Spatial linewidth and density matrix

[KFC]

Hi there,
I am thinking an interesting problem of spatial linewidth of two-level system. Suppose in some way I find out an element of the desinty matrix for the upper state of two-level system, $$\rho_{ee}$$ and it turns out that $$\rho_{ee}$$ is a function of a parameter G, which could be space dependent, i.e.

$$\rho_{ee} = \rho_{ee}(G(x))$$

as we know, $$\rho_{ee}$$ tells the inversed population of the system, when the atoms jump back to ground state from excited state, one line will be observed. Now, suppose I choose G(x) to be a specific function, e.g. Gaussian, such that $$\rho_{ee}(G(x))$$ turns out to be somewhat localized around x=0, so what can I tell about the linewidth of the emitted light?

 

[DrClaude]

The line width is related to the lifetime of the excited state, hence will depend on the details of the coupling between the excited and the ground state in the Hamiltonian. Simply knowing about the excited state is completely insufficient.