Shuyu did you browse in the arXiv? I saw there something about collective excitations, but I am not competent in such a coupling as you say. In your place I would browse in the arXiv under "cond-mat", find a couple of relevant articles, and contact the authors.

Did you try?

For instance, did you see this?

Article Langevin Equation of Collective Modes of Bose–Einstein Conde...

Or, did you see this?

Coupled-mode theory for Bose-Einstein condensates, https://arxiv.org/abs/cond-mat/9910185

(we consider a parabolic double-well potential and derive coupled-mode equations for the complex amplitudes of the BEC macroscopic collective modes.)

Dear Sofia,

Thank you so much for your suggestion. Yes, there are some papers on this topic. But most of them did not mention how to observe collective modes in experiments. Especially, how to measure the different mode components at the same time when they are superposition? This is a question about an experimental technique.

Dear Shuyu,

Are you shy? Ask the authors what is known to them about experiments.

Dear Sofia,

Good idea! I think I need to contact these authors.

Best wish