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 A Mathematical and Theoretical

Quantum dynamics of two-photon quantum Rabi model

Zhiguo Lü, Chunjian Zhao, Hang Zheng

Published Dec 21, 2016• 27 citations
Publisher

Abstract

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.

DOI
10.1088/1751-8121/aa5537

Related papers

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 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

Variational generalized rotating-wave approximation in the two-qubit quantum Rabi model

Bin-Bin Mao, Liangsheng Li, Yimin Wang +4 more

We present an analytical method for the two-qubit quantum Rabi model. While still operating in the frame of the generalized rotating-wave approximation (GRWA), our method further embraces the idea of introducing variational parameters. The optimal value of the variational parameter is determined by minimizing the energy function of the ground state. Comparing with numerically exact results, we show that our method evidently improves the accuracy of the conventional GRWA in calculating fundamental physical quantities, such as energy spectra, mean photon number, and dynamics. Interestingly, the accuracy of our method allows us to reproduce the asymptotic behavior of a mean photon number in a large frequency ratio for the ground state and investigate the quasiperiodical structure of the time evolution, which the GRWA is incapable of predicting. The applicable parameter ranges cover the ultrastrong-coupling regime, which will be helpful to recent experiments.

arXiv:1904.09567•Physical Review A•Mar 18, 2019•16 citations

Extended transformed rotating-wave approximation for the two-photon quantum Rabi model

Jinying Ma, Shaohui Xie, Linjing Liu +2 more

We propose an extended transformed rotating-wave approximation (ETRWA) for the two-photon quantum Rabi model. This method retains the spin-dependent squeezing transformation and the qubit-frequency-dependent parameter-selection condition of the existing TRWA, but replaces its excitation-independent vacuum coefficients with exact photon-number-dependent diagonal and nearest-neighbour two-photon matrix elements. We compute the energy spectra, mean photon numbers, and eigenstate fidelities using the ETRWA, TRWA, and S-GRWA methods. Comparison with exact numerical results shows that the ETRWA improves the 2017 TRWA most clearly at intermediate coupling. The comparison with the S-GRWA is observable-, detuning-, and coupling-dependent. Near spectral collapse, the S-GRWA is superior because it retains the vanishing soft-mode scale, while the finite-squeeze ETRWA omits interblock processes needed for the softening of the continuum and higher excited states. The ETRWA retains closed $$2\times 2$$ diagonalizations, with an excitation-dependent finite-sum cost for generating their coefficients.

Scientific Reports•Sep 8, 2026•0 citations

Symmetric rotating-wave approximation for the generalized single-mode spin-boson system

Victor V. Albert, Gregory D. Scholes, Paul W. Brumer

The single-mode spin-boson model exhibits behavior not included in the rotating-wave approximation (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave approximation that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave approximation both off-resonance and at deep-strong coupling. The symmetric rotating-wave approximation allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime. Additionally, we symmetrize the generalized form of the rotating-wave approximation to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave approximation. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.

arXiv:1106.3800•Physical Review A•Oct 11, 2011•53 citations

Photon squeezing of the Rabi model

Yu Li-Xian, Qifeng Liang, Wang Li-Rong +1 more

Recent experiments about the circuit cavity quantum electrodynamics have realized the ultrastrong couplings between the artificial atom and the photon, in which the coupling strengths have the same order of the photon frequency. In such a regime, the well-known rotating wave approximation is invalid, and the system dynamics is thus governed by the Rabi model. In this paper, we investigate the photon squeezing of the Rabi model. We find numerically that with the increase of the atom-photon coupling strength, the photon squeezing does not increase linearly, but displays a maximum in the ultrastrong coupling regime. In addition, we also reveal that the photonsqueezing can be enhanced by the counter-rotating terms of the Rabi model. Our results are of benefit to preparing the required squeezing state of the photon in experiment.

Acta Physica Sinica•Jan 1, 2013•0 citations
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.
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.
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.
We present an analytical method for the two-qubit quantum Rabi model. While still operating in the frame of the generalized rotating-wave approximation (GRWA), our method further embraces the idea of introducing variational parameters. The optimal value of the variational parameter is determined by minimizing the energy function of the ground state. Comparing with numerically exact results, we show that our method evidently improves the accuracy of the conventional GRWA in calculating fundamental physical quantities, such as energy spectra, mean photon number, and dynamics. Interestingly, the accuracy of our method allows us to reproduce the asymptotic behavior of a mean photon number in a large frequency ratio for the ground state and investigate the quasiperiodical structure of the time evolution, which the GRWA is incapable of predicting. The applicable parameter ranges cover the ultrastrong-coupling regime, which will be helpful to recent experiments.
We propose an extended transformed rotating-wave approximation (ETRWA) for the two-photon quantum Rabi model. This method retains the spin-dependent squeezing transformation and the qubit-frequency-dependent parameter-selection condition of the existing TRWA, but replaces its excitation-independent vacuum coefficients with exact photon-number-dependent diagonal and nearest-neighbour two-photon matrix elements. We compute the energy spectra, mean photon numbers, and eigenstate fidelities using the ETRWA, TRWA, and S-GRWA methods. Comparison with exact numerical results shows that the ETRWA improves the 2017 TRWA most clearly at intermediate coupling. The comparison with the S-GRWA is observable-, detuning-, and coupling-dependent. Near spectral collapse, the S-GRWA is superior because it retains the vanishing soft-mode scale, while the finite-squeeze ETRWA omits interblock processes needed for the softening of the continuum and higher excited states. The ETRWA retains closed 2×22\times 22×2 diagonalizations, with an excitation-dependent finite-sum cost for generating their coefficients.
The single-mode spin-boson model exhibits behavior not included in the rotating-wave approximation (RWA) in the ultra and deep-strong coupling regimes, where counter-rotating contributions become important. We introduce a symmetric rotating-wave approximation that treats rotating and counter-rotating terms equally, preserves the invariances of the Hamiltonian with respect to its parameters, and reproduces several qualitative features of the spin-boson spectrum not present in the original rotating-wave approximation both off-resonance and at deep-strong coupling. The symmetric rotating-wave approximation allows for the treatment of certain ultra- and deep-strong coupling regimes with similar accuracy and mathematical simplicity as does the RWA in the weak-coupling regime. Additionally, we symmetrize the generalized form of the rotating-wave approximation to obtain the same qualitative correspondence with the addition of improved quantitative agreement with the exact numerical results. The method is readily extended to higher accuracy if needed. Finally, we introduce the two-photon parity operator for the two-photon Rabi Hamiltonian and obtain its generalized symmetric rotating-wave approximation. The existence of this operator reveals a parity symmetry similar to that in the Rabi Hamiltonian as well as another symmetry that is unique to the two-photon case, providing insight into the mathematical structure of the two-photon spectrum, significantly simplifying the numerics, and revealing some interesting dynamical properties.
Recent experiments about the circuit cavity quantum electrodynamics have realized the ultrastrong couplings between the artificial atom and the photon, in which the coupling strengths have the same order of the photon frequency. In such a regime, the well-known rotating wave approximation is invalid, and the system dynamics is thus governed by the Rabi model. In this paper, we investigate the photon squeezing of the Rabi model. We find numerically that with the increase of the atom-photon coupling strength, the photon squeezing does not increase linearly, but displays a maximum in the ultrastrong coupling regime. In addition, we also reveal that the photonsqueezing can be enhanced by the counter-rotating terms of the Rabi model. Our results are of benefit to preparing the required squeezing state of the photon in experiment.