Rotating-wave approximation: Difference between revisions

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{{Short description|Model used in atom optics and magnetic resonance}}

{{Refimprove|date=August 2013}}

The '''rotating -wave approximation''' is an approximation used in [[atom optics]] and [[Nuclear magnetic resonance|magnetic resonance]]. In this approximation, terms in a [[Hamiltonian (quantum mechanics)|Hamiltonian]] whichthat 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.<ref name="WuYang2007">{{cite journal |last1=Wu |first1=Ying |last2=Yang |first2=Xiaoxue |title=Strong-Coupling Theory of Periodically Driven Two-Level Systems |journal=Physical Review Letters |volume=98 |issue=1 |year=2007 |issn=0031-9007 |doi=10.1103/PhysRevLett.98.013601 |bibcode = 2007PhRvL..98a3601W |pmid=17358474 |page=013601}}</ref> Explicitly, terms in the Hamiltonians whichthat oscillate with frequencies <math>\omega_L + \omega_0 </math> are neglected, while terms whichthat oscillate with frequencies <math>\omega_L - \omega_0 </math> are kept, where <math> \omega_L </math> is the light frequency, and <math> \omega_0</math> is a transition frequency.

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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: <math>\vec{d} = \vec{d}_\text{eg}|\text{e}\rangle\langle\text{g}| + \vec{d}_\text{eg}^*|\text{g}\rangle\langle\text{e}|</math>

(with <math>^*</math> denoting the [[complex conjugate]]). The [[#DerivationsDerivation|interaction Hamiltonian can then be shown to be]]

: <math>H_1 =

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: <math>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}|,

</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>

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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:

: <math>U =

\begin{align}

U & = e^{iH_0t/\hbar} =\\

e^{i \omega_0 t |\text{e}\rangle \langle\text{e}|} =

& = e^{i \omega_0 t/2 (|\text{ge}\rangle \langle\text{ge}| + e^{i \omega_0 t}- |\text{eg}\rangle \langle\text{eg}|)} \\

& = \cos\left(\frac{\omega_0 t}{2}\right)

</math>,

\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 last3rd step can be seenproved to follow e.g.by fromusing a [[Taylor series]] expansion with the fact that <math>|\text{g}\rangle\langle\text{g}| + |\text{e}\rangle\langle\text{e}| = 1</math>, and due tousing the orthogonality of the states <math>|\text{g}\rangle</math> and <math>|\text{e}\rangle</math>. The substitution for <math>H_0</math> in the second step being different from the definition given in the previous section can be justified either by shifting the overall energy levels such that <math>|\text{g}\rangle</math> has energy <math>0</math> and <math>|\text{e}\rangle</math> has energy <math>\hbar\omega_0</math>, or by notingNote that a multiplication by an overall phase (of <math>e^{i \omega_0 t/2}</math> in this case) on a unitary operator does not affect the underlying physics, so in the further usages of <math>U</math> we will neglect it. WeApplying now<math>U</math> havegives:

: <math>\begin{align}

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&= -\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, and finally transform the approximate Hamiltonian <math>H_{1,I}^{\text{RWA}}</math> back to the Schrödinger picture:

: <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^\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_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}|.