Existence of ground states in translation-invariant non-relativistic QED

We consider a translation-invariant non-relativistic QED model describing  an electron interacting with a quantized electromagnetic field.  Due to the translation-invariance one obtains a Hamiltonian for each  fixed total momentum. For zero total momentum we show the existence  of a ground state for all values of the coupling constant using  a compactness argument. In the case of non-zero momentum one has  to pass to a coherent state representation in order that a ground state  exists. Then we can prove the same result for almost all total momenta  being small enough.