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, [tex]\rho_{ee}[/tex] and it turns out that [tex]\rho_{ee}[/tex] is a function of a parameter G, which could be space dependent, i.e.

[tex]\rho_{ee} = \rho_{ee}(G(x))[/tex]

as we know, [tex]\rho_{ee}[/tex] 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 [tex]\rho_{ee}(G(x))[/tex] turns out to be somewhat localized around x=0, so what can I tell about the linewidth of the emitted light?