It is possible to have photon emission when measuring state of an atom? Suppose in the hydrogen atom we have a state $$\Psi=a\phi_1+b\phi_2$$ where 
$\phi_1$ is the ground state, $\phi_2$  an eigenstate different from the ground state and $a$ + $b$ constant such that  $$a^2+b^2=1$$
Now suppose that we measure the the system and we find it in the ground state $\phi_1$. Is it then possible to have photon emission? 
 A: 
Now suppose that we measure the the system and we find it in the ground state $\phi_1$. Is it then possible to have photon emission? 

No. The previous superposition becomes irrelevant once you perform a projective measurement that results in the system ending up in the ground state. The system is then in the ground state and has no energy to radiate away, so no photon emission is possible.
If your question is whether the measurement process itself can entail a photon emission, then generally speaking this will depend on the measurement procedure. The overall answer is also negative, but it's impossible to provide more details without knowing how you intend to do the measurement.
A: The answer depends whether the hamiltonian of your system takes into account a photon field. From what I have seen, there aren't really great models (hamiltonians) of this in general. 
Quantum Field Theory (the formalism of the standard model) uses a hamiltonian with interactions between the atom and a field of light $A_\mu$. However it only describes the scattering regime and in order to write down the state of a hydrogen atom, the relativistic formalism has to be mixed with the usual Quantum Mechanics formalism, because there is no multi-particle position-space description which is lorentz invariant. That is because >1 particle formalism must have >1 position argument in the wave function, but typically in QM there is only one time argument. However to do a proper Lorentz transformation one needs just as many time as position arguments. Work on "multi-time" wave functions has been done but my vague awareness of the field has led me to believe that a fully consistent formalism hasn't come out of it yet. Nonetheless I consider it a valuable endeavor.
Another alternative, which describes states not just in the scattering regime, is the Jaynes-Cummings Model. This is a sort of toy model for the interactions between particles and light which really works quite well but has limitations due to its simplicity. That's probably more what you're looking for.
