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

Abstract

Abstract The quantum Rabi model (QRM) is widely recognized as an important model in quantum systems, particularly in quantum optics. The Hamiltonian H Rabi is known to have a parity decomposition H Rabi = H + ⊕ H −. In this paper, we give the explicit formulas for the propagator of the Schrödinger equation (integral kernel of the time evolution operator) for the Hamiltonian H Rabi and H ± 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 ±, 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 ± and give some of its basic properties.