The introduction of an asymmetric term into the quantum Rabi model generally lifts energy-level degeneracies. However, when the asymmetry parameter takes specific multiples of the bosonic mode frequency, level degeneracies reappear
−a phenomenon referred to as the hidden symmetry in the asymmetric quantum Rabi model. Identifying the origin of this hidden symmetry and its explicit operator form constitutes two central tasks in studying this system. Here, we investigate the origin of this hidden symmetry using the method of two successive diagonalizations, with a focus on physics in the regime where the ratio between the two-level splitting
Δ and the mode frequency
ω satisfies
Δ/ω≫1. We find that the hidden symmetry stems from energy-level matching within the asymmetric double-well potential, a picture strongly supported by the wavefunctions of both the ground and excited states. Moreover, the emergence of an excited-state quantum phase transition is identified and qualitatively discussed, which arises from the breaking and restoration of this hidden symmetry across different coupling regimes. Our results provide deeper insight into the physics of the asymmetric quantum Rabi model, particularly in the previously less-explored strong-coupling regime where
Δ/ω≫1.
The quantum Rabi model is a standard effective Hamiltonian in studies of light-matter interaction, capturing the simplest nontrivial setting in which a qubit couples to a single harmonic oscillator. Within the broader Rabi family, we focus on two special cases: the Jaynes-Cummings (JC) model, which carries an explicit
U(1) symmetry, and the asymmetric quantum Rabi model (AQRM), which possesses a parameter-dependent "hidden'' symmetry that appears only at integer bias,
ε/ω∈Z, and is not manifest in the Hamiltonian. We use the time-averaged entangling power as an operator-level diagnostic of these symmetry structures. Since the oscillator Hilbert space is infinite-dimensional, we compare two finite input ensembles: a Haar average after Fock-space truncation and a coherent-state average at fixed mean occupation
nˉ. Both diagnostics show peaks at the integer-bias points of the AQRM, where the hidden symmetry resides. In contrast, the manifest
U(1) symmetry at the JC point instead gives a weak dip. Thus, the time-averaged entangling power responds to both the hidden symmetry and the
U(1) symmetry in the Rabi family, with the sign of the response indicating how the symmetry reorganizes the spectral expansion. These results demonstrate that the entangling power can serve as an operator diagnostic to reveal the presence and properties of hidden and manifest symmetries in light-matter systems.
Abstract We study the two-qubit asymmetric quantum Rabi model (AQRM) and find its dark-state solution. Such solutions have at most one photon and constant eigenenergy in the whole coupling regime, causing level crossings in the spectrum, although there is no explicit conserved quantity except for energy. We find an operator in the eigenenergy basis to label all the degeneracies with its eigenvalues, and compare it with the well-known hidden symmetry which exists when bias parameter ϵ is a multiple of half of the resonator frequency ω . Extended to the multimode case, we find symmetries related with conserved bosonic number operators, which also cause level crossings. This paves the way for further symmetry studies on generalized Rabi models.
The asymmetric quantum Rabi model (AQRM) has a broken
Z2 symmetry, with generally a nondegenerate eigenvalue spectrum. In some special cases where the asymmetric parameter is a multiple of the cavity frequency, stable level crossings typical of the
Z2-symmetric quantum Rabi model are recovered, however, without any obvious paritylike symmetry. This unknown ``symmetry'' has thus been referred to as hidden symmetry in the literature. Here, we show that this hidden symmetry is not limited to the AQRM, but exists in various related light-matter interaction models with an asymmetric qubit bias term. Conditions under which the hidden symmetry exists in these models are determined and discussed. By investigating tunneling dynamics in the displaced oscillator basis, a strong connection is found between the hidden symmetry and selective tunneling.
We study the two-qubit asymmetric quantum Rabi model (AQRM) and find its dark-state solution. Such solutions have at most one photon and constant eigenenergy in the whole coupling regime, causing level crossings in the spectrum, although there is no explicit conserved quantity except energy. We find an operator in the eigenenergy basis to label all the degeneracies with its eigenvalues, and compare it with the well-known hidden symmetry which exists when bias parameter
ε is a multiple of half of the resonator frequency
ω. Extended to the multimode case, we find symmetries related with conserved bosonic number operators, which also cause level crossings. This provides a perspective for symmetry studies on generalized Rabi models.
In this paper, we uncover the elusive level crossings in a subspace of the asymmetric two-photon quantum Rabi model (tpQRM) when the bias parameter of qubit is an even multiple of the renormalized cavity frequency. Due to the absence of any explicit symmetry in the subspace, this double degeneracy at the crossing points implies the existence of the hidden symmetry. It is found that the number of the doubly degenerate crossing points in the asymmetric tpQRM is comparable to that in asymmetric one-photon QRM in terms of the same order of the constrained conditions. The bias parameter required for the occurrence of level crossings in the asymmetric tpQRM is characteristically different from that at a multiple of the cavity frequency in the asymmetric one-photon QRM, suggesting the different hidden symmetries in the two asymmetric QRMs.