## Functional methods in the generalized Dicke model |

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annihilation operators approach with functional assume the rotating-wave atomic pseudo-spin operators bilinear combination Bose bosonic mode collective bosonic excitations combination of Fermi commutation relations constants g counter-rotating terms respectively creation and annihilation critical temperature critical transition temperature define the fermion different coupling constants discussed energy eigenstates energy gap equation Euclidean action evaluate the critical Fermi operators fermion Dicke model field mode functional integration method functional method gapless Gaussian Gaussian integral given by Eq Goldstone mode Hamiltonian describing harmonic oscillators Hilbert space identical qubits identical two-level atoms interaction Hamiltonian Jaynes-Cummings model Let us define matrix model describing partition function path integral approach Popov and Fedotov present the spectrum quantized bosonic field quantum coherence quantum computers quantum phase transition qubit coupled reservoir of harmonic rotating rotating-wave approximation scalar field second quantization single mode single quantized mode situation superradiant phase tanh th qubit Hamiltonian thermodynamic quantities vacuum fluctuations virtual processes contribute