Quadrature fluctuation energy, effective Hamiltonians, quasi-particle modes and quantum phase transitions in the Rabi and Dicke models

Abstract

The main purpose of this article is to provide a simple physically motivated exact analytical procedure for determining effective Rabi and Dicke Hamiltonians, which have so far posed serious challenges in studies of quantum phase transition phenomena in the Rabi and Dicke models, where only the lowenergy components have been obtained through approximate unitary transformation, or perturbation methods to determine the critical coupling constant for quantum phase transition. The full effective Hamiltonian automatically reveals the expected scaling of field mode energy and quadrature components near the critical quantum phase transition point and is easily reorganized as a sum of correlated field and quasi-particle spinor modes. A fully quantized Lipkin-Meshkov-Glick Hamiltonian emerges as an effective form of the Dicke Hamiltonian.