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

Journal of Physics B Atomic Molecular and Optical Physics

Berry phase of the Rabi model beyond the rotating-wave approximation

Wenwu Deng, Gao‐xiang Li

Published Nov 7, 2013• 8 citations
Publisher

Abstract

The Berry phase of the Rabi model beyond the rotating-wave approximation is investigated. By use of two unitary transformations, the analytical expressions of the Berry phase are obtained approximately for the Rabi model. The obtained analytical solutions reproduce accurately the high-accuracy numerical results of the Berry phase in the strong coupling regime. It is found that the Berry phase of the ground state is relevant to the second-order terms of the coupling strength, and the Berry phases of the first and second excited states are mainly relevant to the first-, second- and third-order terms of the coupling strength when the effect of the counter-rotating-wave terms is taken into account.

DOI
10.1088/0953-4075/46/22/224018

Related papers

Analytical Berry Phase in Quantum Rabi Model

Chen Qing-h

The quantum Rabi model is solved analytically with photonic Fock state. The effect of counterrotating wave terms are explicitly included in the analytical solutions. The concise analytical expression can be applied to the present-day accessible strong coupling regime. The Berry phase has been further obtained, which agrees quite well with the numerically exact one in a wide coupling regime.

Journal of Anhui Normal University•Jan 1, 2013•0 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

Geometric phase and non-adiabatic resonance of the Rabi model

Sijiang Liu, Zhiguo Lü, Hang Zheng

Abstract We investigate the effects of counterrotating terms on geometric phase and its relation to the resonance of the Rabi model. We apply the unitary transformation with a single parameter to the Rabi model and obtain the transformed Hamiltonian involving multiple harmonic terms. By combining the counter-rotating-hybridized rotating-wave method with time-dependent perturbation theory, we solve systematically time evolution operator and then obtain the geometric phase of the two-level system. Our results are beyond adiabatic approximation and rotating-wave approximation (RWA). Higher-order harmonic resonance happens when driving frequency is equal to higher-order subharmonic of the Rabi frequency. In comparison with numerically exact results, our calculated results are accurate over a wide range of parameters space, especially in higher-order harmonic resonance regimes. In these regimes we demonstrate geometric phases change dramatically while those of the RWA are smooth. The RWA is thoroughly invalid even if the driving strength is extremely weak. We find it is the higher-order harmonic terms that play an important role on the cyclic state and demonstrate the characters of geometric phase in higher-order harmonic resonance regime. We also present analytical formalism of the change rate of geometric phase and quasienergies, which agree well with numerically exact ones even in the strong driving case. The developed method can be applied to explore the dynamics of strongly driven qubits and physical properties of higher-order harmonic processes.

arXiv:2110.01158•Journal of Physics A Mathematical and Theoretical•Sep 24, 2021•3 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

Jaynes-Cummings model: What emerges first beyond the rotating-wave approximation?

S. M. He, Qing‐Hu Chen, Xuezao Ren +2 more

The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by extended Swain's ansatz with the conserved parity. The analytical approximations are then performed at different levels. The well-known rotating-wave approximation is naturally covered in the present zero and first approximations. The effect of the counter rotating-wave term emerges clearly in the second order approximation. The concise analytical expressions are given explicitly and can be applicable up to the ultra-strong coupling regime. The preliminary application to the vacuum Rabi splitting is shown to be very successful.

arXiv:1203.2410•Mar 12, 2012•0 citations
The Berry phase of the Rabi model beyond the rotating-wave approximation is investigated. By use of two unitary transformations, the analytical expressions of the Berry phase are obtained approximately for the Rabi model. The obtained analytical solutions reproduce accurately the high-accuracy numerical results of the Berry phase in the strong coupling regime. It is found that the Berry phase of the ground state is relevant to the second-order terms of the coupling strength, and the Berry phases of the first and second excited states are mainly relevant to the first-, second- and third-order terms of the coupling strength when the effect of the counter-rotating-wave terms is taken into account.
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.
Abstract We investigate the effects of counterrotating terms on geometric phase and its relation to the resonance of the Rabi model. We apply the unitary transformation with a single parameter to the Rabi model and obtain the transformed Hamiltonian involving multiple harmonic terms. By combining the counter-rotating-hybridized rotating-wave method with time-dependent perturbation theory, we solve systematically time evolution operator and then obtain the geometric phase of the two-level system. Our results are beyond adiabatic approximation and rotating-wave approximation (RWA). Higher-order harmonic resonance happens when driving frequency is equal to higher-order subharmonic of the Rabi frequency. In comparison with numerically exact results, our calculated results are accurate over a wide range of parameters space, especially in higher-order harmonic resonance regimes. In these regimes we demonstrate geometric phases change dramatically while those of the RWA are smooth. The RWA is thoroughly invalid even if the driving strength is extremely weak. We find it is the higher-order harmonic terms that play an important role on the cyclic state and demonstrate the characters of geometric phase in higher-order harmonic resonance regime. We also present analytical formalism of the change rate of geometric phase and quasienergies, which agree well with numerically exact ones even in the strong driving case. The developed method can be applied to explore the dynamics of strongly driven qubits and physical properties of higher-order harmonic processes.
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.
The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by extended Swain's ansatz with the conserved parity. The analytical approximations are then performed at different levels. The well-known rotating-wave approximation is naturally covered in the present zero and first approximations. The effect of the counter rotating-wave term emerges clearly in the second order approximation. The concise analytical expressions are given explicitly and can be applicable up to the ultra-strong coupling regime. The preliminary application to the vacuum Rabi splitting is shown to be very successful.