Rotating-wave approximation: Difference between revisions

Revision as of 09:46, 11 December 2020

Revision as of 02:08, 25 March 2022

Line 1:

{{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]] which 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 which oscillate with frequencies <math>\omega_L + \omega_0 </math> are neglected, while terms which 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.

== 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

Line 15 ⟶ 14:

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>\langle\text{e}|\vec{d}|\text{e}\rangle=\langle\text{g}|\vec{d}|\text{g}\rangle=0.</math> This means that defining <math>\vec{d}_{\text{eg}}:=\langle\text{e}|\vec{d}|\text{g}\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 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 [[Interaction 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

-\hbar\left(\tilde{\Omega}^* e^{-i\omega_LtDelta t} + \tilde{\Omega^*}e^{i(\omega_Ltomega_L + \omega_0)t}\right)|\text{ge}\rangle\langle\text{eg}|</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>

</math>

:where <math>H_{1,I}=-\hbar\left(\Omega e^{-i\Delta t}+\tildemathrel{\Omega:=}e^{i( \omega_L+ - \omega_0)t}\right)|\text{e}\rangle\langle\text{g}|</math> is the detuning between the light field and the atom.

-\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 \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 t}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^* e^{i\Delta 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^\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_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"/>

Line 62 ⟶ 64:

: <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>U =

: <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>,

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>,

where the last step can be seen to follow e.g. from 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 to 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 noting that a multiplication by an overall phase (<math>e^{i \omega_0 t/2}</math> in this case) on a unitary operator does not affect the underlying physics. We now have

: <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 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}|\ .

\end{align}</math>

Line 91 ⟶ 96:

: <math>\begin{align}

H_1^{\text{RWA}} &= U^\dagger H_{1,I}^{\text{RWA}} U \\

&= -\hbar\Omega e^{-i\Delta t}e^{-i\omega_0t}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^* e^{i\Delta 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>

Line 101 ⟶ 106:

:<math>

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_Lt}|\text{e}\rangle\langle\text{g}| -

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

</math>