Hi Valeria — Thanks for you question. The M and N matrices are the normal ordered moments of the destruction operators.

As described in lines 39-40 of the file you mentioned they are defined as

N_{i,j} =\langle a_i^\dagger a_j \rangle

M_{i,j} = \langle a_i a_j \rangle

They have the nice property of being independent of \hbar and are zero for the multimode vacuum state.

They are also related to the A and B matrices in Eq. 2.37 of the classical paper by Simon, Mukunda and Dutta (https://journals.aps.org/pra/abstract/10.1103/PhysRevA.49.1567)

If you want to convert between M and N and the standard quadrature covariance matrix V you can use the method `scovmaxp`

to go from M and N to V and you can use `fromscovmat`

to do the opposite.

Also, if you want to play directly with the standard covariance matrices in the quadrature formalism I’d recommend you to have a look at the `symplectic`

module of The Walrus (cf. https://the-walrus.readthedocs.io/en/latest/code/symplectic.html)

Hope this answers your question,

Nicolas