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

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7. Quantum simulations of evaporatively cooled Bose condensatesFigure 7.9: A three-dimensional GP calculation with strongly interacting atoms, showing thetime evolution of the atom density in (a) Fourier space and (b) coordinate space. Time has beennormalised by t 0 =0.79ms, momentumbyk 0 =1.32 × 10 6 m −1 and position by x 0 =0.76µm.the total number of atoms finally is very small, due to the greater damping. Of course theabsorbing scalpel can be reversed or turned off before it cuts into the minimum uncertaintystate. But when that happens, the distribution spreads out again in both x and k space.This does not occur when the stochastic terms in the simulation are turned off, and somay be regarded as a spontaneous quantum heating effect, due perhaps to anticorrelationsin the atoms’ positions. Since the nonlinear (mean-field) term is quite small in thesesimulations, there is no bosonic-stimulation incentive to stay in the minimum uncertaintystate.Gross-Pitaevskii simulations indicate that a condensate might form in this situationif the interaction strength were much stronger. It is an indication only, because for theseparameters a mean-field treatment, in which spontaneous correlation effects are neglected,is not physically justified. Figure 7.9 gives the GP results for atoms with a scatteringlength of 60nm. A macroscopic occupation of the minimum uncertainty state occurs earlyin the simulation, which persists after the scalpel is turned off at t = 40. Whether theBose stimulation evident in this situation would overcome the quantum heating effect isuncertain without carrying out the fully quantum calculation.Truncating the noise terms in the positive-P equation like this in order to produce theGP equation cannot be justified for a first-principles calculation. The GP equation may be165
7. Quantum simulations of evaporatively cooled Bose condensatesregarded as the limit of the positive-P or Wigner equations when there are many particlesper mode. In this limit, the physical situation being simulated has changed, from one inwhich begins in an uncondensed state with on average less than one atom per mode, toone which starts in an already condensed state with many atoms distributed among a fewmodes. So instead of modelling evaporative cooling from an uncondensed to an condensedstate, the simulation models the relaxation of a condensate in some form of excited state(not a canonical ensemble in this case) to a zero-temperature ground state. This resemblesthe simulations presented in [114].Neglecting the noise terms in the phase-space equations (Eq. (7.4)) constrains a systemthat begins in a coherent state (or a mixture of coherent states) to remain so. If theassumption of many particles per mode is not made, then this fully coherent description isinconsistent with the usual formulation of the GP equation, which does take into accountcertain atom correlations. This is because a state which is diagonal in a coherent stateexpansion cannot include any atom correlations. However it may be possible to derivephase-space equations using an expansion in which atom correlations are included in thebasis. With such equations, atom correlations effects may still be retained even if the noiseterms are neglected.7.6 ConclusionsIn summary, in this chapter we have demonstrated a three-dimensional real-time quantumdynamical simulation of Bose condensation with mesoscopic numbers of interacting atomson a large lattice. Sampling errors and lattice size restrictions impose strong limitations onthese simulations. The main limitation preventing a description of current experiments isthe weak nonlinearity required by the method. Also, the possible appearance of boundaryterms in the positive-P distribution prohibits simulations over long times. The results showevidence for highly nonclassical behaviour in a first principles simulation of condensateformation. For certain situations, the simulations reveal a unique form of condensatewhich, as distinct from current experiments, has a relatively low atom density. The mainmechanism of condensate formation here would seem to be a spontaneous emission intothe condensate mode rather than bosonic stimulation. The final state in these cases, forwhich there is no canonical condensate, is far from thermal equilibrium. Spontaneous166
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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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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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Page 123 and 124: Part IIQuantum evolution of a Bose
Page 125 and 126: 6. Quantum effects in optical fibre
Page 141 and 142: Chapter 7Quantum simulations of eva
Page 143 and 144: 7. Quantum simulations of evaporati
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Page 171 and 172: A. Stochastic calculus in outlineco
Page 173 and 174: Appendix BQuasiprobability distribu
Page 175 and 176: B. Quasiprobability distributions u
Page 177 and 178: Bibliography[1] Agrawal, G. P. Nonl
Page 179 and 180: BIBLIOGRAPHY[22] Carter,S.J.Quantum
Page 181 and 182: BIBLIOGRAPHY[45] Drummond, P. D., C
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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