The quantum Rabi model (QRM), describing a two-level system coupled to a quantum harmonic oscillator, is one of the simplest and most ubiquitous interaction models in quantum mechanics. Despite its simplicity, the QRM has found applications in various fields of physics. An exciting more recent realization is in circuit quantum electrodynamics (cQED), where an on-chip artificial atom (or a qubit) is coupled to a transmission line or a waveguide. The qubits in cQED systems contain a bias term, which breaks the Z_2 symmetry in the QRM. Therefore, the precise model realized in cQED systems is referred to as the asymmetric quantum Rabi model (AQRM).  

In this presentation, we explore theoretical methods to treat the AQRM and develop a complete approach to describe its energy landscape. For the ground state, we present a physically motivated variational wave function, the non-orthogonal qubit states expansion. The results show that the variational expansion is a significant improvement over the existing approximations. For excited states, we propose a generalized adiabatic approximation (GAA) to calculate the eigenvalues. The GAA is based on the similarity between the exact exceptional solutions and Laguerre polynomials. Importantly, the GAA correctly recovers the conical intersections in the energy landscape. The geometric phases around these conical intersections are calculated analytically. We also investigate the hidden symmetry of the AQRM in detail. The AQRM has a broken Z_2 symmetry, with generally a non-degenerate eigenvalue spectrum. In some special cases, stable level crossings typical of the Z_2 symmetric quantum Rabi model are recovered. This has thus been referred to as hidden symmetry in the literature. We show that this hidden symmetry is not limited to the AQRM, but exists in various related light matter interaction models. Finally, the equivalence between the generalized Poschl-Teller potential and the spectrum of the AQRM is also revealed.  

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