Rotating-wave approximation: Difference between revisions

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The '''rotating wave approximation''' is an approximation used in [[atom optics]] and [[magnetic resonance]]. In this approximation, terms in a Hamiltonian which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic resonance, and the intensity is low. Explicitly, terms in the Hamiltonians which oscillate with frequencies <math>\omega_lomega_L+\omega_0 </math>, are neglected, while terms which oscillate with frequencies <math>\omega_lomega_L-\omega_0 </math> are kept, where <math> \omega_lomega_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.

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

[[Category:Chemical physics]]

==Mathematical formulation==

For simplicity consider a [[two-state quantum system|two-level atomic system]] with [[excited state|excited]] and ground states <math>|\text{e}\rangle</math> and <math>|\text{g}\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]] of the atom can be written as

<math>H_0=\hbar\omega_0|\text{e}\rangle\langle\text{e}|</math>

{{atomic-physics-stub}}

Suppose the atom is placed at <math>z=0</math> in an external (classical) [[electric field]] of frequency <math>\omega_L</math>, given by <math>\vec{E}(z,t)=\vec{E}_0(z)e^{-i\omega_Lt}+\vec{E}_0(z)^*e^{i\omega_Lt}</math> (so that the field contains both positive- and negative-frequency modes in general). Then under the [[dipole approximation]] the [[interaction Hamiltonian]] can be expressed as

<math>H_I=-\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_I.</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> 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}|+\text{H.c.}</math>

(with `H.c.' denoting the [[Hermitean conjugate]]). The interaction Hamiltonian can now be shown to be (see the Derivations section below)

<math>H_I=-\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</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 or Dirac picture where the transformed Hamiltonian <math>\bar{H}</math> is given by

<math>\bar{H}=-\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>

===Making the 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 laser 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 are negligible and the Hamiltonian can be written in the interaction picture as

<math>\bar{H}_\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, in the [[Schrödinger picture]] the Hamiltonian is given by

<math>

H_\text{RWA}=\hbar\omega_0|\text{e}\rangle\langle\text{e}|

-\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>

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.

==Derivations==

Given the above definitions the interaction Hamiltonian is

<math>\begin{align}

H_I &= -\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}_0e^{-i\omega_Lt}+\vec{E}_0^*e^{i\omega_Lt}\right) \\

&=-\left(\vec{d}_\text{eg}\cdot\vec{E}_0e^{-i\omega_Lt}

+\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^{-i\omega_Lt}

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

&=-\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}|,

\end{align}</math>

as stated. The next stage is to find the Hamiltonian in the interaction picture, <math>\bar{H}.</math> The unitary operator required for the transformation is

<math>U=e^{iH_0t/\hbar}</math>

and an arbitrary state <math>|\psi\rangle</math> transforms to <math>|\bar{\psi}\rangle=U|\psi\rangle.</math> The [[Schrödinger equation]] must still hold in this new picture, so

<math>

\bar{H}|\bar{\psi}\rangle

=i\hbar\partial_t|\bar{\psi}\rangle

=i\hbar\dot{U}|\psi\rangle+Ui\hbar\partial_t|\psi\rangle

=\left(i\hbar\dot{U}+UH\right)|\psi\rangle

=\left(i\hbar\dot{U}U^\dagger+UHU^\dagger\right)|\bar{\psi}\rangle,

</math>

where a dot denotes the [[time derivative]]. This shows that the new Hamiltonian is given by

<math>\begin{align}

\bar{H}&=i\hbar\dot{U}U^\dagger+UHU^\dagger

=i\hbar\left(\frac{i}{\hbar}UH_0\right)U^\dagger+U(H_0+H_I)U^\dagger

=UH_IU^\dagger \\

&=-e^{i\omega_0t|\text{e}\rangle\langle\text{e}|}\left(

\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}|\right)

e^{-i\omega_0t|\text{e}\rangle\langle\text{e}|}\\

&=-\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>

where <math>\Delta:=\omega_L-\omega_0</math> is the detuning of the light field.

The penultimate equality can be easily seen from the [[series expansion]] of the exponential map and the fact that

<math>\langle\text{i}|\text{j}\rangle=\delta_\text{ij}</math> for i and j each equal to e or g (and <math>\delta_\text{ij}</math> the [[Kronecker delta]]).

The final step is to transform the approximate Hamiltonian back to the Schrödinger picture. The first line of the previous calculation shows that

<math>\bar{H}=UH_IU^\dagger</math>, so in the same manner as the last calculation,

<math>\begin{align}

H_{I,\text{RWA}}&=U^\dagger\bar{H}_\text{RWA}U \\

&=e^{-i\omega_0t|\text{e}\rangle\langle\text{e}|}

\left(-\hbar\Omega e^{-i\Delta t}|\text{e}\rangle\langle\text{g}|

-\hbar\Omega^*e^{i\Delta t}|\text{g}\rangle\langle\text{e}|\right)

e^{i\omega_0t|\text{e}\rangle\langle\text{e}|} \\

&=-\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>

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_{I,\text{RWA}} = \hbar\omega_0|\text{e}\rangle\langle\text{e}|

-\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>

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

[[Category:Chemical physics]]