Light-matter Interaction Models: Symmetry and Non-Hermiticity

Abstract

This thesis studies light-matter interaction models related to the quantum Rabi model (QRM) in two aspects: the hidden $\mathbb{Z}_2$ symmetry and non-Hermiticity, together with some investigation on the integrable boundary conditions of a statistical mechanical model generalising the six vertex model.

It is known that bias terms break their parity symmetries in asymmetric generalisations of QRM-related models. Based on observations of crossings restorations, it was suspected that hidden $\mathbb{Z}_2$ symmetries exist in these models. The method of calculating corresponding symmetry operators for the asymmetric QRM (AQRM) was constructed in \cite{Mangazeev2021}, and we propose ansatzes for AQRM-related models based on this result. Example symmetry operators are calculated for the anisotropic AQRM, the asymmetric Rabi-Stark model (ARSM), the anisotropic ARSM and the biased Dicke model. We also propose and prove the expression of symmetry operators with an arbitrary number of qubits for the biased Dicke model and its quadratic relation with the Hamiltonian. These results show that the presence of hidden $\mathbb{Z}_2$ symmetry is a general property in AQRM-related models, although its precise physical meaning is still undetermined.

The non-Hermitian part of this thesis is inspired by the work \cite{Lee2015} of $\mathcal{PT}$-symmetric semi-classical Rabi model. In this study, we construct the model by coupling a $\mathcal{PT}$-symmetric qubit with a bosonic field. We derive the spectrum of this PT-QRM and look at the $\mathcal{PT}$-phase boundaries. The $\mathcal{PT}$-phase boundaries are found to form an infinite number of exceptional surfaces (ESs) in 3-dimensional parameter space, which is not observed for any other models. We numerically calculate some of them and derive their approximations at small parameter regimes.

In the last part, we study the stochastic higher spin six vertex model and the $\mathcal{PT}$-symmetric QRM. To be more specific, we look at its integrable boundary conditions by solving the reflection equation. The stochastic R-matrix between a weight 1 and a weight $J$ particle was constructed in \cite{Kuniba2016} and admits a factorised form \cite{Bosnjak2016}. The boundary matrix $K$ is known for a weight 1 particle but not for higher weights. We find triangular solutions for the $K_J$-matrix, which correspond to only input/output allowed at boundaries. The general solution of $K$-matrix is also found as a double sum of $q$-Pochhammer symbols.

The thesis is organised as follows: in Chapter 1, we introduce some basics of quantum optics, including photons and bosonic operators, together with their realisations and applications. In Chapter 2, we review the QRM, its related models and some useful approximations for later use. The three topics mentioned above are presented in Chapters 3, 4 and 5, respectively. The concluding remarks are presented in Chapter 6.

This thesis is based on my following publications:\\ \textit{2022 Chinese Phys. B 31 014210}: Hidden symmetry operators for asymmetric generalized quantum Rabi models,\\ \textit{2021 J. Phys. A: Math. Theor. 54 325202}: Hidden symmetry in the biased Dicke model,\\ \textit{2019 Nuc. Phys. B, 945 114665}: Boundary matrices for the higher spin six vertex model,\\ \textit{In preparation}: Quantum Rabi model with $\mathcal{PT}$-symmetry. Show more