Macroscopic quantum states in Dicke model of [formula omitted]-symmetric non-Hermitian Hamiltonian and superradiant phase with imaginary atomic population

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We in this paper study the macroscopic quantum states of a PT-symmetric Non-Hermitian Hamiltonian for N quantum-dots of two imaginary-levels in an optical cavity. The eigenstate energies are obtained in terms of spin-coherent-state variational method, in which PT-symmetric but non-unitary operator is used to diagonalize the effective non-Hermitian Hamiltonian. Biorthogonal sets of the trial wave functions necessarily appear as a consequence of non-Hermitian Hamiltonian. In the normal phase of zero photon-number average energies are imaginary called the PT-symmetry broken state, while the atomic population is real showing the unstable nature of the system. The energy of superradiant phase (SP) is real called unbroken symmetry state, however the atomic population is pure imaginary. The electrons in SP oscillate between upper and lower levels stimulated by the cavity field. The quantum phase transition from broken to unbroken symmetry states occurs at critical values of the atom-field coupling and nonlinear atom-photon interaction.