  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.
