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arXiv:2311.02670 · Journal of Physics A Mathematical and Theoretical

Rotating wave approximation and renormalized perturbation theory

Peng Wang, Erik Orvehed Hiltunen, John C. Schotland

Published Mar 20, 2026• 1 citation
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Abstract

Abstract The rotating wave approximation (RWA) plays a central role in the quantum dynamics of two-level systems. In this paper we derive corrections to the RWA using the renormalization group approach to asymptotic analysis. We study both the semiclassical and fully quantum Rabi models and compare our analytical results with numerical calculations in the ultrastrong coupling regime, where the atom-coupling is comparable to the driving frequency.

DOI
10.1088/1751-8121/ae551f

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Abstract The rotating wave approximation (RWA) plays a central role in the quantum dynamics of two-level systems. In this paper we derive corrections to the RWA using the renormalization group approach to asymptotic analysis. We study both the semiclassical and fully quantum Rabi models and compare our analytical results with numerical calculations in the ultrastrong coupling regime, where the atom-coupling is comparable to the driving frequency.
Ultrastrong coupling (USC) in the quantum Rabi model, characterized by the breakdown of the rotating-wave approximation (RWA) has become a topic of considerable interest and study. This critical reevaluation of the validity of the RWA concludes that the accepted definition of USC in terms of a fixed ratio of coupling to field frequency is inadequate. Connecting an improved spectral validity criterion with the derivation of the semiclassical limit predicts that the dynamical validity of the quantum RWA should be linked to that of the corresponding semiclassical model. This, however, is not supported by numerical calculations of coherent-state dynamics, which unambiguously demonstrate that spectral validity does not imply dynamical validity and reveal surprisingly complicated dependence on coupling and field amplitude.
Ultrastrong coupling (USC) in the quantum Rabi model, characterized by the breakdown of the rotating-wave approximation (RWA), has emerged as a topic of considerable interest and study. This critical reevaluation of the validity of the RWA concludes that the accepted definition of USC in terms of a fixed ratio of coupling to field frequency is inadequate. Connecting an improved spectral validity criterion with the derivation of the semiclassical limit suggests that the dynamical validity of the quantum RWA should be linked to that of the corresponding semiclassical model. This, however, is not supported by numerical calculations of coherent-state dynamics, which unambiguously demonstrate that spectral validity does not imply dynamical validity and reveal surprisingly complicated dependence on coupling and field amplitude.
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.
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 Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by an extended Swain ansatz with conserved parity. Analytical approximations are then performed at different levels. The well-known rotating-wave approximation (RWA) is naturally covered in the present zeroth- and first-order approximations. A first-order correction to the RWA can be obtained in a second-order approximation, by which the effect of the counter-rotating-wave term emerges clearly. Concise analytical expressions are given explicitly and can be applicable up to the ultrastrong-coupling regime. A preliminary application to vacuum Rabi splitting is shown to be very successful.
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.