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

arXiv:1502.00487

Solutions to the anisotropic quantum Rabi model

Liwei Duan, Qing‐Hu Chen

Published Feb 2, 2015• 1 citation
arXivHTMLPDF

Abstract

In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called $G$-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.

DOI
10.48550/arxiv.1502.00487

Related papers

Multiple ground-state instabilities in the anisotropic quantum Rabi model

Xiang-You Chen, Liwei Duan, Daniel Braak +1 more

In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counterrotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding $G$-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities, and they are doubly degenerate. We find analytically that the ground state and the first excited state can cross several times, indicating multiple first-order phase transitions as a function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model, which are caused by a hidden symmetry.

arXiv:2101.12396•Physical Review A•Apr 9, 2021•24 citations

Generalized rotating-wave approximation for the two-qubit quantum Rabi model

Yuyu Zhang, Qing‐Hu Chen

The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.

arXiv:1410.1965•Physical Review A•Jan 8, 2015•37 citations

Analytical Solution for the Anisotropic Rabi Model: Effects of Counter-Rotating Terms

Guofeng Zhang, Han‐Jie Zhu

The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.

arXiv:1503.06538•Scientific Reports•Mar 4, 2015•22 citations

Analytical solution and spectral structure of the two-photon anisotropic Rabi-Stark model

Zhan-Yuan Yan, Jiaxin Cheng, Fangcheng Qiu +3 more

Abstract Since the realization of the strong coupling between light and matter in experimental setups, the quantum Rabi model and its generalized models describing the interaction between the boson field and the two-level system have attracted extensive interest again. The study of anisotropic generalized Rabi models enables us to better understand the novel physical properties of the interaction between light and matter in the ultra-strong and deep-strong coupling regions. In this work, the two-photon anisotropic Rabi-Stark model (tpARSM) is analytically solved by using the Bogoliubov operator approach and the su(1, 1) Lie algebra. We derive the G-function, whose zeros give the regular spectrum of the system. By studying the pole structure of the G-function and the coefficients in the function, exceptional solutions, including the first-order quantum phase transition points, doubly degenerate exceptional solutions and nondegenerate exceptional solutions, are obtained. By discussing the spectral structure, we give the conditions for the first-order quantum phase transition of tpARSM. Furthermore, we find that the property that all of the lowest doubly degenerate crossing points in the two-photon Rabi-Stark model have the same energy only holds for the special case of the tpARSM in which the anisotropy parameter is equal to 1. Finally, from the perspective of first-order quantum phase transitions, concise conditions for the ground state energy level to collapse to or escape from the collapse point for the tpARSM are presented. A good understanding of the tpARSM will lay a good foundation for studying the extended two-photon systems involving multiple levels and multiple bosonic modes, and even the relevant open quantum systems.

Physica Scripta•May 28, 2024•3 citations

Solution of two-qubit Rabi model with extended generalized rotating-wave approximation

Zhan-Yuan Yan, Peihua Qu, Bingbing Xu +2 more

The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.

Modern Physics Letters B•Feb 9, 2021•4 citations

Generalized quantum Rabi model with both one- and two-photon terms: A concise analytical study

You-Fei Xie, Liwei Duan, Qing‐Hu Chen

A generalized quantum Rabi Hamiltonian with both one- and two-photon terms has emerged in the circuit quantum electrodynamics system for a decade. The usual parity symmetry is broken naturally in the simultaneous presence of both couplings, which complicates analytical treatments, even in the rotating-wave approximations. In this paper, we propose an adiabatic approximation to this generic model by using Bogoliubov operators and obtain a very concise analytical solution for both eigenvalues and eigenstates. Although the adiabatic approximation is only exact in the vanishing limit of the qubit frequency, the results for some physical observables nevertheless agree well with the numerical ones in a wide parameter regime. In the rotating-wave approximations, we also derive an analytical eigensolution. Two dominant Rabi frequencies are found in the Rabi oscillations of this generalized model. We also apply the present analytical theory to the vacuum Rabi splitting. It is found that some different phenomena emerge just because of the presence of the additional two-photon coupling term.

arXiv:1806.05082•Physical Review A•Jan 3, 2019•31 citations
In this work, the anisotropic quantum Rabi model with different coupling strengths of the rotating-wave and counter-rotating wave terms is studied by using two kinds of extended coherent states (ECS). By the first kind of ECS, we can derive a so-called GGG-function, by which both the regular and exceptional solutions can be given. The exceptional solution are just corresponding to the crossing points of two energy levels with different parities, so is doubly degenerate. By the second kind of ECS, a general scheme for the eigensolutions is derived analytically in a unified way. The zero-order approximation is just the adiabatic approximation, and the first-order approximation is actually a generalized rotating-wave approximation. The algebraic formulae for the eigensolutions are given explicitly in two approximations. The generalized rotating-wave approximations work well in a wide range of two different coupling strengths and the qubit detunings.
In this work, the anisotropic variant of the quantum Rabi model with different coupling strengths of the rotating and counterrotating wave terms is studied by the Bogoliubov operator approach. The anisotropy preserves the parity symmetry of the original model. We derive the corresponding GGG-function, which yields both the regular and exceptional eigenvalues. The exceptional eigenvalues correspond to the crossing points of two energy levels with different parities, and they are doubly degenerate. We find analytically that the ground state and the first excited state can cross several times, indicating multiple first-order phase transitions as a function of the coupling strength. These crossing points are related to manifest parity symmetry of the Hamiltonian, in contrast to the level crossings in the asymmetric quantum Rabi model, which are caused by a hidden symmetry.
The generalized rotating-wave approximation (GRWA) is presented for the two-qubit quantum Rabi model. The analytical expressions in the zeroth-order approximation recover the previous adiabatic ones. The counter-rotating-wave terms can be eliminated by performing the first-order corrections. An effective solvable Hamiltonian with the same form as the ordinary RWA one is then obtained, giving very accurate eigenvalues and eigenstates. Energy levels in the present GRWA are in accordance with the numerical exact diagonalization ones in a wide range of coupling strengths. The population dynamics in the GRWA are in quantitative agreement with the numerical results and exhibit the absence of collapses clearly, revealing the effects of the counter-rotating wave.
The anisotropic Rabi model, which was proposed recently, differs from the original Rabi model: the rotating and counter-rotating terms are governed by two different coupling constants. This feature allows us to vary the counter-rotating interaction independently and explore the effects of it on some quantum properties. In this paper, we eliminate the counter-rotating terms approximately and obtain the analytical energy spectrums and wavefunctions. These analytical results agree well with the numerical calculations in a wide range of the parameters including the ultrastrong coupling regime. In the weak counter-rotating coupling limit we find out that the counter-rotating terms can be considered as the shifts to the parameters of the Jaynes-Cummings model. This modification shows the validness of the rotating-wave approximation on the assumption of near-resonance and relatively weak coupling. Moreover, the analytical expressions of several physics quantities are also derived, and the results show the break-down of the U(1)-symmetry and the deviation from the Jaynes-Cummings model.
Abstract Since the realization of the strong coupling between light and matter in experimental setups, the quantum Rabi model and its generalized models describing the interaction between the boson field and the two-level system have attracted extensive interest again. The study of anisotropic generalized Rabi models enables us to better understand the novel physical properties of the interaction between light and matter in the ultra-strong and deep-strong coupling regions. In this work, the two-photon anisotropic Rabi-Stark model (tpARSM) is analytically solved by using the Bogoliubov operator approach and the su(1, 1) Lie algebra. We derive the G-function, whose zeros give the regular spectrum of the system. By studying the pole structure of the G-function and the coefficients in the function, exceptional solutions, including the first-order quantum phase transition points, doubly degenerate exceptional solutions and nondegenerate exceptional solutions, are obtained. By discussing the spectral structure, we give the conditions for the first-order quantum phase transition of tpARSM. Furthermore, we find that the property that all of the lowest doubly degenerate crossing points in the two-photon Rabi-Stark model have the same energy only holds for the special case of the tpARSM in which the anisotropy parameter is equal to 1. Finally, from the perspective of first-order quantum phase transitions, concise conditions for the ground state energy level to collapse to or escape from the collapse point for the tpARSM are presented. A good understanding of the tpARSM will lay a good foundation for studying the extended two-photon systems involving multiple levels and multiple bosonic modes, and even the relevant open quantum systems.
The generalized rotating-wave approximation (GRWA) method is extended to the two-qubit quantum Rabi model. In the first-order approximation (one photon exchange), the Hamiltonian matrix in photon number space is simplified by introducing two variational parameters. However, the Hamiltonian matrix is not a diagonalizable matrix yet. Furthermore, by presenting a constraint condition on coupling strength and atomic transition frequency, the Hamiltonian matrix is simplified and an effective solvable Hamiltonian with block diagonal form is obtained. In the even and odd parity space, the energy spectra and eigenstates of the two-qubit quantum Rabi model are achieved analytically. Most of the energy spectra, especially the lower energy levels, agree well with the numerical exact results in ultra-strong coupling region, and the ground state wave function can gives a fairly accurate result of mean photon number.
A generalized quantum Rabi Hamiltonian with both one- and two-photon terms has emerged in the circuit quantum electrodynamics system for a decade. The usual parity symmetry is broken naturally in the simultaneous presence of both couplings, which complicates analytical treatments, even in the rotating-wave approximations. In this paper, we propose an adiabatic approximation to this generic model by using Bogoliubov operators and obtain a very concise analytical solution for both eigenvalues and eigenstates. Although the adiabatic approximation is only exact in the vanishing limit of the qubit frequency, the results for some physical observables nevertheless agree well with the numerical ones in a wide parameter regime. In the rotating-wave approximations, we also derive an analytical eigensolution. Two dominant Rabi frequencies are found in the Rabi oscillations of this generalized model. We also apply the present analytical theory to the vacuum Rabi splitting. It is found that some different phenomena emerge just because of the presence of the additional two-photon coupling term.