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Journal of Physics A Mathematical and Theoretical

Analytical solution and hidden symmetry operators of asymmetric two-mode quantum Rabi model

Zhan-Yuan Yan, Yifei Xia, Jurihui Li, Jinying Ma, Yan Liu

Published Mar 2, 2022• 7 citations
Publisher

Abstract

Abstract The asymmetric two-mode quantum Rabi model (tmQRM) breaks Z 2 -symmetry and also implies hidden symmetry. In this paper, the energy spectrum are analytically computed with Bogoliubov transformation and SU (1, 1) Lie algebra. The analytical results are in good agreement with those obtained by numerical calculation. We identify the condition for the bias parameter such that level crossings appear in the spectrum which are caused by the hidden symmetry. This condition is related not only to the field frequency, but also to the coupling strength between the field and qubit. The level crossing points could be determined by the pole energy and recurrence coefficients. This is tested and verified in the low energy levels. The generators of the hidden Z 2 -symmetry, commuting with the Hamiltonian, are derived within the Bogoliubov operator approach. The present results may provide more information to the hidden symmetry in the asymmetric tmQRM.

DOI
10.1088/1751-8121/ac5a22

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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 The true level crossing in the asymmetric quantum Rabi model without any obvious symmetry can be exhibited in the energy spectrum if the qubit bias is a multiple of the cavity frequency, which should imply the existence of the hidden symmetry. In this work, within a Bogoliubov operator approach, we can readily derive the symmetry operators associated with the hidden symmetry hierarchically for arbitrary multiples. The symmetry operators for small multiples in the literature can be extremely easily reproduced in our general scheme. In addition, a general parity operator is defined through the symmetry operator, which naturally includes the well-known parity operator of the symmetric model. We believe that the present approach can be straightforwardly extended to other asymmetric Rabi models to find the relevant symmetry operators.
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.
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.
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.