Four-mode squeezed states: two-field quantum systems and the symplectic group $\mathrm {Sp}(4,{\mathbb {R}})$ - Archive ouverte HAL Access content directly
Journal Articles Eur.Phys.J.C Year : 2022

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

(1, 2) , (2) , (1)
1
2
Julien Grain
• Function : Author
Vincent Vennin

#### 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.

### Dates and versions

hal-03229409 , version 1 (20-06-2022)

### Licence

Attribution - CC BY 4.0

### Identifiers

• HAL Id : hal-03229409 , version 1
• ARXIV :
• DOI :
• INSPIRE :

### Cite

Thomas Colas, Julien Grain, Vincent Vennin. Four-mode squeezed states: two-field quantum systems and the symplectic group $\mathrm {Sp}(4,{\mathbb {R}})$. Eur.Phys.J.C, 2022, 82 (1), pp.6. ⟨10.1140/epjc/s10052-021-09922-y⟩. ⟨hal-03229409⟩

### Export

BibTeX TEI Dublin Core DC Terms EndNote Datacite

69 View
9 Download

### Share

Gmail Facebook Twitter LinkedIn More