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<title>Rotating-wave approximation</title>

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Rotating-wave approximation

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The rotating-wave approximation is an approximation used in <a href="Atom_optics" title="Atom optics">atom optics</a> and <a href="Nuclear_magnetic_resonance" title="Nuclear magnetic resonance">magnetic resonance</a>. In this approximation, terms in a <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> that oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic transition, and the intensity is low.<a href="#cite_note-WuYang2007-1">[1]</a> Explicitly, terms in the Hamiltonians that oscillate with frequencies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{L}+\omega _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{L}+\omega _{0}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \omega _{L}-\omega _{0}}</annotation>

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</math><img src="./912728ff6d52518d204c8c9037eaf934d877e85a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.138ex; height:2.343ex;" alt="{\displaystyle \omega _{L}-\omega _{0}}" loading="lazy"> are kept, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{L}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{L}}</annotation>

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</math><img src="./7284e7df80758a0946c74413f95c81c1750c755c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.009ex;" alt="{\displaystyle \omega _{L}}" loading="lazy"> is the light frequency, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}">

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The name of the approximation stems from the form of the Hamiltonian in the <a href="Interaction_picture" title="Interaction picture">interaction picture</a>, as shown below. By switching to this picture the evolution of an atom due to the corresponding atomic Hamiltonian is absorbed into the system <a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">ket</a>, 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 <a href="Redfield_equation#Secular_approximation" title="Redfield equation">secular approximation</a>.<a href="#cite_note-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_formulation">Mathematical formulation</h2></div>

For simplicity consider a <a href="Two-state_quantum_system" title="Two-state quantum system">two-level atomic system</a> with <a href="Ground_state" title="Ground state">ground</a> and <a href="Excited_state" title="Excited state">excited</a> states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\text{g}}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |{\text{g}}\rangle }</annotation>

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</math><img src="./7188bd246f4d6900447b3c44082c1e5476c9a94f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.714ex; height:2.843ex;" alt="{\displaystyle |{\text{g}}\rangle }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\text{e}}\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle |{\text{e}}\rangle }</annotation>

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</math><img src="./5618e4caf7b6b03a51eca2636da84d59fd6b7aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.584ex; height:2.843ex;" alt="{\displaystyle |{\text{e}}\rangle }" loading="lazy">, respectively (using the <a href="Bra%E2%80%93ket_notation" title="Bra–ket notation">Dirac bracket notation</a>). Let the energy difference between the states be <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar \omega _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \hbar \omega _{0}}</annotation>

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</math><img src="./88965d38ca9b0a66958bffaacea92f2d33741cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.807ex; height:2.509ex;" alt="{\displaystyle \hbar \omega _{0}}" loading="lazy"> so that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation>

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</math><img src="./9a713d16c489051d4f515e12b1f86061c6be799b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}" loading="lazy"> is the transition frequency of the system. Then the unperturbed <a href="Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a> of the atom can be written as

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<annotation encoding="application/x-tex">{\displaystyle H_{0}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|}</annotation>

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</math><img src="./9995bd7250948817288b3261c341e7baab1b155e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:28.805ex; height:5.343ex;" alt="{\displaystyle H_{0}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|}" loading="lazy">.</dd></dl>

Suppose the atom experiences an external classical <a href="Electric_field" title="Electric field">electric field</a> of frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{L}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {E}}(t)={\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}}</annotation>

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</math><img src="./b37e75b20c8e0b1fb7d470611a32d9e8dcae73c4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.446ex; height:3.509ex;" alt="{\displaystyle {\vec {E}}(t)={\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}}" loading="lazy">; e.g., a <a href="Plane_wave" title="Plane wave">plane wave</a> propagating in space. Then under the <a href="Dipole#Torque_on_a_dipole" title="Dipole">dipole approximation</a> the interaction Hamiltonian between the atom and the electric field can be expressed as

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<annotation encoding="application/x-tex">{\displaystyle H_{1}=-{\vec {d}}\cdot {\vec {E}}}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {d}}}">

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</math><img src="./3e1eb48ef8121a782f64c8453fa6f80a9ca26e2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.547ex; height:2.843ex;" alt="{\displaystyle {\vec {d}}}" loading="lazy"> is the <a href="Transition_dipole_moment" title="Transition dipole moment">dipole moment operator</a> of the atom. The total Hamiltonian for the atom-light system is therefore <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=H_{0}+H_{1}.}">

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<annotation encoding="application/x-tex">{\displaystyle H=H_{0}+H_{1}.}</annotation>

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</math><img src="./a8c82612f5aaa4f99d2e23f49c3541303a378b45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.62ex; height:2.509ex;" alt="{\displaystyle H=H_{0}+H_{1}.}" loading="lazy"> The atom does not have a dipole moment when it is in an <a href="Energy_eigenstate" class="mw-redirect" title="Energy eigenstate">energy eigenstate</a>, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{e}}\right\rangle =\left\langle {\text{g}}\left|{\vec {d}}\right|{\text{g}}\right\rangle =0.}">

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<annotation encoding="application/x-tex">{\displaystyle \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{e}}\right\rangle =\left\langle {\text{g}}\left|{\vec {d}}\right|{\text{g}}\right\rangle =0.}</annotation>

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</math><img src="./00351e266de5c99723934509e286d43f263af764.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.307ex; height:4.843ex;" alt="{\displaystyle \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{e}}\right\rangle =\left\langle {\text{g}}\left|{\vec {d}}\right|{\text{g}}\right\rangle =0.}" loading="lazy"> This means that defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {d}}_{\text{eg}}\mathrel {:=} \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{g}}\right\rangle }">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {d}}_{\text{eg}}\mathrel {:=} \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{g}}\right\rangle }</annotation>

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</math><img src="./69a7234116bbf2b8fd90695cf5bda573343ae561.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.728ex; height:4.843ex;" alt="{\displaystyle {\vec {d}}_{\text{eg}}\mathrel {:=} \left\langle {\text{e}}\left|{\vec {d}}\right|{\text{g}}\right\rangle }" loading="lazy"> allows the dipole operator to be written as

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<annotation encoding="application/x-tex">{\displaystyle {\vec {d}}={\vec {d}}_{\text{eg}}|{\text{e}}\rangle \langle {\text{g}}|+{\vec {d}}_{\text{eg}}^{*}|{\text{g}}\rangle \langle {\text{e}}|}</annotation>

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</math><img src="./a227fb034aaea1a4a76c5c56f5bf986f0f3721ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:24.745ex; height:3.843ex;" alt="{\displaystyle {\vec {d}}={\vec {d}}_{\text{eg}}|{\text{e}}\rangle \langle {\text{g}}|+{\vec {d}}_{\text{eg}}^{*}|{\text{g}}\rangle \langle {\text{e}}|}" loading="lazy"></dd></dl>

(with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ^{*}}">

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</math><img src="./5156ec8960b3bb95881ad7179c891c47fdfa8b9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.054ex; height:2.176ex;" alt="{\displaystyle ^{*}}" loading="lazy"> denoting the <a href="Complex_conjugate" title="Complex conjugate">complex conjugate</a>). The <a href="#Derivation">interaction Hamiltonian can then be shown to be</a>

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<annotation encoding="application/x-tex">{\displaystyle H_{1}=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|}</annotation>

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</math><img src="./7d1e36f70bae66ab998f3d3e445eb6488f2fd112.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:65.742ex; height:4.843ex;" alt="{\displaystyle H_{1}=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \Omega =\hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}}</annotation>

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</math><img src="./1dc15b705985edc4a3226dc7477474e7d0ab5d9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.272ex; height:3.509ex;" alt="{\displaystyle \Omega =\hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}}" loading="lazy"> is the <a href="Rabi_frequency" title="Rabi frequency">Rabi frequency</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\Omega }}\mathrel {:=} \hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\Omega }}\mathrel {:=} \hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}}</annotation>

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</math><img src="./e1ee1319aca944f20cbb2030c40b59b095fc6cbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.919ex; height:3.676ex;" alt="{\displaystyle {\tilde {\Omega }}\mathrel {:=} \hbar ^{-1}{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}}" loading="lazy"> is the counter-rotating frequency. To see why the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\Omega }}}">

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<annotation encoding="application/x-tex">{\displaystyle {\tilde {\Omega }}}</annotation>

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</math><img src="./890c1ff8958e549253d2d69fa34a30acb771d5d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.676ex;" alt="{\displaystyle {\tilde {\Omega }}}" loading="lazy"> terms are called counter-rotating consider a <a href="Unitary_transformation" title="Unitary transformation">unitary transformation</a> to the <a href="Interaction_picture" title="Interaction picture">interaction or Dirac picture</a> where the transformed Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1,I}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{1,I}}</annotation>

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</math><img src="./38e168da4b06bc279f767da91d699d7e31488097.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.271ex; height:2.843ex;" alt="{\displaystyle H_{1,I}}" loading="lazy"> is given by

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<annotation encoding="application/x-tex">{\displaystyle H_{1,I}=-\hbar \left(\Omega e^{-i\Delta \omega t}+{\tilde {\Omega }}e^{i(\omega _{L}+\omega _{0})t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i(\omega _{L}+\omega _{0})t}+\Omega ^{*}e^{i\Delta \omega t}\right)|{\text{g}}\rangle \langle {\text{e}}|,}</annotation>

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</math><img src="./39215fa9712f82490598edfc9d8924277de0658a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:77.736ex; height:4.843ex;" alt="{\displaystyle H_{1,I}=-\hbar \left(\Omega e^{-i\Delta \omega t}+{\tilde {\Omega }}e^{i(\omega _{L}+\omega _{0})t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i(\omega _{L}+\omega _{0})t}+\Omega ^{*}e^{i\Delta \omega t}\right)|{\text{g}}\rangle \langle {\text{e}}|,}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \Delta \omega \mathrel {:=} \omega _{L}-\omega _{0}}</annotation>

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</math><img src="./4851085940281c3eebb7b1c780bca32e7b614403.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.265ex; height:2.509ex;" alt="{\displaystyle \Delta \omega \mathrel {:=} \omega _{L}-\omega _{0}}" loading="lazy"> is the detuning between the light field and the atom.

<div class="mw-heading mw-heading3"><h3 id="Making_the_approximation">Making the approximation</h3></div>

This is the point at which the rotating wave approximation is made. The dipole approximation has been assumed, and for this to remain valid the electric field must be near <a href="Resonance" title="Resonance">resonance</a> with the atomic transition. This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta \omega \ll \omega _{L}+\omega _{0}}">

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</math><img src="./53854094c38d8eba570efd8be969fa3f02067598.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.134ex; height:2.509ex;" alt="{\displaystyle \Delta \omega \ll \omega _{L}+\omega _{0}}" loading="lazy"> and the complex exponentials multiplying <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\Omega }}}">

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</math><img src="./fe57838941d0dde2d33b5050db67aa99b744c20d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.843ex;" alt="{\displaystyle {\tilde {\Omega }}^{*}}" loading="lazy"> can be considered to be rapidly oscillating. Hence on any appreciable time scale, the oscillations will quickly average to 0. The rotating wave approximation is thus the claim that these terms may be neglected and thus the Hamiltonian can be written in the interaction picture as

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<annotation encoding="application/x-tex">{\displaystyle H_{1,I}^{\text{RWA}}=-\hbar \Omega e^{-i\Delta \omega t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|.}</annotation>

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</math><img src="./20a236d2b4919deaa1853cbfb23ac39bbbe43486.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:43.494ex; height:3.676ex;" alt="{\displaystyle H_{1,I}^{\text{RWA}}=-\hbar \Omega e^{-i\Delta \omega t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|.}" loading="lazy"></dd></dl>

Finally, transforming back into the <a href="Schr%C3%B6dinger_picture" title="Schrödinger picture">Schrödinger picture</a>, the Hamiltonian is given by

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<annotation encoding="application/x-tex">{\displaystyle H^{\text{RWA}}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.}</annotation>

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</math><img src="./1e08ec44c01e3d53ee67797c924406af1f378815.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:66.656ex; height:5.343ex;" alt="{\displaystyle H^{\text{RWA}}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.}" loading="lazy"></dd></dl>

Another criterion for rotating wave approximation is the weak coupling condition, that is, the Rabi frequency should be much less than the transition frequency.<a href="#cite_note-WuYang2007-1">[1]</a>

At this point the rotating wave approximation is complete. A common first step beyond this is to remove the remaining time dependence in the Hamiltonian via another unitary transformation.

<div class="mw-heading mw-heading2"><h2 id="Derivation">Derivation</h2></div>

Given the above definitions the interaction Hamiltonian is

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{1}=-{\vec {d}}\cdot {\vec {E}}&=-\left({\vec {d}}_{\text{eg}}|{\text{e}}\rangle \langle {\text{g}}|+{\vec {d}}_{\text{eg}}^{*}|{\text{g}}\rangle \langle {\text{e}}|\right)\cdot \left({\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)\\&=-\left({\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\left({\vec {d}}_{\text{eg}}^{*}\cdot {\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {d}}_{\text{eg}}^{*}\cdot {\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|\\&=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|,\end{aligned}}}</annotation>

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</math><img src="./1c2259793312f069c25aae7e0b9a069436137d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:96.589ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}H_{1}=-{\vec {d}}\cdot {\vec {E}}&=-\left({\vec {d}}_{\text{eg}}|{\text{e}}\rangle \langle {\text{g}}|+{\vec {d}}_{\text{eg}}^{*}|{\text{g}}\rangle \langle {\text{e}}|\right)\cdot \left({\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)\\&=-\left({\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {d}}_{\text{eg}}\cdot {\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\left({\vec {d}}_{\text{eg}}^{*}\cdot {\vec {E}}_{0}e^{-i\omega _{L}t}+{\vec {d}}_{\text{eg}}^{*}\cdot {\vec {E}}_{0}^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|\\&=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|,\end{aligned}}}" loading="lazy"></dd></dl>

as stated. The next step is to find the Hamiltonian in the <a href="Interaction_picture" title="Interaction picture">interaction picture</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1,I}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}U&=e^{iH_{0}t/\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{aligned}}}</annotation>

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</math><img src="./b63b0305a79bbb6815de8b87aeaed6745b7cf625.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.338ex; width:62.088ex; height:19.843ex;" alt="{\displaystyle {\begin{aligned}U&=e^{iH_{0}t/\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{aligned}}}" loading="lazy">

,where the 3rd step can be proved by using a <a href="Taylor_series" title="Taylor series">Taylor series</a> expansion, and using the orthogonality of the states <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\text{g}}\rangle }">

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</math><img src="./5618e4caf7b6b03a51eca2636da84d59fd6b7aa1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.584ex; height:2.843ex;" alt="{\displaystyle |{\text{e}}\rangle }" loading="lazy">. Note that a multiplication by an overall phase of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e^{i\omega _{0}t/2}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{1,I}&\equiv UH_{1}U^{\dagger }\\&=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)e^{i\omega _{0}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|e^{-i\omega _{0}t}\\&=-\hbar \left(\Omega e^{-i\Delta \omega t}+{\tilde {\Omega }}e^{i(\omega _{L}+\omega _{0})t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i(\omega _{L}+\omega _{0})t}+\Omega ^{*}e^{i\Delta \omega t}\right)|{\text{g}}\rangle \langle {\text{e}}|\ .\end{aligned}}}</annotation>

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</math><img src="./ca5ccbf84b6fc6058c26d8330c15b5496cd58320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:79.068ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}H_{1,I}&\equiv UH_{1}U^{\dagger }\\&=-\hbar \left(\Omega e^{-i\omega _{L}t}+{\tilde {\Omega }}e^{i\omega _{L}t}\right)e^{i\omega _{0}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i\omega _{L}t}+\Omega ^{*}e^{i\omega _{L}t}\right)|{\text{g}}\rangle \langle {\text{e}}|e^{-i\omega _{0}t}\\&=-\hbar \left(\Omega e^{-i\Delta \omega t}+{\tilde {\Omega }}e^{i(\omega _{L}+\omega _{0})t}\right)|{\text{e}}\rangle \langle {\text{g}}|-\hbar \left({\tilde {\Omega }}^{*}e^{-i(\omega _{L}+\omega _{0})t}+\Omega ^{*}e^{i\Delta \omega t}\right)|{\text{g}}\rangle \langle {\text{e}}|\ .\end{aligned}}}" loading="lazy"></dd></dl>

Now we apply the RWA by eliminating the counter-rotating terms as explained in the previous section:

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<annotation encoding="application/x-tex">{\displaystyle H_{1,I}^{\text{RWA}}=-\hbar \Omega e^{-i\Delta \omega t}|{\text{e}}\rangle \langle {\text{g}}|+-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|}</annotation>

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</math><img src="./2c4d3c0b36c13e903a1fa710bc577a2e61769de9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:44.656ex; height:3.676ex;" alt="{\displaystyle H_{1,I}^{\text{RWA}}=-\hbar \Omega e^{-i\Delta \omega t}|{\text{e}}\rangle \langle {\text{g}}|+-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|}" loading="lazy"></dd></dl>

Finally, we transform the approximate Hamiltonian <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{1,I}^{\text{RWA}}}">

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<annotation encoding="application/x-tex">{\displaystyle H_{1,I}^{\text{RWA}}}</annotation>

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</math><img src="./ffd83cc05e560964bf4cfc13eb226828a471e5e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:6.467ex; height:3.676ex;" alt="{\displaystyle H_{1,I}^{\text{RWA}}}" loading="lazy"> back to the Schrödinger picture:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H_{1}^{\text{RWA}}&=U^{\dagger }H_{1,I}^{\text{RWA}}U\\&=-\hbar \Omega e^{-i\Delta \omega t}e^{-i\omega _{0}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|e^{i\omega _{0}t}\\&=-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.\end{aligned}}}</annotation>

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</math><img src="./e87f553af38aad01583d0dd8ec5fe6dbf6c6dc57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.381ex; margin-bottom: -0.29ex; width:53.539ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}H_{1}^{\text{RWA}}&=U^{\dagger }H_{1,I}^{\text{RWA}}U\\&=-\hbar \Omega e^{-i\Delta \omega t}e^{-i\omega _{0}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\Delta \omega t}|{\text{g}}\rangle \langle {\text{e}}|e^{i\omega _{0}t}\\&=-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.\end{aligned}}}" loading="lazy"></dd></dl>

The atomic Hamiltonian was unaffected by the approximation, so the total Hamiltonian in the Schrödinger picture under the rotating wave approximation is

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<annotation encoding="application/x-tex">{\displaystyle H^{\text{RWA}}=H_{0}+H_{1}^{\text{RWA}}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.}</annotation>

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</math><img src="./851b9b66d18ce8ab7445ab8ad9461c59921cd0b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:82.047ex; height:5.343ex;" alt="{\displaystyle H^{\text{RWA}}=H_{0}+H_{1}^{\text{RWA}}={\frac {\hbar \omega _{0}}{2}}|{\text{e}}\rangle \langle {\text{e}}|-{\frac {\hbar \omega _{0}}{2}}|{\text{g}}\rangle \langle {\text{g}}|-\hbar \Omega e^{-i\omega _{L}t}|{\text{e}}\rangle \langle {\text{g}}|-\hbar \Omega ^{*}e^{i\omega _{L}t}|{\text{g}}\rangle \langle {\text{e}}|.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWuYang2007" class="citation journal cs1">Wu, Ying; Yang, Xiaoxue (2007). "Strong-Coupling Theory of Periodically Driven Two-Level Systems". Physical Review Letters. 98 (1): 013601. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007PhRvL..98a3601W">2007PhRvL..98a3601W</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.98.013601">10.1103/PhysRevLett.98.013601</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0031-9007">0031-9007</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/17358474">17358474</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFMäkeläMöttönen2013" class="citation journal cs1">Mäkelä, H.; Möttönen, M. (13 November 2013). <a rel="nofollow" class="external text" href="https://link.aps.org/doi/10.1103/PhysRevA.88.052111">"Effects of the rotating-wave and secular approximations on non-Markovianity"</a>. Physical Review A. 88 (5): 052111. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1306.6301">1306.6301</a>. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2013PhRvA..88e2111M">2013PhRvA..88e2111M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevA.88.052111">10.1103/PhysRevA.88.052111</a>.</cite>

</li>

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