 2.13: Energy spectrum, scaled by Ω, forN = 400 and Θ=2.500400300200E1000−100−2000 100 200 300 400 500m44
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 and 22: Chapter 2Properties of an atomic Bo
Page 23 and 24: 2. Properties of an atomic Bose con
Page 43: 2. Properties of an atomic Bose con
Page 62 and 63: 3. Homodyne measurements on a Bose-
Page 70: 3. Homodyne measurements on a Bose-
Page 79 and 80: Chapter 4Continuously monitored con
Page 81 and 82: 4. Continuously monitored Bose cond
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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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. . . but this book is already too
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