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arXiv:2003.04257

Squeeze operator: a classical view

Angel García-Chung

Published Mar 9, 2020• 0 citations
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Abstract

In this paper we show the relation between $sp(4,\mathbb{R})$, the Lie algebra of the symplectic group, and the elements of the symplectic group $Sp(4,\mathbb{R})$. We use this relation to provide a classical analog of the squeeze operator $\widehat{S}(ζ)$. This classical squeeze matrix shares some similarities with the correlation matrix ${\bf V}^{(2)}$ and its amount of squeezing is half of that in the correlation matrix.

DOI
10.48550/arxiv.2003.04257

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In this paper we show the relation between sp(4,R)sp(4,\mathbb{R})sp(4,R), the Lie algebra of the symplectic group, and the elements of the symplectic group Sp(4,R)Sp(4,\mathbb{R})Sp(4,R). We use this relation to provide a classical analog of the squeeze operator S^(ζ)\widehat{S}(ζ)S(ζ). This classical squeeze matrix shares some similarities with the correlation matrix V(2){\bf V}^{(2)}V(2) and its amount of squeezing is half of that in the correlation matrix.
In this paper, we show the relation between sp(4,R)sp(4,\mathbb{R})sp(4,R), the Lie algebra of the symplectic group, and the elements of Sp(4,R)Sp(4,\mathbb{R})Sp(4,R). We use this result to obtain some special cases of symplectic matrices relevant to the study of squeezed states. In this regard, we provide some applications in quantum mechanics and analyze the squeezed polymer states obtained from the polymer representation of the symplectic group. Remarkably, the polymer's dispersions are the same as those obtained for the squeezed states in the usual representation.
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Four-mode squeezed states: two-field quantum systems and the symplectic group Sp(4,R)\mathrm {Sp}(4,{\mathbb {R}})Sp(4,R)
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 Sp(4,R)\mathrm {Sp}(4,{\mathbb {R}})Sp(4,R) Sp ( 4 , 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.
Coordinate wave functions for the one-mode squeezed states produced by the quantum analog of the general linear transformation in phase space are calculated. The probability density [\ensuremath{\Vert}\ensuremath{\psi}(q)${\ensuremath{\Vert}}^{2}$] for these states is Gaussian with center predicted by the classical transformation. The quantum image (which includes the traditional two-mode squeeze operator) of a three-parameter symplectic map in two-mode phase space equally generates squeezed states having Gaussian \ensuremath{\Vert}\ensuremath{\psi}(q1{q}_{1}q1​,q1{q}_{1}q1​)${\ensuremath{\Vert}}^{2}$. The center of the two-mode Gaussian is again predicted by the classical mapping.