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

TL;DR

This paper constructs and analyzes four-mode squeezed states representing two coupled quantum fields, exploring their properties, structure via symplectic groups, and environmental effects, with relevance to cosmology.

Contribution

It introduces a method to build four-mode squeezed states using symplectic geometry and studies their physical and decoherence properties in a two-field quantum system.

Findings

Four-mode squeezed states generalize two-mode states with quantum transfer between fields.

Decoherence can occur without significantly altering observable power spectra.

The symplectic group $ ext{Sp}(4, ext{R})$ provides a framework for state construction.

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.