Four-mode squeezed states: two-field quantum systems and the symplectic
  group $\mathrm{Sp}(4,\mathbb{R})$
  We construct the four-mode squeezed states and study their physical
properties. These states describe two linearly-coupled quantum scalar fields,
which makes them physically relevant in various contexts such as cosmology.
They are shown to generalise the usual two-mode squeezed states of single-field
systems, with additional transfers of quanta between the fields. To build them
in the Fock space, we use the symplectic structure of the phase space. For this
reason, we first present a pedagogical analysis of the symplectic group
$\mathrm{Sp}(4,\mathbb{R})$ and its Lie algebra, from which we construct the
four-mode squeezed states and discuss their structure. We also study the
reduced single-field system obtained by tracing out one of the two fields. This
procedure being easier in the phase space, it motivates the use of the Wigner
function which we introduce as an alternative description of the state. It
allows 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.
