Abstract
The quasi-energy states have been found analytically for single quantum particles and an atomic Bose-Einstein condensate in a trap with periodically oscillating walls with a small modulation depth. A resonance is shown to exist as the modulation frequency approaches the difference of the frequencies corresponding to the levels in the unperturbed problem. Quasi-energy splitting and, accordingly, beats with a periodic population exchange between two levels in resonance have been found in the resonant case. Bistability of the response to trap size modulation, when the sustenance (depending on the initial conditions) of various quasi-energy states is possible under the same conditions, has been found for a Bose-Einstein condensate under resonance conditions.
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