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

Atom–field interaction without the rotating-wave approximation: quasi-classical trajectories

G. A. Finney, Julio R. Gea-Banacloche

Published May 8, 1994• 0 citations
OpenAlex

Abstract

We study the interaction between a two-level system and a harmonic oscillator without the rotating-wave approximation (RWA). In quantum optics, this model (sometimes called the spin-boson model) is traditionally thought of as a two-Ievel atom coupled to a single-mode radiation field, but it could also describe (1) a two-level ion in a trap coupled to the motion of its own center of mass or (2) a two-level particle in a solid coupled to a phonon mode. For sufficiently strong coupling, this model is known to exhibit chaos in a variety of semi-classical approximations.1

Related papers

Interaction of a two-level atom with single-mode optical field beyond the rotating wave approximation

Ju Liu, Zhiyuan Li

One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.

Optics Express•Nov 10, 2014•12 citations

Coherent Interactions

Stephen Mark Barnett, PAUL M. RADMORE

Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.

•Nov 14, 2002•0 citations

Coupled modes locally interacting with qubits: Critical assessment of the rotating-wave approximation

Paulo C. Cárdenas, Wallace S. Teixeira, F. L. Semião

The interaction of qubits with quantized modes of electromagnetic fields has been largely addressed in the quantum optics literature under the rotating wave approximation (RWA), where rapid oscillating terms in the qubit-mode interaction picture Hamiltonian can be neglected. At the same time, it is generally accepted that, provided the interaction is sufficiently strong or for long times, the RWA tends to describe physical phenomena incorrectly. In this work, we extend the investigation of the validity of the RWA to a more involved setup where two qubit-mode subsystems are brought to interaction through their harmonic coordinates. Our treatment is all analytic thanks to a sequence of carefully chosen unitary transformations, which allows us to diagonalize the Hamiltonian within and without the RWA. By also considering qubit dephasing, we find that the purity of the two-qubit state presents non-Markovian features which become more pronounced as the coupling between the modes gets stronger and the RWA loses its validity. In the same regime, there occurs fast generation of entanglement between the qubits, which is also not correctly described under the RWA. The setup and results presented here clearly show the limitations of the RWA in a scenario amenable to exact description and free from numerical uncertainties. Consequently, it may be of interest for the community working with cavity or circuit quantum electrodynamic systems in the strong coupling regime.

arXiv:1701.01104•Physical Review A•Apr 12, 2017•3 citations

Collective behavior ofMbosonic modes interacting with a single two-level atom

G. Benivegna, ANTONINO MESSINA

The Hamiltonian describing, without the rotating-wave approximation (RWA), the linear interaction between M bosonic modes with an Einstein spectrum and a single two-level atom is exactly and canonically transformed introducing M suitable collective independent field modes, in such a way that only one among them is coupled to the atom. Some physical consequences of this fact are analyzed and, in particular, the existence of radiation-trapping phenomena together with the possibility of atomic absorption suppression is established. The applicability of the RWA to this system is discussed and the importance of the effective-field statistics for the time evolution of the system is pointed out.

Physical Review A•Jun 1, 1988•14 citations

Quantum effects versus the semiclassical approximation

Luca Bonci, Paolo Grigolini A

We study a two-level quantum system interacting through a coupling strength g with a single-mode radiation field. We show that the rotating-wave-approximation (RWA) Hamiltonian produces quantum-mechanical effects more persistent than those generated by the full Hamiltonian. Within the full Hamiltonian the semiclassical prediction is recovered upon increase of the radiation field intensity, whereas with the RWA Hamiltonian no reconciliation with the predictions of the semiclassical approximation is realized by fields of any intensity, if the interaction strength g is kept unchanged.

Physical Review A•Oct 1, 1992•5 citations

Atom-field interaction without the rotating-wave approximation: A path-integral approach

Khalid Zaheer, Muhammad Suhail Zubairy

We use the path-integral technique to solve the quantum-mechanical problem of interaction of a single two-level atom with a single-mode quantized field without the rotating-wave approximation (RWA). We calculate the full spin-Bose propagator in the form of a matrix as an expansion around a path corresponding to the RWA. The effect of energy-nonconserving terms on the atomic population inversion is studied, and we show that even under the conditions in which the RWA is considered to be valid there is a significant effect on the atomic inversion due to these terms.

Physical Review A•Mar 1, 1988•88 citations
We study the interaction between a two-level system and a harmonic oscillator without the rotating-wave approximation (RWA). In quantum optics, this model (sometimes called the spin-boson model) is traditionally thought of as a two-Ievel atom coupled to a single-mode radiation field, but it could also describe (1) a two-level ion in a trap coupled to the motion of its own center of mass or (2) a two-level particle in a solid coupled to a phonon mode. For sufficiently strong coupling, this model is known to exhibit chaos in a variety of semi-classical approximations.1
One of the simplest models involving the atom-field interaction is the coupling of a single two-level atom with single-mode optical field. Under the rotating wave approximation, this problem is reduced to a form that can be solved exactly. But the approximation is only valid when the two levels are resonant or nearly resonant with the applied electromagnetic radiation. Here we present an analytical solution without the rotating wave approximation and applicable to general atom-field interaction far away from the resonance. We find that there exists remarkable influence of the initial phase of optical field on the Rabi oscillations and Rabi splitting, and this issue cannot be explored in the context of the rotating wave approximation. Due to the retention of the counter-rotating terms, higher-order harmonic appears during the Rabi splitting. The analytical solution suggests a way to regulate and control the quantum dynamics of a two-level atom and allows for exploring more essential features of the atom-field interaction.
Abstract This chapter presents some of the models describing the time-dependent interaction between a simple atom and a field, which may be described either classically or fully quantum mechanically. A number of equivalent interaction pictures and the important rotating-wave approximation are introduced. The two-state dynamics of an atom coupled to a classical field is solved, starting with perturbation theory. Non-perturbative behaviour is analysed, leading to Rabi oscillations of the atomic probabilities. The Jaynes–Cummings model of the dynamics of a single two-state atom coupled to a single quantized mode of the electromagnetic field is solved leading to collapses and revivals of the excited state probability. An exactly soluble model of the coupling of a single state to a Bixon–Jortner quasi-continuum of discrete levels is analysed. Finally, the coherent interactions between two field modes, of the type common in nonlinear optics, are discussed.
The interaction of qubits with quantized modes of electromagnetic fields has been largely addressed in the quantum optics literature under the rotating wave approximation (RWA), where rapid oscillating terms in the qubit-mode interaction picture Hamiltonian can be neglected. At the same time, it is generally accepted that, provided the interaction is sufficiently strong or for long times, the RWA tends to describe physical phenomena incorrectly. In this work, we extend the investigation of the validity of the RWA to a more involved setup where two qubit-mode subsystems are brought to interaction through their harmonic coordinates. Our treatment is all analytic thanks to a sequence of carefully chosen unitary transformations, which allows us to diagonalize the Hamiltonian within and without the RWA. By also considering qubit dephasing, we find that the purity of the two-qubit state presents non-Markovian features which become more pronounced as the coupling between the modes gets stronger and the RWA loses its validity. In the same regime, there occurs fast generation of entanglement between the qubits, which is also not correctly described under the RWA. The setup and results presented here clearly show the limitations of the RWA in a scenario amenable to exact description and free from numerical uncertainties. Consequently, it may be of interest for the community working with cavity or circuit quantum electrodynamic systems in the strong coupling regime.
The Hamiltonian describing, without the rotating-wave approximation (RWA), the linear interaction between M bosonic modes with an Einstein spectrum and a single two-level atom is exactly and canonically transformed introducing M suitable collective independent field modes, in such a way that only one among them is coupled to the atom. Some physical consequences of this fact are analyzed and, in particular, the existence of radiation-trapping phenomena together with the possibility of atomic absorption suppression is established. The applicability of the RWA to this system is discussed and the importance of the effective-field statistics for the time evolution of the system is pointed out.
We study a two-level quantum system interacting through a coupling strength g with a single-mode radiation field. We show that the rotating-wave-approximation (RWA) Hamiltonian produces quantum-mechanical effects more persistent than those generated by the full Hamiltonian. Within the full Hamiltonian the semiclassical prediction is recovered upon increase of the radiation field intensity, whereas with the RWA Hamiltonian no reconciliation with the predictions of the semiclassical approximation is realized by fields of any intensity, if the interaction strength g is kept unchanged.
We use the path-integral technique to solve the quantum-mechanical problem of interaction of a single two-level atom with a single-mode quantized field without the rotating-wave approximation (RWA). We calculate the full spin-Bose propagator in the form of a matrix as an expansion around a path corresponding to the RWA. The effect of energy-nonconserving terms on the atomic population inversion is studied, and we show that even under the conditions in which the RWA is considered to be valid there is a significant effect on the atomic inversion due to these terms.