Rotating-wave approximation: Difference between revisions

Revision as of 06:11, 19 April 2007

Revision as of 19:39, 19 April 2007

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

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> DefiningThis 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}|+\text{H.c.}</math>

(with `H.c.' denoting the [[Hermitean conjugate]]). The interaction Hamiltonian can nowthen 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}|

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===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 laserelectric 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 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}|

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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

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<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)

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

</math>

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