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

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7. Quantum simulations of evaporatively cooled Bose condensatesvortex formation in small evaporatively cooled condensates is also observed.Simulations of the now traditional form of evaporative cooling (using the RF scalpeland maintaining a high density) appear intractable with this method. This is due tothe large size of the stochastic terms, which must be included to treat accurately theinteratomic collisions. Removing the stochastic terms results in mean-field simulationsthat show considerable differences with the first-principles quantum calculations for theseparameters. The quantum simulations show that the phase-space distribution becomesquite broad, indicating that the corresponding quantum state evolves into superpositionsof distant coherent states. Calculations of higher-order correlations also suggest thatthe condensate is far from being in a coherent state. Thus the customary phase-spacerepresentations, which involve coherent-state expansions, are not the natural choice withwhich to treat this problem.7.6.1 Future directionsWe shall conclude this thesis with a discussion of a possible way forward. As we saw in Sec.4.6, other phase-space techniques besides those involving coherent states are possible. Anyovercomplete set of states could be used to generate an appropriate expansion, althoughthe utility of such a basis in generating phase-space equations is another matter. With theinitial choice of candidates so large, some physical insight can be helpful to find prospectivebasis sets.The Bogolubov transformation[107, 137], used to describe perturbations to the meanfield, is similar to a squeezing transformation. So perhaps squeezed states provide a closerfit to the true ground state, in which case an expansion of the density operator in termsof squeezed states[27, 155, 156] might produce more compact realisations in phase space.As an illustration of how squeezing enters the picture once we move beyond coherentstates, consider a weakly interacting condensate occupying a single mode of the trappingpotential. The annihilation operator for this mode can be expanded around a coherentstate:â = α + δâ. (7.25)The condensate will occupy the state which minimises the free energy[140]:ˆF =(E 0 − µ)â † â + κâ † â † ââ, (7.26)167
7. Quantum simulations of evaporatively cooled Bose condensateswhere µ is the chemical potential, which can be written in terms of the coherent amplitudeα = |α|e iθ as µ = E 0 +2κ|α| 2 . Substituting the expansion for â into Eq. (7.26) andretaining terms quadratic in δâ gives()ˆF ≃ κ −|α| 4 + |α| 2 (e 2iθ δâ † δâ † +2δâ † δâ + e −2iθ δâδâ(= κ −|α| 4 + |α| 2 ( ˆX)2 − 1) , (7.27)where the quadrature operators have been defined asˆX = δâe −iθ + δâ † e iθ (7.28a)(Ŷ = −iδ âe −iθ − iδâ † e iθ) .(7.28b)Thus Eq. (7.27) shows that minimising the free energy minimises the fluctuations in the ˆXquadrature, and hence maximises the fluctuations in the Ŷ quadrature. Now the numbervariance is∆n 2 = 〈 (â † â) 2〉 − 〈 â † â 〉 2( 〈X= |α| 2 2 〉 − 〈 X 〉 (2 〈X〉3 ) )+ O , (7.29)|α|which means that with minimised fluctuations in the ˆX quadrature, the ground state isamplitude squeezed.Appendix B overviews the properties of squeezed states and suggests how suitablephase-space representations can be generated from them. As these preliminary investigationsshow, the price to pay for going beyond coherent states is a higher-dimensionalphase space and an increase in complexity. Generalisation of the phase-space representationsinvolves extra parameters (the squeezing parameter ξ in this case), which must beeither chosen apriorior evolved through simulation to obtain a more compact phase-spacerepresentation. There are a myriad of possible choices for the type of representation, eachwith a large parameter space. No comprehensive survey of representations has been undertaken,so the future in this direction is open. The work presented in this thesis is onlya tentative beginning; but perhaps a seed has been sown that will one day reap a harvestof powerful multimode quantum dynamical simulation techniques.168
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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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3. Homodyne measurements on a Bose-
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Chapter 4Continuously monitored con
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Chapter 5Weak force detection using
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5. Weak force detection using a dou
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Page 123 and 124: Part IIQuantum evolution of a Bose
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Page 173 and 174: Appendix BQuasiprobability distribu
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Page 177 and 178: Bibliography[1] Agrawal, G. P. Nonl
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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Open Quantum Dynamics of Mesoscopic Bose-Einstein ... - Physics

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