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

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 Schrödinger 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.

arXiv comment22 pages, 2 figures. Sections of this article were originally part of earlier versions of arXiv:1906.09597 (up to [v4])

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Cited by in corpus (3)

• Remarks on the hidden symmetry of the asymmetric quantum Rabi model
• The heat kernel of the asymmetric quantum Rabi model
• General symmetry operators of the asymmetric quantum Rabi model