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We study the normal modes and the stability of two component condensates in aphase separated regime. In such a regime the system can undergo a quantum phasetransition upon the change of interaction strength between bosons of the samespecies or the variation of the trap frequencies. In this transition, thedistribution of the two components changes from a symmetric to an asymmetricshape. We discuss the nature of the phase transition, the role of the interfacetension and the phase diagram. The symmetric to asymmetric transition is thesimplest quantum phase transition that one can imagine. We found new branchesof normal modes which are specific for trapped multicomponent condensates andare analogous to the waves at the interface between two layers of immisciblefluids under gravity. At the point of the phase transition the frequencies ofthose modes go to zero and become imaginary which causes an instability. Theinterface tension shifts the normal mode frequencies and changes the stabilityregion of the system.Comment: 17 pages, 10 eps figure

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