Raman transition - Index of refraction

[Oamitt]

When one calculate the real part of the index of refraction for a specific multilevel atom one would use the following formula:
$$Re(n(\Delta))=1+\frac{N e^2}{8\pi \epsilon_0 m\omega} \sum_i \frac{\Delta_i}{\Delta_i^2+(\gamma/2)^2}K_i$$
Where $K_i$ is the C-G coefficient.

My question is as follow:
How can I calculate this index of refraction in the case of a Raman transition? a.e. in the case where there is a coupling between two fields to create an otherwise forbidden transition.

 

[eljose79]

Calculating the index of refraction for a Raman transition involves a slightly different approach compared to a regular multilevel atom. In a Raman transition, the atom is excited by two laser fields with different frequencies, creating an intermediate state that then decays to a final state through spontaneous emission. This process results in a change in the index of refraction, which can be calculated using the following formula:

Re(n(\Delta))=1+\frac{N e^2}{8\pi \epsilon_0 m\omega_1\omega_2} \sum_i \frac{\Delta_i}{\Delta_i^2+(\gamma_1/2)^2}K_i \frac{\Delta_i}{\Delta_i^2+(\gamma_2/2)^2}

Where \omega_1 and \omega_2 are the frequencies of the two laser fields, and \gamma_1 and \gamma_2 are the corresponding decay rates of the intermediate state. The summation in the formula takes into account all the possible transitions between the intermediate and final states.

In addition to the C-G coefficients, the formula also includes the frequencies and decay rates of the two laser fields. These factors determine the strength of the Raman transition and thus affect the resulting index of refraction.

It is important to note that the index of refraction for a Raman transition can also have an imaginary component, which accounts for the absorption of light by the atom. This can be calculated using the imaginary part of the index of refraction formula, which includes the same factors as the real part formula but with a negative sign in front of the summation.

In summary, calculating the index of refraction for a Raman transition involves considering the frequencies and decay rates of the two laser fields in addition to the C-G coefficients. This formula can then be used to determine the change in the index of refraction and absorption of light in the system.