Normal modes and stability of phase separated trapped Bose-Einstein condensates

We study the normal modes and the stability of two component condensates in a phase separated regime. In such a regime the system can undergo a quantum phase transition upon the change of interaction strength between bosons of the same species or the variation of the trap frequencies. In this transition, the distribution of the two components changes from a symmetric to an asymmetric shape. We discuss the nature of the phase transition, the role of the interface tension and the phase diagram. The symmetric to asymmetric transition is the simplest quantum phase transition that one can imagine. We found new branches of normal modes which are specific for trapped multicomponent condensates and are analogous to the waves at the interface between two layers of immiscible fluids under gravity. At the point of the phase transition the frequencies of those modes go to zero and become imaginary which causes an instability. The interface tension shifts the normal mode frequencies and changes the stability region of the system.