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LucasBrown (talk | contribs) Adding local short description: "Model used in atom optics and magnetic resonance", overriding Wikidata description "mathematical simplification used in atom optics and magnetic resonance" |
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The name of the approximation stems from the form of the Hamiltonian in the [[interaction picture]], as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system [[bra–ket notation|ket]], leaving only the evolution due to the interaction of the atom with the light field to consider. It is in this picture that the rapidly oscillating terms mentioned previously can be neglected. Since in some sense the interaction picture can be thought of as rotating with the system ket only that part of the electromagnetic wave that approximately co-rotates is kept; the counter-rotating component is discarded.
The rotating-wave approximation is closely related to, but different from, the [[Redfield_equation#Secular_approximation|secular approximation]].<ref>{{cite journal |first1=H. |last1=Mäkelä |first2=M. |last2=Möttönen |title=Effects of the rotating-wave and secular approximations on non-Markovianity |url=https://link.aps.org/doi/10.1103/PhysRevA.88.052111 |journal=Physical Review A |date=13 November 2013 |pages=052111 |volume=88 |issue=5| doi=10.1103/PhysRevA.88.052111|arxiv=1306.6301 |bibcode=2013PhRvA..88e2111M }}</ref>
== Mathematical formulation ==
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\end{align}</math>
as stated. The next step is to find the Hamiltonian in the [[interaction picture]], <math>H_{1,I}</math>. The required unitary transformation is:
\begin{align} U & = e^{iH_0t/\hbar} \\ & = e^{i \omega_0 t/2 (|\text{e}\rangle \langle\text{e}| - |\text{g}\rangle \langle\text{g}|)} \\
& = \cos\left(\frac{\omega_0 t}{2}\right)
\left(|\text{e}\rangle \langle\text{e}| + |\text{g}\rangle \langle\text{g}|\right) + i \sin\left(\frac{\omega_0 t}{2}\right) \left(|\text{e}\rangle \langle\text{e}| - |\text{g}\rangle \langle\text{g}|\right) \\ & = e^{-i\omega_0 t/2}|\text{g}\rangle \langle\text{g}| + e^{i \omega_0 t/2} |\text{e}\rangle \langle\text{e}| \\
& = e^{-i\omega_0 t/2}\left(|\text{g}\rangle \langle\text{g}| + e^{i \omega_0 t} |\text{e}\rangle \langle\text{e}|\right)
\end{align}
</math> ,where the 3rd step can be proved by using a [[Taylor series]] expansion, and using the orthogonality of the states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>. Note that a multiplication by an overall phase of <math>e^{i \omega_0 t/2}</math> on a unitary operator does not affect the underlying physics, so in the further usages of <math>U</math> we will neglect it. Applying <math>U</math> gives:
: <math>\begin{align}
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