What is the most general definition of a bosonic Gaussian state?

Question

I am reading this paper where the definition of the bosonic state is mentioned on page 2 here :- http://arxiv.org/pdf/0806.1625.pdf . From a general definition of any density operator in terms of Wigner functions, it reduces to a Bosonic Gaussian for Gaussian Wigner functions. Is this a more general definition of the Bosonic state or is it equivalent to $\rho=\exp(\sum_{ij}M_{ij}a_{i}^{\dagger}a_{j}) $ where $a_{i}^{\dagger}, a_{j}$ are bosonic creation and annihilation operators. It will be sufficient to get a reference that can describe the Wigner representation more explicitly and relate it to my representation- all the references I found start with the Wigner function.

This post imported from StackExchange Physics at 2014-08-07 15:37 (UCT), posted by SE-user cleanplay