Condensed exciton polaritons in a two-dimensional trap: Elementary excitations and shaping by a Gaussian pump beam

An exciton-polariton condensate (EPC) confined in a parabolic two-dimensional trap is considered theoretically. In the realistic limit of weakly interacting polaritons, the nonlinear term in the Gross-Pitaevskii equation describing the properties of the condensate can be considered as a perturbation with respect to the trapping potential, which allows for a convenient analytical description of the EPC ground state and Bogolyubov-type elementary excitations around it. The excitation modes with the energies and wave functions depending on the polariton-polariton coupling strength are derived for the condensate, neglecting interaction with uncondensed polaritons can be neglected. The energies of these modes are shown to be almost equidistant, even for a rather strong polariton-polariton interaction inside the condensate. This makes lateral parabolic traps promising candidates for realization of bosonic cascade lasers based on exciton polaritons. Another physical scenario is also considered where the interaction with a reservoir of uncondensed polaritons is more important than that inside the EPC. In this case, it is shown that the condensate is “reshaped” by the repulsive interaction with the reservoir, namely, pushed out from the center of the trap in real space and blue-shifted in energy, in agreement with the results obtained in a number of recent experiments.