$Sp(4;\mathbb{R})$ Squeezing for Bloch Four-Hyperboloid via The Non-Compact Hopf Map

Abstract

We explore the hyperbolic geometry of squeezed states in the perspective of the non-compact Hopf map. Based on analogies between squeeze operation and $Sp(2,\mathbb{R})$ hyperbolic rotation, two types of the squeeze operators, the (usual) Dirac- and the Schwinger-types, are introduced. We clarify the underlying hyperbolic geometry and $SO(2,1)$ representations of the squeezed states along the line of the 1st non-compact Hopf map. Following to the geometric hierarchy of the non-compact Hopf maps, we extend the $Sp(2; \mathbb{R})$ analysis to $Sp(4; \mathbb{R})$ --- the isometry of an split-signature four-hyperboloid. We explicitly construct the $Sp(4; \mathbb{R})$ squeeze operators in the Dirac- and Schwinger-types and investigate the physical meaning of the four-hyperboloid coordinates in the context of the Schwinger-type squeezed states. It is shown that the Schwinger-type $Sp(4;\mathbb{R})$ squeezed one-photon state is equal to an entangled superposition state of two $Sp(2;\mathbb{R})$ squeezed states and the corresponding concurrence has a clear geometric meaning. Taking advantage of the group theoretical formulation, basic properties of the $Sp(4;\mathbb{R})$ squeezed coherent states are also investigated. In particular, we show that the $Sp(4; \mathbb{R})$ squeezed vacuum naturally realizes a generalized squeezing in a 4D manner.