Calculation of mode coupling for quadrupole excitations in a Bose-Einstein condenstate

In this paper, we give a theoretical description of resonant coupling between two collective excitations of a Bose-condensed gas on, or close to, a second-harmonic resonance. Using analytic expressions for the quasi-particle wave functions, we show that the coupling between quadrupole modes is strong, leading to a coupling time of a few milliseconds (for a TOP trap with radial frequency similar to100 Hz and similar to10(4) atoms). Using the hydrodynamic approximation, we derive an analytic expression for the coupling matrix element. These can be used with an effective Hamiltonian (that we also derive) to describe the dynamics of the coupling process and the associated squeezing effects.