Relaxation time and spatial width

[KFC]

Consider a two-level system, let T1 be the relaxation time for the excited system and T2 is the relaxation time for coherence. If we use a Gaussian (with small constant waist w) $$exp(-x^2/w^2)$$ to shine on the sample, because the waist is small, propably only the atoms around x=0 will be excited. Note that the excited atoms will emit lights eventually. If we collect the lights from the sample, we will get a Gaussian profile with constant waist eventually. What interesting is if we do the same thing on two medium, for one T1/T2 is small but for the other, T1/T2 is big. You will find that the waist of the emitted Gaussian spatial profile for big T1/T2 is smaller than that for small T1/T2, why's that? What's the significance for big T1 and small T2? How does this affect the waist of the emitted light?

 

[azntoon]

The significance of having a large T1 and a small T2 is that it leads to a shorter lifetime of the excited state. This means that the atoms around x=0 will emit light faster than atoms in other regions, resulting in a narrower Gaussian profile. The smaller T2 means that the coherence between the excited and ground state will quickly diminish, which also contributes to a narrower Gaussian profile. In summary, having a large T1 and a small T2 leads to a faster relaxation from the excited state, resulting in a narrower Gaussian profile.