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Scientific Reports

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

Jinying Ma, Shaohui Xie, Linjing Liu, Luyao Lu, Zhanyuan Yan

Published Sep 8, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1038/s41598-026-70334-7

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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 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.
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.
We revisit the theoretical description of the ultrastrong light-matter interaction in terms of exactly solvable effective Hamiltonians. A perturbative approach based on polaronic and spin-dependent squeezing transformations provides an effective Hamiltonian for the quantum Rabi model up to the second order in the expansion parameter. The model consistently includes both rotating and counter-rotating terms, going therefore beyond the rotating-wave approximation. Analytical and numerical results show that the proposed Hamiltonian performs better than the Bloch-Siegert model when calculating operator averages (e.g., the mean photon number and number of excitations). This improvement is due to a refined calculation of the dressed states within the present model. Regarding the frequency shift induced by the qubit-photon interaction, we find a different sign from the Bloch-Siegert value. This influences the structure of the eigenstates in a nontrivial way and ensures the correct calculation of the number of excitations associated with a given dressed state. As a consistency check, we show that the exactly solvable independent boson model is reproduced as a special limit case of the perturbative Hamiltonian.
In this report, we have studied exceptional spectra of the two-qubit quantum Rabi model in two situations. Firstly, an exceptional spectra is achieved in resonant condition, in which the frequencies of two qubit and photon field satisfy resonant relation. With transformed rotating-wave approximation (TRWA) method, the Rabi model can be mapped into the solvable formation in qubit-photon Fock states space. Based on the quantum supperconducting circuits experiment setup, the best value range of the system parameters are discussed. In the resonant station working window, the energy spectrums are calculated. Secondly, a special quasi-exact solution of two qubits Rabi model in reservoir is achieved. The algebraic structure of Hamiltonian is analyzed in the photon number space, a closed quasi-exact eigenstates space is found, and the quasi-exact solution can be clearly found from the algebraic structure of Hamiltonian. The results could be used to test quantum Rabi model.