Heat kernel for the quantum Rabi model II: propagators and spectral determinants

TL;DR

This paper derives explicit formulas for the propagators of the quantum Rabi model's Hamiltonian and its parity components, linking spectral determinants to Braak's G-function and advancing understanding of the model's spectral properties.

Contribution

It provides explicit propagator formulas for the quantum Rabi model and its parity sectors, and relates spectral determinants to the Braak G-function through meromorphic continuation.

Findings

Explicit propagator formulas for H_Rabi and H_{±}

Spectral determinants linked to Braak G-function

Meromorphic continuation of spectral zeta functions

Abstract

The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian $H_{\text{Rabi}}$ is known to have a parity decomposition $H_{\text{Rabi}} = H_{+} \oplus H_{-}$. In this paper, we give the explicit formulas for the propagator of the Schr\"odinger equation (integral kernel of the time evolution operator) for the Hamiltonian $H_{\text{Rabi}}$ and $H_{\pm}$ by the Wick rotation (meromorphic continuation) of the corresponding heat kernels. In addition, as in the case of the full Hamiltonian of the QRM, we show that for the Hamiltonians $H_{\pm}$, the spectral determinant is, up to a non-vanishing entire function, equal to the Braak $G$-function (for each parity) used to prove the integrability of the QRM. To do this, we show the meromorphic continuation of the spectral zeta function of the Hamiltonians $H_{\pm}$ and give some of its basic properties.