Abstract
The Dicke model is derived in the contraction limit of a pseudo-deformation of the quasispin algebra in the su(2)-based Richardson-Gaudin models. Likewise, the integrability of the Dicke model is established by constructing the full set of conserved charges, the form of the Bethe Ansatz state, and the associated Richardson-Gaudin equations. Thanks to the formulation in terms of the pseudo-deformation, the connection from the su(2)-based Richardson-Gaudin model towards the Dicke model can be performed adiabatically.
Highlights
The interaction of a single quantized mode of electromagnetic radiation with a twostate system, such as a nuclear spin or two-level atom, can be modeled by means of the Rabi Hamiltonian [1]
The Dicke model is derived in the contraction limit of a pseudo-deformation of the quasispin algebra in the su(2)-based Richardson-Gaudin models
The integrability of the Dicke model is established by constructing the full set of conserved charges, the form of the Bethe Ansatz state, and the associated Richardson-Gaudin equations
Summary
The interaction of a single quantized mode of electromagnetic radiation (photons) with a twostate system, such as a nuclear spin or two-level atom, can be modeled by means of the Rabi Hamiltonian [1]. This contraction maps the su(2) quasispin directly onto a bosonic Heisenberg-Weyl algebra hw(1) It has been shown how the Bethe Ansatz states in the Richardson-Gaudin (RG) solution [14, 15] of the reduced Bardeen-Cooper-Schriefer (BCS) Hamiltonian for conventional superconductivity [16] can be connected adiabatically to a product state of generalized bosons [17] employing a pseudo deformation of the quasispin algebra. We will reassess the derivation of the conserved charges of the Dicke from those of the Richardson-Gaudin (RG) models in the framework of the pseudo-deformed quasispin algebra. The connection can be made consistently on the level of the Hamiltonian, the conserved charges, the Bethe Ansatz state, as well as the Bethe (or RichardsonGaudin) equations, and sheds light on how the Dicke model can be embedded within the larger class of Richardson-Gaudin integrable models. The RG equations are solved in the tractable full contraction limit (18), and are adiabatically brought into the original form (6) by tuning ξ → 1
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