Resonant Interactions Of Collective Modes In A Quasi-one-dimensional Attractive Bose-Einstein Condensate

The realization of Bose-Einstein condensate (Bose-Einstein condensate, BEC)in dilute atomic gases, it not only proved Einstein’s prediction, but also provides anexcellent opportunity for the study of condensed body and collective modes intrapped, weakly interacting many-body systems. Elementary excitations and theirinteractions are of primary important subjects in quantum many-body physics, andcollective modes are the most basic excitation modes in BEC. The interparticleinteraction may result in dramatic effects and many nonlinear excitations can appearin certain experimental conditions. The most spectacular experimental progress forthe nonlinear excitations achieved in BECs are including observation of solitons andvortices. On the other hand, there is growing interest on the mode-mode resonantinteractions of the excitations in trapped condensed Bose gases. In this thesis westudy the interaction of the elementary excitations generated in BECs.The study of the quasi-one-dimensional (Q1d) system is of primary importantsubject in BEC, which is relatively simple and easy to research. It has been shownthat, in the case of a quasi-one-dimension, the GP equation supports dark solitons fora repulsive interaction and bright solitons for an attractive interatomic interaction. Upto now, there appear many research studies of collective modes and their interactionsin BEC with repulsive interactions, such as second-harmonic generation (SHG) andthree-mode resonant interaction (TMRI). However, a similar study of BEC withattractive interactions is lacking.In this thesis, by mean-field frame, the resonant interaction between collectivemodes in a quasi-one-dimensional attractive Bose-Einstein condensate is investigated.We derive nonlinearly coupled amplitude equations using a method of multiplescales. By using mean-field approximation theory we analytically calculate thecoupling matrix elements describing mode-mode interactions based on theorthonormal and complete eigenfunctions of the collective modes. The couplingmechanisms of the collective modes have two types, namely, the Landau and Beliaevones. In particular, we investigate the SHG and TMRI of collective modes in Q1D（including attractive Q1D BEC and repulsive ones）BEC in detail. Our result showsthat the physics of the resonant mode couplings in attractive Q1D BEC is richer thanthat of repulsive Q1D BEC.