Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physica Scripta

Hidden Lie algebraic structures and exact exceptional spectrum of the two-photon quantum Rabi model with asymmetric static bias

Qiongtao Xie

Published Aug 1, 2025• 1 citation
Publisher

Abstract

Abstract We systematically investigate the quadratic and linear Lie algebraic structures of the two-photon quantum Rabi model (TPQRM) with an asymmetric static bias in the coordinate representation. The exact existence conditions for the two distinct Lie algebras are analytically derived and confirmed numerically. These algebraic structures result in two classes of exact eigenenergies parameterized by integers and half-integers, respectively. Furthermore, we investigate the role of the static bias term in modifying the spectral properties of the TPQRM. The level crossings corresponding to exact integer energies remain robust against static bias perturbations, while those associated with half-integer energies undergo a characteristic transition from level crossings to anti-crossings. The derived exact wave functions provide explicit evidence for interaction-controlled phase segregation phenomena. Finally, we demonstrate the versatility of our analytical framework through successful applications to the TPQRM in two alternative Fock-Bargmann representations.

DOI
10.1088/1402-4896/adfca9

Related papers

Symmetry operators of the asymmetric two-photon quantum Rabi model

You-Fei Xie, Qing‐Hu Chen

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.

arXiv:2106.05817•Journal of Physics A Mathematical and Theoretical•Oct 7, 2022•3 citations

Double degeneracy associated with hidden symmetries in the asymmetric two-photon Rabi model

You-Fei Xie, Qing‐Hu Chen

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.

arXiv:2102.03944•Physical Review Research•Jul 16, 2021•19 citations

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

Zhan-Yuan Yan, Yifei Xia, Jurihui Li +2 more

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.

Journal of Physics A Mathematical and Theoretical•Mar 2, 2022•7 citations

Analytical solutions to asymmetric two-photon quantum Rabi model and its exceptional spectrum

Marzieh Baradaran, Luis Miguel Nieto, S. Zarrinkamar

Within the Segal-Bargmann representation, a generalized Rabi model is considered that includes both two-photon and asymmetric (bias) terms. It is shown that, through a suitable transformation, exact solutions can be obtained using the Bethe ansatz approach. Applying this approach to the meromorphic structure of the resulting fourth-order differential equation, arbitrary-state analytical solutions of the problem are presented for restricted values of parameters. Thereby, the exceptional spectrum of the qubit system is obtained. These exceptional points in the spectrum are of great experimental importance and directly associated with enhanced sensitivity and nontrivial dynamical behavior of the system since in the vicinity of these special parameter values, eigenstates become highly structured, making observables extremely sensitive to small changes in coupling or bias, an important feature for precise quantum control.

Annals of Physics•Jul 14, 2026•0 citations

Dark-state solution and symmetries of the two-qubit multimode asymmetric quantum Rabi model

Ze-Feng Lei, Junlong Tian, Jie Peng

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.

Europhysics Letters (EPL)•Aug 29, 2024•1 citation

Dark-state solution and symmetries of the two-qubit multimode asymmetric quantum Rabi model

Lei, Ze-Feng, Junlong Tian, Peng, Jie

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.

arXiv:2311.00174•Oct 31, 2023•0 citations
Abstract We systematically investigate the quadratic and linear Lie algebraic structures of the two-photon quantum Rabi model (TPQRM) with an asymmetric static bias in the coordinate representation. The exact existence conditions for the two distinct Lie algebras are analytically derived and confirmed numerically. These algebraic structures result in two classes of exact eigenenergies parameterized by integers and half-integers, respectively. Furthermore, we investigate the role of the static bias term in modifying the spectral properties of the TPQRM. The level crossings corresponding to exact integer energies remain robust against static bias perturbations, while those associated with half-integer energies undergo a characteristic transition from level crossings to anti-crossings. The derived exact wave functions provide explicit evidence for interaction-controlled phase segregation phenomena. Finally, we demonstrate the versatility of our analytical framework through successful applications to the TPQRM in two alternative Fock-Bargmann representations.
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.
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.
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.
Within the Segal-Bargmann representation, a generalized Rabi model is considered that includes both two-photon and asymmetric (bias) terms. It is shown that, through a suitable transformation, exact solutions can be obtained using the Bethe ansatz approach. Applying this approach to the meromorphic structure of the resulting fourth-order differential equation, arbitrary-state analytical solutions of the problem are presented for restricted values of parameters. Thereby, the exceptional spectrum of the qubit system is obtained. These exceptional points in the spectrum are of great experimental importance and directly associated with enhanced sensitivity and nontrivial dynamical behavior of the system since in the vicinity of these special parameter values, eigenstates become highly structured, making observables extremely sensitive to small changes in coupling or bias, an important feature for precise quantum control.
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.