Abstract
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.
Highlights
Let us stress that the explicit construction of the squeezed quantum states, especially in the Fock’s space, is of formal interest
By considering that one of the two fields represents the observed system and the other field stands for the environment, the formalism we develop will allow us to go beyond those methods and present exact results
We are inspired by problems formulated in the context of cosmology, it is worth mentioning that the formalism we develop here is generic and broad in applicability
Summary
It provides important insight into the physical mechanisms at play in the dynamics of those states and in the emergence of peculiar properties such as quantum entanglement It is required in a number of concrete computations 2, we introduce the physical setup describing two free scalar fields, both at the classical and quantum levels, and we highlight the symplectic structure that underlies its phase space. This leads us to introducing the symplectic group in four dimensions, Sp(4, R), which we formally describe in Sect.
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