Four-mode squeezed states: two-field quantum systems and the symplectic group $\mathrm{Sp}(4,\mathbb{R})$

We present a pedagogical analysis of the symplectic group $\mathrm{Sp}(4,\mathbb{R})$ and its Lie algebra, and derive new factorised forms of the group elements. These results are then used to describe two linearly-coupled quantum scalar fields. Such systems are found to be placed in four-mode squeezed states, which are constructed explicitly in the Fock space. They are shown to generalise the two-mode squeezed states of single-field systems, with additional transfers of quanta between the two fields. The structure of the state is also investigated in phase space by means of the Wigner function. Finally, we study the reduced single-field system obtained by tracing out one of the two fields. This analysis is done both in the Fock space and in the phase space, and allow us to discuss environmental effects in the case of linear interactions. In particular, we find that there is always a range of interaction coupling for which decoherence occurs without substantially affecting the power spectra (hence the observables) of the system. Applications in the context of cosmology are also discussed