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{{Short description|Model used in atom optics and magnetic resonance}}
{{Refimprove|date=August 2013}}
The '''rotating
The name of the approximation stems from the form of the Hamiltonian in the [[interaction picture]], as shown below.
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 ==▼
▲== Mathematical formulation ==
For simplicity consider a [[two-state quantum system|two-level atomic system]] with [[ground state|ground]] and [[excited state|excited]] states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>, respectively (using the [[bra–ket notation|Dirac bracket notation]]). Let the energy difference between the states be <math>\hbar\omega_0</math> so that <math>\omega_0</math> is the transition frequency of the system. Then the unperturbed [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the atom can be written as
: <math>H_0 = \frac{\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}|-\frac{\hbar\omega_0}{2}|\text{g}\rangle\langle\text{g}|</math>.
Suppose the atom experiences an external classical [[electric field]] of frequency <math>\omega_L</math>, given by
<math>\vec{E}(t) = \vec{E}_0 e^{-i\omega_Lt} +\vec{E}_0^* e^{i\omega_Lt}</math>; e.g., a [[plane wave]] propagating in space. Then under the [[dipole#Torque on a dipole|dipole approximation]] the interaction Hamiltonian between the atom and the electric field can be expressed as
: <math>H_1 = -\vec{d} \cdot \vec{E}</math>,
where <math>\vec{d}</math> is the [[transition dipole moment|dipole moment operator]] of the atom. The total Hamiltonian for the atom-light system is therefore <math>H = H_0 + H_1.</math> The atom does not have a dipole moment when it is in an [[energy eigenstate]], so <math>\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.</math> This means that defining <math>\vec{d}_
: <math>\vec{d} = \vec{d}_
(with <math>^*</math> denoting the [[complex conjugate]]). The [[#Derivation|interaction Hamiltonian can then be shown to be
: <math>H_1 =
-\hbar\left(\Omega e^{-i\omega_Lt} + \tilde{\Omega}e^{i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}| -\hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt} + \Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|
</math>
where <math>\Omega = \hbar^{-1}\vec{d}_\text{eg} \cdot \vec{E}_0</math> is the [[Rabi frequency]] and <math>\tilde{\Omega} \mathrel{:=} \hbar^{-1}\vec{d}_\text{eg} \cdot \vec{E}_0^*</math> is the counter-rotating frequency. To see why the <math>\tilde{\Omega}</math> terms are called
: <math>H_{1,I} =
▲where <math>\Omega=\hbar^{-1}\vec{d}_\text{eg}\cdot\vec{E}_0</math> is the [[Rabi frequency]] and <math>\tilde{\Omega}:=\hbar^{-1}\vec{d}_\text{eg}\cdot\vec{E}_0^*</math> is the counter-rotating frequency. To see why the <math>\tilde{\Omega}</math> terms are called `counter-rotating' consider a [[unitary transformation]] to the [[Interaction picture|interaction or Dirac picture]] where the transformed Hamiltonian <math>H_{1,I}</math> is given by
-\hbar\left(
</math>
▲ -\hbar\left(\tilde{\Omega}^*e^{-i(\omega_L+\omega_0)t}+\Omega^*e^{i\Delta t}\right)|\text{g}\rangle\langle\text{e}|,</math>
where <math>\Delta \omega \mathrel{:=} \omega_L - \omega_0</math> is the detuning between the light field and the atom.
=== Making the approximation ===
[[File:TLSRWA.gif|thumb|Two-level-system on resonance with a driving field with (blue) and without (green) applying the rotating-wave approximation.]]
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 [[resonance]] with the atomic transition. This means that <math>\Delta \omega \ll \omega_L + \omega_0</math> and the complex exponentials multiplying <math>\tilde{\Omega}</math> and <math>\tilde{\Omega}^*</math> 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
: <math>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}|.
</math> Finally, transforming back into the [[Schrödinger picture]], the Hamiltonian is given by
:<math>H^\text{RWA} =
- \frac{\hbar\omega_0}{2}|\text{g}\rangle\langle\text{g}|
- \hbar\Omega e^{-i\omega_Lt}|\text{e}\rangle\langle\text{g}|
- \hbar\Omega^* e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.
</math>
Another criterion for rotating wave approximation is the weak coupling condition, that is, the Rabi frequency should be much less than the transition frequency.<ref name="WuYang2007"/>
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: <math>\begin{align}
▲H_1 &= -\vec{d}\cdot\vec{E} \\
&= -\hbar\left(\Omega e^{-i\omega_Lt} + \
▲ -\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|,
\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:
<math>
▲: <math>U = e^{iH_0t/\hbar} = e^{i \omega_0 t |\text{e}\rangle \langle\text{e}|} = |\text{g}\rangle \langle\text{g}| +e^{i \omega_0 t} |\text{e}\rangle \langle\text{e}|</math>,
\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
: <math>\begin{align}
H_{1,I} &\equiv U H_1 U^\dagger \\
&= -\hbar\left(\Omega e^{-i\omega_Lt} + \tilde{\Omega}e^{i\omega_Lt}\right)e^{i\omega_0t}|\text{e}\rangle\langle\text{g}|
-\hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt} + \Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|e^{-i\omega_0t} \\
&= -\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{align}</math>
Now we apply the RWA by eliminating the counter-rotating terms as explained in the previous section
: <math>
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}|
</math>
Finally, we transform the approximate Hamiltonian <math>H_{1,I}^{\text{RWA}}</math> back to the Schrödinger picture:
: <math>\begin{align}
H_1^
&= -\hbar\Omega e^{-i\Delta \omega t}e^{-i\
-\hbar\Omega^* e^{i\Delta \omega t}|\text{g}\rangle\langle\text{e}|e^{i\omega_0t} \\
&= -\hbar\Omega e^{-i\omega_Lt}|\text{e}\rangle\langle\text{g}|
-\hbar\Omega^* e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.
\end{align}</math>
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:<math>
H^\text{RWA} = H_0 + H_1^
\frac{\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}| - \frac{\hbar\omega_0}{2}|\text{g}\rangle\langle\text{g}| -
</math>
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