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arXiv:2010.02496 · Journal of Physics A Mathematical and Theoretical

The hidden symmetry of the asymmetric quantum Rabi model

Vladimir V. Mangazeev, Murray T. Batchelor, Vladimir V. Bazhanov

Published Feb 9, 2021• 39 citations
arXivHTMLPDF

Abstract

Abstract The asymmetric quantum Rabi model (AQRM) exhibits level crossings in the eigenspectrum for the values ϵ ∈ 1 2 Z of the bias parameter ϵ . Such level crossings are expected to be associated with some hidden symmetry of the model. The origin of this hidden symmetry is established by finding the operators which commute with the AQRM Hamiltonian at these special values. The construction is given explicitly for the first several cases ϵ = 0 , 1 2 , 1 and can be applied to other related light–matter interaction models for which similar level crossings have been observed in the presence of a bias term.

DOI
10.1088/1751-8121/abe426

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Abstract The asymmetric quantum Rabi model (AQRM) exhibits level crossings in the eigenspectrum for the values ϵ ∈ 1 2 Z of the bias parameter ϵ . Such level crossings are expected to be associated with some hidden symmetry of the model. The origin of this hidden symmetry is established by finding the operators which commute with the AQRM Hamiltonian at these special values. The construction is given explicitly for the first several cases ϵ = 0 , 1 2 , 1 and can be applied to other related light–matter interaction models for which similar level crossings have been observed in the presence of a bias term.
The asymmetric quantum Rabi model (AQRM) has a broken Z2{\mathbb{Z}}_{2}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{\mathbb{Z}}_{2}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.
Abstract The symmetry operators generating the hidden Z 2 symmetry of the asymmetric quantum Rabi model (AQRM) at bias ϵ ∈ 1 2 Z have recently been constructed by Mangazeev et al (2021 J. Phys. A: Math. Theor. 54 12LT01). We start with this result to determine symmetry operators for the N -qubit generalisation of the AQRM, also known as the biased Dicke model, at special biases. We also prove for general N that the symmetry operators, which commute with the Hamiltonian of the biased Dicke model, generate a Z 2 symmetry.
Abstract The true level crossings in a subspace of the asymmetric two-photon quantum Rabi model (tpQRM) have been observed when the bias parameter of qubit is an even multiple of the renormalized cavity frequency. Generally, such level crossings imply the existence of the hidden symmetry because the bias term breaks the obvious symmetry exactly. In this work, we propose a Bogoliubov operator approach for the asymmetric tpQRM to derive the symmetry operators associated with the hidden symmetry hierarchically. The explicit symmetry operators consisting of Lie algebra at low biases can be easily obtained in our general scheme. We believe the present approach can be extended for other asymmetric Rabi models to find the relevant hidden symmetry.
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.
It has been observed that the asymmetric quantum Rabi model (QRM), which does not possess any obvious symmetry, exhibits energy level crossings, which are often associated with symmetries. This observation suggests that there is in fact a symmetry in the asymmetric QRM, even though simple inspection of the model and its Hamiltonian does not reveal the nature of this symmetry. Here we present the results of numerical calculations on the energy eigenstates of the asymmetric QRM in an attempt to elucidate the nature of the symmetry. In particular, we note that the distribution of states in the Hilbert space among different symmetry classes is normally independent of system parameters and test whether this property holds for the asymmetric QRM. We find that it does not, which both helps explain why the symmetry is hidden and adds more intrigue as to what its nature might be.
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 εεε is a multiple of half of the resonator frequency ωωω. 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.