Rotating-wave approximation: Difference between revisions

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

{{Unreferenced|date=December 2009}}

{{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 resonancetransition, 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-ketbra–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 ==

== 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-ketbra–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

e.g. a [[plane wave]] propagating in space. Then under the [[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}_{\text{eg}} \mathrel{:=} \left\langle\text{e}\left|\vec{d}\right|\text{g}\right\rangle</math> allows the dipole operator to be written as

: <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 [[#Derivation|interaction Hamiltonian can then be shown to be (see the Derivation section below)]]

: <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 `''counter-rotating'' consider a [[unitary transformation]] to the [[Interactioninteraction picture|interaction or Dirac picture]] where the transformed Hamiltonian <math>H_{1,I}</math> is given by

-\hbar\left(\tilde{\Omega}^*e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|</math>

: <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

: <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>

</math>

where <math>\Delta \omega \mathrel{:=} \omega_L - \omega_0</math> is the detuning between the light field and the atom.

: <math>H_{1,I}=-\hbar\left(\Omega e^{-i\Delta 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 t}\right)|\text{g}\rangle\langle\text{e}|,</math>

where <math>\Delta:=\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} =

H^ \textfrac{RWA}=\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}|

- \frac{\hbar\Omega e^omega_0}{-i\omega_Lt2}|\text{eg}\rangle\langle\text{g}|

- \hbar\Omega^* e^{-i\omega_Lt}|\text{ge}\rangle\langle\text{eg}|.

- \hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt}+\Omega^*e^{i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}|</math>.

</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"/>

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.

Line 57 ⟶ 66:

: <math>\begin{align}

H_1 & = -\vec{d}\cdot\vec{E} \\

: <math>U = e^{iH_0t/\hbar} &= e^{i -\omega_0 t |left(\textvec{ed}\rangle \langle_\text{eeg}|} = |\text{ge}\rangle \langle\text{g}| +e^{i \omega_0 tvec{d} _\text{eg}^*|\text{eg}\rangle \langle\text{e}|</math>,\right)

H_1 &= -\vec{d}\cdot\vec{E} \\

&=- \cdot \left(\vec{dE}_\text_0 e^{eg}|-i\text{e}\rangle\langle\text{gomega_Lt}| + \vec{d}_\text{egE}_0^*|\text e^{g}i\rangle\langle\text{eomega_Lt}|\right) \\

\cdot &= -\left(\vec{Ed}_0e^_\text{-i\omega_Lteg}+ \cdot \vec{E}_0^* e^{-i\omega_Lt}\right) \\

&=-\left( +\vec{d}_\text{eg} \cdot \vec{E}_0e_0^* e^{-i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

+-\left(\vec{d}_\text{eg}^* \cdot \vec{E}_0^* e^{-i\omega_Lt}\right)|\text{e}\rangle\langle\text{g}|

-\left( +\vec{d}_\text{eg}^* \cdot \vec{E}_0e_0^* e^{-i\omega_Lt}\right)|\text{g}\rangle\langle\text{e}| \\

&= -\hbar\left(\Omega e^{-i\omega_Lt} + \vectilde{d}_\text{egOmega}^*\cdot\vec{E}_0^* e^{i\omega_Lt}\right)|\text{ge}\rangle\langle\text{eg}| \\

&= -\hbar\left(\tilde{\Omega}^* e^{-i\omega_Lt} +\tilde{ \Omega}^* e^{i\omega_Lt}\right)|\text{eg}\rangle\langle\text{ge}|,

-\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 last3rd step can be seenproved toby follow e.g. fromusing a [[Taylor series]] expansion, and due tousing 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 havewill neglect it. Applying <math>U</math> gives:

: <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, 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_0tomega_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}|.

\end{align}</math>

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

: <math>

H^\text{RWA} = H_0 + H_1^{\text{RWA}} =
\frac{\hbar\omega_0}{2}|\text{e}\rangle\langle\text{e}| -

- \frac{\hbar\Omega e^omega_0}{-i\omega_Lt2}|\text{eg}\rangle\langle\text{g}| -

- \hbar\Omega^* e^{-i\omega_Lt}|\text{ge}\rangle\langle\text{eg}|. -

\hbar\Omega^*e^{i\omega_Lt}|\text{g}\rangle\langle\text{e}|.

</math>

==References==

{{Reflist}}

{{DEFAULTSORT:Rotating Wave Approximation}}

[[Category:Atomic, molecular, and optical physics]]

[[Category:Chemical physics]]

[[ko:회전파 근사]]