The quantum Rabi model (QRM) plays a central role in fundamental quantum optics, which describes a two-level system (qubit) interacting with a fully quantised light field. In the weak coupling regime ($g \ll 1$), counter-rotating terms can be neglected, and the QRM reduces to the celebrated Jaynes-Cummings model (JCM). The JCM agrees with most optical realisations very well. However, recent realisations of such systems on superconducting electrical circuits (circuit QED) can reach $g\sim 1$, where the JCM does not hold anymore. To explore possible applications of this regime, such as quantum computing, we need to study the QRM and its generalisations. For example, a bias term appears naturally in circuit QED systems, which results in the asymmetric QRM (AQRM). In this talk, we will study the AQRM from 3 aspects: the hidden $\mathbb{Z}_2$ symmetry in AQRM-related models, the exact solvability of the AQRM and the $\mathcal{PT}$-symmetric AQRM. 

The bias term in AQRM Hamiltonian breaks the $\mathbb{Z}_2$ parity symmetry in the QRM. However, it was observed (D. Braak 2011, \textit{etc.}) that level crossings are restored when the bias field takes particular values, indicating the presence of symmetry. As it can't be observed from the Hamiltonian directly, this symmetry was usually called "hidden symmetry" before V. Mangazeev \textit{et al.} constructed its symmetry operator in \textit{J. Phys. A 54 12LT01}. In the first part, based on this work, we will show how to calculate symmetry operators from our ansatz. Anisotropic AQRM/biased Dicke model will be the example for the 1-/multi-qubit model.

The (A)QRM spectra were solved by Braak in 2011, more than 70 years after the model was introduced to this field. Inspired by the method used in \textit{J. Phys. A 49 194002}, we try to re-derive the (A)QRM spectra from its monodromy data. In the second part, we will find the Painlev\'e equation associated with the (A)QRM using isomonodromic deformation and obtain the monodromy data. We managed to derive the $\tau$-function, hence the energies, at Judd points of the spectra.

C. Bender found in the 1990s that $\mathcal{PT}$-symmetric Hamiltonians can have real energies while not being Hermitian, where real-complex energy phase transitions occur at "exceptional points". The AQRM turns into $\mathcal{PT}$-symmetric when bias becomes purely imaginary ($\epsilon\rightarrow i\epsilon$). In the last part, we will explore the phase boundaries in the parameter space and propose approximations to calculate them at different parameter regions.

https://anu.zoom.us/j/89697093251?pwd=OElRVmVNanhoQmw0di9JbmE3WjZpZz09
Meeting ID: 896 9709 3251
Password: 767520