Coherent Atomic Oscillations and Resonances between Coupled   Bose-Einstein Condensates with Time-Dependent Trapping Potential
  We study the quantum coherent-tunneling between two Bose-Einstein condensates separated through an oscillating trap potential. The cases of slowly and rapidly varying in time trap potential are considered. In the case of a slowly varying trap we study the nonlinear resonances and chaos in the oscillations of the relative atomic population. Using the Melnikov function approach, we find the conditions for chaotic macroscopic quantum-tunneling phenomena to exists. Criteria for the onset of chaos are also given. We find the values of frequency and modulation amplitude which lead to chaos on oscillations in the relative population, for any given damping and the nonlinear atomic interaction. In the case of a rapidly varying trap we use the multiscale expansion method in the parameter epsilon = 1/Omega, where Omega is the frequency of modulations and we derive the averaged system of equations for the modes. The analysis of this system shows that new macroscopic quantum self trapping regions, in comparison with the constant trap case, exist. 
F.Kh. Abdullaev and R.A. Kraenkel