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

7. Quantum simulations of evaporatively cooled Bose condensatesto the next, thus increasing the sampling error in the calculations of the moments. Thenumerical algorithm may still converge, although the rapidly changing fields at long timesdo preclude fast simulations.The underlying problem here is with the phase-space representation. Highly nonclassicalstates are not well represented by the coherent states by which the positive-P expansionis defined: off-diagonal elements, where ψ 1 differs from ψ 2 , become large and the state canno longer be mapped onto a compact region of phase space. In fact, for longer times thedistribution may not vanish at the phase-space boundaries, leading to a breakdown of thepositive-P method. Other basis states may be used for the expansion, but the resultingphase-space equations will not be so simple as Eq. (7.4), if they exist at all. The upshot isthat for the positive-P technique to give a tractable numerical problem, relatively weaklyinteracting atoms must be considered, with scattering lengths much less than the usual s-wave scattering lengths for the alkali-metal atoms. A consequence of the weak nonlinearityis that, as revealed in the simulations, the dominant mechanism of condensate growth isspontaneous emission into the ground state, rather than the stimulated emission as foundin other studies[63].7.5 Numerical resultsFor the small-trap parameters used in the simulations, the effect of the stochastic termson the dynamics is very large. In fact the quantum fluctuations that these stochasticterms introduce are much larger than the initial thermal fluctuations, such that the initialfeatures of the distribution do not persist. This means that the choice of the initial stateof the system is not critical, and also that in order to determine properties of the finalquantum ground state of the system, the stochastic terms are vital. To illustrate this, wecan remove the quantum noise terms, so that the simulations are of the GP equation, withinitial conditions corresponding to a thermal state. For our parameters, these simulationsdid not show strong Bose condensation effects, in contrast to the fully quantum-mechanicalsimulations. This demonstrates the highly nonclassical nature of the early stages of Bosecondensation, in which spontaneous transitions to the lowest energy states clearly play animportant role.For the simulations shown in the figures, a 0 =0.6nm and the mass, corresponding153