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arXiv:1301.3585 · Journal of Modern Physics

Introduction to the Rotating Wave Approximation (RWA): Two Coherent Oscillations

Kazuyuki Fujii

Published Jan 1, 2017• 33 citations
arXivHTMLPDF

Abstract

In this note, I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics.In Quantum Optics, it plays a very important role in order to obtain an analytic approximate solution of some Schrödinger equation, while it is curious from the mathematical point of view.I explain it carefully with two coherent oscillations for them and expect that they will overcome the problem in the near future.

DOI
10.4236/jmp.2017.812124

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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

•May 8, 1994•0 citations
In this note, I introduce a mysterious approximation called the rotating wave approximation (RWA) to undergraduates or non-experts who are interested in both Mathematics and Quantum Optics.In Quantum Optics, it plays a very important role in order to obtain an analytic approximate solution of some Schrödinger equation, while it is curious from the mathematical point of view.I explain it carefully with two coherent oscillations for them and expect that they will overcome the problem in the near future.
Rotating wave approximation (RWA) plays a key rule in quantum optics to solve some Schrödinger equation approximately. For example, it is well known that RWA has been used to calculate the transition probability. However, so far no one shows the validity of RWA. Recently Larson [J. Larson, Phys. Rev. Lett. 108, 033601 (2012)] gave the first example showing that the application of RWA in the Jaynes-Cummings model may lead to false conclusions. In this paper, we analytically and numerically show that the application of RWA leads to false conclusions about the transition probability for the near or exact resonance.
Developing an analytical theory for atomic coherence driven by ultrashort laster pulses has proved to be challenging due to the breakdown of the rotating wave approximation (RWA). In this paper, we present an approximate, closed-form solution to the Schrodinger equation that describes a two-level atom under the excitation of a far-off-resonance, few-cycle pulse of arbitrary shape without invoking the RWA. As an example of its applicability, an analytical solution for Gaussian pulses is explicitly given. Comparisons with numerical solutions validate the accuracy our solution within the scope of the approximation. Finally, we outline an alternative approach that can lead to a more accurate solution by capturing the nonlinear behaviors of the system.
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.
We consider a forced harmonic oscillator in one-dimension. Using coherent states, we show that the treatment of the system is simplified, that the relationship between the classical and quantum solutions becomes transparent, and that the evolution operator of the system can be calculated easily as the free evolution operator of the harmonic oscillator followed by a displacement operator that depends on the classical solution. In addition, we consider the system in the rotating-wave-approximation (RWA), an application of the Averaging Theorem. We determine the relationship between the exact solution and the one in the RWA, test the accuracy of the RWA, and explain why the RWA gives accurate results in the realm of cavity quantum electrodynamics. Finally, we apply the results to a charged particle interacting with an electromagnetic field.
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