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

Quantum phase transition in an effective three-mode model of interacting bosons