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

Physical Review A

Rotating-Wave Approximation in Coherent Interactions

P. L. Knight, L. Allen

Published Jan 1, 1973• 36 citations
Publisher

Abstract

Some consequences of the rotating-wave approximation (RWA) for spontaneous emission are explored. The Winger---Weisskopf theory of the spontaneous exponential decay of a single excited atom is shown to contain an implicit form of the RWA. However, to describe the cooperative superradiant level shift, the RWA should not be used, and care should be taken over the choice of interaction Hamiltonians.

DOI
10.1103/physreva.7.368

Related papers

Spectrum in spontaneous emission: Beyond the Weisskopf-Wigner approximation

P. R. Berman, G. W. Ford

The theory of spontaneous emission presented by Weisskopf and Wigner [V. Weisskopf and E. Wigner, Z. Phys. 63, 54 (1930)] provides an excellent approximation of the actual decay that atoms undergo on optically allowed transitions. However, the theory cannot be rigorously correct since it leads to a Lorentzian spectrum that extends to negative frequencies. Within the rotating-wave approximation, we derive a closed-form expression for the spectrum that is valid for all frequencies.

Physical Review A•Aug 27, 2010•12 citations

An exact analysis of spontaneous emission by a single two level atom in the rotating wave approximation II. Numerical results

S. Swain

For Pt. I see ibid. vol.5, p.1587(1972). The problem considered is that of spontaneous emission by a single, two level atom which interacts with N modes of the electromagnetic field in the electric dipole, rotating wave approximation. The equations derived in Pt. I are solved numerically for N = 4,5,10,11,20,21 and 201 to find the eigenvalues and various time dependent properties of the system. The connection between this approach and conventional Wigner-Weisskopf theory is discussed.

Journal of physics. A, Proceedings of the Physical Society. General•Nov 1, 1972•12 citations

Spontaneous emission in dipole approximation revisited

P. R. Berman, Peter W. Milonni

Spontaneous emission in dipole approximation is studied theoretically using both source-field theory and a Schrodinger picture approach. Using source-field theory we obtain formal equations for the Poynting vector and energy density without making the rotating wave approximation (RWA) and Weisskopf-Wigner approximation (WWA). The initial condition at t=0 is one in which the atom is in an excited state and the field in the vacuum state. The source-field expressions are evaluated within the the RWA and WWA and are found to satisfy Poynting's theorem. To explore the consequences of not making the RWA and WWA, the Poynting vector and energy density are calculated using perturbation theory. We use a Schrodinger picture approach and essentially reproduce and complement the results of Compagno, Passante, and Persico [J. Mod. Optics 37:8, 1377 (2007)] and those of Power and Thirunamachandran [Phys. Rev. A 45, 54 (1992)] obtained using a Heisenberg picture approach. The theory involves a sum over field mode frequencies and both finite cutoffs and convergence factors are used to carry out the sums. It is shown that the perturbation theory calculation leads to unphysical values for atomic state populations for all times when a sum over all field frequencies is taken, even if a convergence factor is used. It is also proved that the fields calculated using source-field theory always satisfy Poynting's theorem for ct not equal to R, where R is the distance from the atom.

arXiv:2507.14075•Physical Review A•Jan 26, 2026•1 citation

Simple and consistent spontaneous emission rate derivation with a physically justified frequency cutoff

Wojciech Górecki

The exact determination of the spontaneous emission coefficient for an excited atom is an extremely complex problem so various approximations are typically used. One of the most popular ones is the use of the dipole approximation of a two-level atom followed by rotating wave approximation (RWA). However, such an approach applied to the entire frequency spectrum results in the appearance of divergent integrals in the derivations, which are not treated rigorously in typical student textbooks. It is known from the literature that the introduction of cutoff for frequencies, justified by the finite size of the atom, may solve this problem. For didactic purposes, in this paper, we introduce a mathematically simple cutoff, which allows for a straightforward yet mathematically consistent rederivation of the Weisskopf-Wigner spontaneous emission rate (up to the small correction) within RWA. Importantly, this cutoff is not a mathematical trick to make calculations easier but is related to a real feature of the physical system, the neglect of which leads to inconsistency. More precise analysis demand going beyond RWA and dipole approximation.

arXiv:2303.02212•Mar 3, 2023•0 citations

Ultrasharp lines without the rotating wave approximation

Qing Yang Xu, Xiangming Hu, Jian-wu Yin

We study the effects of the non-rotating wave approximation (non-RWA) on the spontaneous emission of a Λ-type three-level atom, where the metastable states are coupled to a common excited state by a weak laser field and the lower-level doublet is driven by a strong microwave field. When the non-RWA is applied to the interaction of the atom with the microwave field, the spontaneous emission spectrum is comprised of a series of ultrasharp lines. The physical interpretation of the spectral characteristics is given in light of the dressed states.

Journal of Physics B Atomic Molecular and Optical Physics•Jul 4, 2008•0 citations

Quantifying the breakdown of the rotating-wave approximation in single-photon superradiance

Mads Anders Jørgensen, Martijn Wubs

Abstract We study quantitatively the breakdown of the rotating-wave approximation (RWA) when calculating collective light emission by quantum emitters, in particular in the weak-excitation limit. Our starting point is a known multiple-scattering formalism where the full light–matter interaction leads to induced inter-emitter interactions described by the classical Green function of inhomogeneous dielectric media. When making the RWA in the light–matter interaction, however, these induced interactions differ from the classical Green function, and for free space we find a reduction of the interatomic interaction strength by up to a factor of two. By contrast, for the corresponding scalar model the relative RWA error for the inter-emitter interaction even diverges in the near field. For two identical emitters, the errors due to the RWA in collective light emission will show up in the emission spectrum, but not in the sub- and superradiant decay rates. In case of two non-identical emitters, also the collective emission rates will differ by making the RWA. For three or more identical emitters, the RWA errors in the interatomic interaction in general affect both the collective emission spectra and the collective decay rates. Ring configurations with discrete rotational symmetry are an interesting exception. Interestingly, the maximal errors in the collective decay rates due to making the RWA occur for finite emitter separations.

arXiv:2112.04952•Journal of Physics B Atomic Molecular and Optical Physics•Aug 15, 2022•11 citations
Some consequences of the rotating-wave approximation (RWA) for spontaneous emission are explored. The Winger---Weisskopf theory of the spontaneous exponential decay of a single excited atom is shown to contain an implicit form of the RWA. However, to describe the cooperative superradiant level shift, the RWA should not be used, and care should be taken over the choice of interaction Hamiltonians.
Spontaneous emission in dipole approximation is studied theoretically using both source-field theory and a Schrodinger picture approach. Using source-field theory we obtain formal equations for the Poynting vector and energy density without making the rotating wave approximation (RWA) and Weisskopf-Wigner approximation (WWA). The initial condition at t=0 is one in which the atom is in an excited state and the field in the vacuum state. The source-field expressions are evaluated within the the RWA and WWA and are found to satisfy Poynting's theorem. To explore the consequences of not making the RWA and WWA, the Poynting vector and energy density are calculated using perturbation theory. We use a Schrodinger picture approach and essentially reproduce and complement the results of Compagno, Passante, and Persico [J. Mod. Optics 37:8, 1377 (2007)] and those of Power and Thirunamachandran [Phys. Rev. A 45, 54 (1992)] obtained using a Heisenberg picture approach. The theory involves a sum over field mode frequencies and both finite cutoffs and convergence factors are used to carry out the sums. It is shown that the perturbation theory calculation leads to unphysical values for atomic state populations for all times when a sum over all field frequencies is taken, even if a convergence factor is used. It is also proved that the fields calculated using source-field theory always satisfy Poynting's theorem for ct not equal to R, where R is the distance from the atom.
The exact determination of the spontaneous emission coefficient for an excited atom is an extremely complex problem so various approximations are typically used. One of the most popular ones is the use of the dipole approximation of a two-level atom followed by rotating wave approximation (RWA). However, such an approach applied to the entire frequency spectrum results in the appearance of divergent integrals in the derivations, which are not treated rigorously in typical student textbooks. It is known from the literature that the introduction of cutoff for frequencies, justified by the finite size of the atom, may solve this problem. For didactic purposes, in this paper, we introduce a mathematically simple cutoff, which allows for a straightforward yet mathematically consistent rederivation of the Weisskopf-Wigner spontaneous emission rate (up to the small correction) within RWA. Importantly, this cutoff is not a mathematical trick to make calculations easier but is related to a real feature of the physical system, the neglect of which leads to inconsistency. More precise analysis demand going beyond RWA and dipole approximation.
We study the effects of the non-rotating wave approximation (non-RWA) on the spontaneous emission of a Λ-type three-level atom, where the metastable states are coupled to a common excited state by a weak laser field and the lower-level doublet is driven by a strong microwave field. When the non-RWA is applied to the interaction of the atom with the microwave field, the spontaneous emission spectrum is comprised of a series of ultrasharp lines. The physical interpretation of the spectral characteristics is given in light of the dressed states.
Abstract We study quantitatively the breakdown of the rotating-wave approximation (RWA) when calculating collective light emission by quantum emitters, in particular in the weak-excitation limit. Our starting point is a known multiple-scattering formalism where the full light–matter interaction leads to induced inter-emitter interactions described by the classical Green function of inhomogeneous dielectric media. When making the RWA in the light–matter interaction, however, these induced interactions differ from the classical Green function, and for free space we find a reduction of the interatomic interaction strength by up to a factor of two. By contrast, for the corresponding scalar model the relative RWA error for the inter-emitter interaction even diverges in the near field. For two identical emitters, the errors due to the RWA in collective light emission will show up in the emission spectrum, but not in the sub- and superradiant decay rates. In case of two non-identical emitters, also the collective emission rates will differ by making the RWA. For three or more identical emitters, the RWA errors in the interatomic interaction in general affect both the collective emission spectra and the collective decay rates. Ring configurations with discrete rotational symmetry are an interesting exception. Interestingly, the maximal errors in the collective decay rates due to making the RWA occur for finite emitter separations.