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m Moved the definition of Delta to *before* its first use :) |
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Line 24:
<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}|
where <math>\Delta:=\omega_L-\omega_0</math> is the detuning of the light field.▼
===Making the approximation===
Line 84 ⟶ 86:
-\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]]).
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