The heat kernel and spectral zeta function for the quantum Rabi model

Cid Reyes-Bustos, Masato Wakayama

Published 2019-06-23Version 1

The quantum Rabi model (QRM) is widely recognized as a particularly important model in quantum optics. It is considered to be the simplest and most fundamental system describing quantum light-matter interaction. Its Hamiltonian is known to have a $\mathbb{Z}_2$-symmetry, called parity. The purpose of the present paper is twofold. Firstly, we describe the heat kernel of the Hamiltonian explicitly using the Trotter-Kato product formula. To the best knowledge of the authors, this is the first explicit derivation of a closed formula of the heat kernel for any non-trivial interacting quantum system. Further, the heat kernel for this model is given by a two-by-two matrix of operators and is expressed as a direct sum of two heat kernels representing the parity decomposition. Secondly, we investigate basic properties of the spectral zeta function for the QRM (and with each parity) via the Mellin transform of the partition function of the QRM, that is, the trace of the integral operator defined by the heat kernel. These properties show the meromorphic continuation, describe special values at negative integers of the spectral zeta function, and form the basis for advancing potential studies and further number theoretic investigation. We expect that the methods developed in this paper may be applicable to other quantum interaction systems.