# AC Stark shift in a 3 level system

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• BillKet
BillKet
Hello! I want to make sure I got this right. Say we have a 3 level system ordered such that ##E_1<E_2<<E_3## with ##\omega_0 = E_2-E_1##. I use a laser to measure ##\omega_0## (e.g. I scan the laser frequency across the expected location of the resonance and fit that with a Voigt profile, we can ignore any systematic effects of the measurement). If I had just a 2 level system, the center of the peak would be exactly at ##\omega_0##. Now, let's say that the electric field can also couple ##E_2## and ##E_3## (but ##E_1## and ##E_3## can't be coupled in the electric dipole approximation). The formula for the AC Stark shift in this case is given by

$$E_{\mathrm{Stark}} = \frac{\Omega^2\omega}{2(\omega^2-\omega_0^2)}$$

where ##\Omega## is the Rabi frequency of the field between ##E_2## and ##E_3## and ##\omega = E_3-E_2##. In our limit this becomes:

$$E_{\mathrm{Stark}} = \frac{\Omega^2}{2\omega}$$
So the ##E_2## levels gets pushed down (towards the ##E_1##) by this amount. So the resonant frequency in this case will appear to be at:

$$\omega_0-\frac{\Omega^2}{2\omega}$$

Is this right (at least to first order in perturbation theory as the 2 pair of levels are treated independently)? Thank you!

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