  In this work we study an effective three-mode model describing interacting
bosons. These bosons can be considered as exciton-polaritons in a semiconductor
microcavity at the magic angle. This model exhibits quantum phase transition
(QPT) when the parameters of the corresponding Hamiltonian are continuously
varied. The properties of the Hamiltonian spectrum (e.g., the distance between
two adjacent energy levels) and the phase space structure of the thermodynamic
limit of the model are used to indicate QPT. The relation between spectral
properties of the Hamiltonian and the corresponding classical frame of the
thermodynamic limit of the model is established as indicative of QPT . The
average number of bosons in a specific mode and the entanglement properties of
the ground state as functions of the parameters are used to characterize the
order of the transition and also to construct a phase diagram. Finally, we
verify our results for experimental data obtained for a setting of
exciton-polaritons in a semiconductor microcavity.
