Theoretical Investigation On Three-mode Approximation

Double well problem is one of the basic problems in theoretical physics. As one important method, two-mode approximation plays an important role to understand the behaviors governed by double well, whatever linear or nonlinear quantum system. Triple wells, as a bridge from double well to periodic wells, could be a simplest model and has attracted more attentions from various aspects, such as adiabatic transformation of electrons in quantum dots, nonlinear quantum dynamics of BECs with triple wells and quantum phase transition. In the point of quantum theory, all of those quantum phenomena could be understood based three modes approximation, which is anology with two-mode approximation for double wells. In this thesis, we are going to develop one kind analytical formula for three mode approximation, and then present one picture to understand the lowest three energy levels for triple wells.Firstly, we briefly reminded the double well and two modes approximation. As one application, the effect of the size of barrier on the dynamics and coupling strength between two weak linked BEC are investigated.Analogy with the two mode approximation, we analytically develop the three mode approximation. One transformation,analogy to two mode approximation, is presented for triple-square wells. The energy splitting is determined by the strength of the tunneling coupling between nearest neighbour wells, while the next-nearest neighbour tunnelling coupling plays crucial role to the invariant first excited state with the maximum entanglement states for the far separated square-wells.