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)