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note
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At the time of writing this manuscript we found out that the problem of going beyond the 2-mode dynamics had been also addressed in M. Holland, J. Park, and R. Walser, cond-mat/0005062. The authors point out an effect of condensate depletion due to a thermal cloud. However, in their treatment they neglect all collisions.
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We do recover the 2-mode dynamics in the semiclassical approximation starting from the pure atomic condensate (very low temperature). However, as we see in Fig. 1, even in the case of a 100% initial occupation of the atomic condensate, the quantum initiation makes the 2-mode model invalid after ∼0.6 ms.
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