Phase diagram of microcavity polariton condensates with a harmonic potential trap

We theoretically explore the phase transition in inhomogeneous exciton-polariton condensates with variable pumping conditions. Through Bogoliubov excitations to the radial-symmetric solutions of complex Gross-Pitaevskii equation, we determine not only the bifurcation of stable and unstable modes by the sign of fluid compressibility but also two distinct stable modes which are characterized by the elementary excitations and the stability of singly quantized vortex. One state is the quasi-condensate BKT phase with Goldstone flat dispersion; the other state is the localized-BEC phase which exhibits linear-type dispersion and has an excitation energy gap at zero momentum.