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

Communications in Theoretical Physics

Solutions and quantum dynamics of the quantum Rabi model with A-square terms

Xiang-You Chen, Tian Ye, Qing‐Hu Chen

Published Dec 5, 2024• 1 citation
Publisher

Abstract

Abstract It is well known that the A-square term must be considered in both cavity and circuit quantum electrodynamics systems, because it arises in the derivation from the minimal coupling Hamiltonian at any finite coupling strength. In this paper, we study the quantum Rabi model with A-square terms using the Bogoliubov operator approach. After a unitary transformation, the A-square terms can be eliminated, resulting in a modified quantum Rabi model with renormalized parameters. A transcendental function responsible for the exact solution is then derived. The presence of the A-square terms is found to significantly alter the energy spectrum. The dynamics are also studied using the obtained exact wave function, which is sensitive to the strength of the A-square terms at strong coupling. We believe that these results could be observed in future light–matter interaction systems in the ultra-strong and deep strong coupling regimes.

DOI
10.1088/1572-9494/ad9a89

Related papers

Exact solution for the quantum Rabi model with the $\boldsymbol{\mathsf{A}}^{2}$ term

И. Д. Феранчук, N. Q. San, A. U. Leonau +1 more

Quantum Rabi model (QRM) is widely used for the analysis of the radiation-matter interaction at the fundamental level in cavity quantum electrodynamics. Typically the QRM Hamiltonian includes only $\boldsymbol{\mathsf{p}} \cdot \boldsymbol{\mathsf{A}}$ term, however, the complete nonrelativistic Hamiltonian of quantum electrodynamics includes $\boldsymbol{\mathsf{A}}^{2}$ term as well. Here we find an exact solution and demonstrate with the help of the exact canonical transformations that the QRM Hamiltonian with the $\boldsymbol{\mathsf{A}}^{2}$ term (QRMA) is reduced to the standard QRM model Hamiltonian with the renormalized frequency and the coupling constant and the eigenstates are expressed through the squeezed states of the field. As a result, the $\boldsymbol{\mathsf{A}}^{2}$ term qualitatively changes the behavior of the QRM with purely electromagnetic interaction in the strong coupling regime: the value of the ground state energy of an atom inside the cavity is higher than in vacuum and the number of crossing of energy levels with different quantum numbers decreases.

arXiv:2002.03702•Feb 10, 2020•0 citations

Quantum Rabi–Stark model: solutions and exotic energy spectra

You-Fei Xie, Liwei Duan, Qing‐Hu Chen

Abstract The quantum Rabi–Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling are present on an equal footing, are studied within the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived in a compact way, much simpler than previous ones obtained in the Bargmann representation. The zeros of transcendental functions reproduce completely the regular spectra. In terms of the explicit pole structure of these functions, two kinds of exceptional eigenvalues are obtained and distinguished in a transparent manner. Very interestingly, a first-order quantum phase transition indicated by level crossing of the ground state and the first excited state is induced by the positive nonlinear Stark-like coupling, which is however absent in any previous isotropic quantum Rabi models. When the absolute value of the nonlinear coupling strength is equal to twice the cavity frequency, this model can be reduced to an effective quantum harmonic oscillator, and solutions are then obtained analytically. The spectra collapse phenomenon is observed at a critical coupling, while below this critical coupling, infinite discrete spectra accumulate into a finite energy from below.

arXiv:1811.04431•Journal of Physics A Mathematical and Theoretical•Apr 26, 2019•39 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

Solutions to the mixed quantum Rabi model

Liwei Duan, You-Fei Xie, Qing‐Hu Chen

The analytically exact solutions to the mixed quantum Rabi model including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of $4 \times 4$ determinants responsible for the exact solutions are derived. These so-called $G$-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of $G$-functions reproduce completely the regular spectra. The exceptional eigenvalues can be also obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point $1/2$. Surprisingly, all energy levels only collapse to the lower one, which diverges negatively. All the level crossings in the unmixed models are relaxed to avoided crossings in the present mixed model due to absence of parity symmetry. Very interestingly, the effective one-photon coupling is enhanced in the presence of the two-photon coupling, which paves a highly efficient and economic way to access the deep-strong coupling regime even though the original one-photon coupling is weak.

•Jul 7, 2018•2 citations

Analog quantum simulation of the Rabi model in the ultra-strong coupling regime

Jochen Braumüller, Michael Marthaler, Andre Schneider +4 more

The quantum Rabi model describes the fundamental mechanism of light-matter interaction. It consists of a two-level atom or qubit coupled to a quantized harmonic mode via a transversal interaction. In the weak coupling regime, it reduces to the well-known Jaynes-Cummings model by applying a rotating wave approximation. The rotating wave approximation breaks down in the ultra-strong coupling regime, where the effective coupling strength g is comparable to the energy ω of the bosonic mode, and remarkable features in the system dynamics are revealed. Here we demonstrate an analog quantum simulation of an effective quantum Rabi model in the ultra-strong coupling regime, achieving a relative coupling ratio of g/ω ~ 0.6. The quantum hardware of the simulator is a superconducting circuit embedded in a cQED setup. We observe fast and periodic quantum state collapses and revivals of the initial qubit state, being the most distinct signature of the synthesized model.An analog quantum simulation scheme has been explored with a quantum hardware based on a superconducting circuit. Here the authors investigate the time evolution of the quantum Rabi model at ultra-strong coupling conditions, which is synthesized by slowing down the system dynamics in an effective frame.

arXiv:1611.08404•Nature Communications•Sep 27, 2017•170 citations

Quantum dynamics of two-photon quantum Rabi model

Zhiguo Lü, Chunjian Zhao, Hang Zheng

Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.

Journal of Physics A Mathematical and Theoretical•Dec 21, 2016•27 citations
Abstract It is well known that the A-square term must be considered in both cavity and circuit quantum electrodynamics systems, because it arises in the derivation from the minimal coupling Hamiltonian at any finite coupling strength. In this paper, we study the quantum Rabi model with A-square terms using the Bogoliubov operator approach. After a unitary transformation, the A-square terms can be eliminated, resulting in a modified quantum Rabi model with renormalized parameters. A transcendental function responsible for the exact solution is then derived. The presence of the A-square terms is found to significantly alter the energy spectrum. The dynamics are also studied using the obtained exact wave function, which is sensitive to the strength of the A-square terms at strong coupling. We believe that these results could be observed in future light–matter interaction systems in the ultra-strong and deep strong coupling regimes.
Exact solution for the quantum Rabi model with the A2\boldsymbol{\mathsf{A}}^{2}A2 term
Quantum Rabi model (QRM) is widely used for the analysis of the radiation-matter interaction at the fundamental level in cavity quantum electrodynamics. Typically the QRM Hamiltonian includes only p⋅A\boldsymbol{\mathsf{p}} \cdot \boldsymbol{\mathsf{A}}p⋅A term, however, the complete nonrelativistic Hamiltonian of quantum electrodynamics includes A2\boldsymbol{\mathsf{A}}^{2}A2 term as well. Here we find an exact solution and demonstrate with the help of the exact canonical transformations that the QRM Hamiltonian with the A2\boldsymbol{\mathsf{A}}^{2}A2 term (QRMA) is reduced to the standard QRM model Hamiltonian with the renormalized frequency and the coupling constant and the eigenstates are expressed through the squeezed states of the field. As a result, the A2\boldsymbol{\mathsf{A}}^{2}A2 term qualitatively changes the behavior of the QRM with purely electromagnetic interaction in the strong coupling regime: the value of the ground state energy of an atom inside the cavity is higher than in vacuum and the number of crossing of energy levels with different quantum numbers decreases.
Abstract The quantum Rabi–Stark model, where the linear dipole coupling and the nonlinear Stark-like coupling are present on an equal footing, are studied within the Bogoliubov operators approach. Transcendental functions responsible for the exact solutions are derived in a compact way, much simpler than previous ones obtained in the Bargmann representation. The zeros of transcendental functions reproduce completely the regular spectra. In terms of the explicit pole structure of these functions, two kinds of exceptional eigenvalues are obtained and distinguished in a transparent manner. Very interestingly, a first-order quantum phase transition indicated by level crossing of the ground state and the first excited state is induced by the positive nonlinear Stark-like coupling, which is however absent in any previous isotropic quantum Rabi models. When the absolute value of the nonlinear coupling strength is equal to twice the cavity frequency, this model can be reduced to an effective quantum harmonic oscillator, and solutions are then obtained analytically. The spectra collapse phenomenon is observed at a critical coupling, while below this critical coupling, infinite discrete spectra accumulate into a finite energy from below.
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.
The analytically exact solutions to the mixed quantum Rabi model including both one- and two-photon terms are found by using Bogoliubov operators. Transcendental functions in terms of 4×44 \times 44×4 determinants responsible for the exact solutions are derived. These so-called GGG-functions with pole structure can be reduced to the previous ones in the unmixed models. The zeros of GGG-functions reproduce completely the regular spectra. The exceptional eigenvalues can be also obtained by another transcendental function. From the pole structure, we can derive two energy limits when the two-photon coupling strength tends to the collapse point 1/21/21/2. Surprisingly, all energy levels only collapse to the lower one, which diverges negatively. All the level crossings in the unmixed models are relaxed to avoided crossings in the present mixed model due to absence of parity symmetry. Very interestingly, the effective one-photon coupling is enhanced in the presence of the two-photon coupling, which paves a highly efficient and economic way to access the deep-strong coupling regime even though the original one-photon coupling is weak.
The quantum Rabi model describes the fundamental mechanism of light-matter interaction. It consists of a two-level atom or qubit coupled to a quantized harmonic mode via a transversal interaction. In the weak coupling regime, it reduces to the well-known Jaynes-Cummings model by applying a rotating wave approximation. The rotating wave approximation breaks down in the ultra-strong coupling regime, where the effective coupling strength g is comparable to the energy ω of the bosonic mode, and remarkable features in the system dynamics are revealed. Here we demonstrate an analog quantum simulation of an effective quantum Rabi model in the ultra-strong coupling regime, achieving a relative coupling ratio of g/ω ~ 0.6. The quantum hardware of the simulator is a superconducting circuit embedded in a cQED setup. We observe fast and periodic quantum state collapses and revivals of the initial qubit state, being the most distinct signature of the synthesized model.An analog quantum simulation scheme has been explored with a quantum hardware based on a superconducting circuit. Here the authors investigate the time evolution of the quantum Rabi model at ultra-strong coupling conditions, which is synthesized by slowing down the system dynamics in an effective frame.
Abstract We apply a simple analytical method based on a unitary transformation to calculate the ground state, its excitation spectrum and quantum dynamic evolution of physical quantities for the double-photon quantum Rabi Hamiltonian over the wide coupling-strength range. The concise analytical method possesses the same mathematical simplicity as the approach of the rotating wave approximation (RWA). By quantitative comparison with the numerically exact result obtained by matrix diagonalization, we confirm that our calculated results obtained by transformed rotating-wave method are not only accurate in the weak coupling regime but also correct in intermediate strong-coupling case. In the intermediate ultrastrong-coupling regime, the calculated values of the ground state and lower lying excited states are nearly the same as the exact ones. It turns out that our calculation for the energy spectrum is beyond the ordinary-RWA. Meanwhile, we demonstrate the signatures resulting from the counter-rotating wave terms by monitoring the population, the coherence, the squeezing of the photon under the ultra-strong conditions. In particular, we find that when the frequency of the photon is much larger than the transition frequency of the system, the lineshape of the time evolution becomes complicated with the increase of the coupling strength, which may be verified experimentally.