Rotating-wave approximation: Difference between revisions

Revision as of 02:08, 25 March 2022

Revision as of 13:54, 25 March 2022

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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#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>\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 interaction Hamiltonian can then be shown to be (see the Derivation section below)

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</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

: <math>H_{1,I} =

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:<math>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}|

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</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}

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: <math>\begin{align}

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

&= -\hbar\Omega e^{-i\Delta t}e^{-i\omega_0tomega_0 t}|\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}|