In the context of Saturated absorption spectroscopy, I'm having trouble modeling stimulated emission, and getting the result that is written in articles, such as this article. I tried to use a non-relativistic calculation as a start, for an atom with 2-level separated by $\hbar\omega_0$:
- A 2 level atom of momentum $p_1$ in the ground state is absorbing a photon of frequency $\omega_1$. After absorption the atom has momentum $p_2$.
- A photon of frequency $\omega_1$ from the other direction is stimulating the excited atom to decay to the ground state, and two photons of frequencies $\omega_2$ and $\omega_3$ are emitted, and the atom ends in a momentum $p_3$.
What I'm not sure of is the following:
- Is one of the 2 photons emitted eventually is one of the original photons? Meaning, is $\omega_1 = \omega_2 \ne \omega_3$? Or perhaps $\omega_2=\omega_3$
- I know that $\omega_2$ and(or) $\omega_3$ should be Doppler shifted from $\omega_0$ due to the atom's momentum during the stimulated emission. I'm not sure whether I should Doppler shift these(this) $\omega$(s) relative to momentum $p_2$ or $p_3$ - the momentum before or after the stimulated emission?
I tried to write energy and momentum conservation for both of the above events, and calculate $\omega_1$ as a function of $\omega_0$, using Mathematica, and I got very complicated results, that even when Taylor expanded for small $R=\frac{\hbar\omega_0}{2mc^2}$, don't converge to the known and commonly cited result of:
$$ \nu_{measured}= \nu_{real}(1\pm R)$$
Where $+$ is for absorption and $-$ is for stimulated emission.
Thankfully, I did manage though to compute the above result for absorption.
