Analytic Solution of the Rabi model

Abstract

The Rabi model (Rabi, Phys Rev 51:652–654, 1937) describes the simplest interaction between a cavity mode with a frequency ω and a two-level system with a resonance frequency ω 0. The model is characterized by the Hamiltonian (Rabi, Phys Rev 51:652–654, 1937; Schweber, Ann Phys 41:205–229, 1967) $$\displaystyle{ \hat{H}_{R} = \hslash \omega 1\!\!1 \hat{a}^{\dag }\hat{a} + \hslash g\sigma _{ 1}(\hat{a}^{\dag } +\hat{ a}) +\mu \sigma _{ 3}, }$$ (30.1) where \(\hat{a}\) and \(\hat{a}^{\dag }\) are the conventional boson annihilation and creation operators satisfying commutation relation \([\hat{a},\hat{a}^{\dag }] = 1\), g is a coupling constant, \(\mu = \hslash \omega _{0}/2\), \( 1\!\!1 \) is the unit matrix, σ j are the Pauli matrices in their standard representation, and we set the reduced Planck constant \(\hslash = 1\). In the Bargmann space of entire functions (Bargmann, Commun Pure Appl Math 14:187–214, 1961), the eigenfunctions of the Rabi model can be determined in terms of an entire function $$\displaystyle{ \upvarphi (z) =\sum _{ n=0}^{\infty }\phi _{ n}z^{n}. }$$ (30.2)