The vanilla description of stimulated emission (e.g. in the context of an inverted population laser gain medium) says that a photon with some state vector specifying its energy / polarization / direction / etc. will approach an atom or molecule in an excited state, and (via induction of a dipole?) induce the production of a second photon with the same state vector (i.e. the same energy / polarization / direction / etc.). Ignoring anything exotic, like two- or multiphoton emission, if the emitted photon has some energy $\Delta E$, then the atom in the excited state must have a ground state or other eigenstate $\approx \Delta E$ below the excited state's energy range (which, for example, makes stimulated emission in some materials more efficient or higher yield than in others).
Taken to the extreme, this statement seems to become ridiculous, since it implies that the atom or molecule interacting with the EM field of the incident photon can somehow perfectly measure the components of this photon's state vector. So I suppose we can modify the statement to say that this measurement can be performed with an accuracy given by Heisenberg's uncertainty relation. But is the distribution for the difference in energy / direction / polarization actually much broader than this? How can we characterize it?