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We construct the four-mode squeezed states and study their physicalproperties. 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-fieldsystems, with additional transfers of quanta between the fields. To build themin the Fock space, we use the symplectic structure of the phase space. For thisreason, we first present a pedagogical analysis of the symplectic group$\mathrm{Sp}(4,\mathbb{R})$ and its Lie algebra, from which we construct thefour-mode squeezed states and discuss their structure. We also study thereduced single-field system obtained by tracing out one of the two fields. Thisprocedure being easier in the phase space, it motivates the use of the Wignerfunction which we introduce as an alternative description of the state. Itallows us to discuss environmental effects in the case of linear interactions.In particular, we find that there is always a range of interaction coupling forwhich decoherence occurs without substantially affecting the power spectra(hence the observables) of the system.

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