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$Sp(4;\mathbb{R})$ Squeezing for Bloch Four-Hyperboloid via The Non-Compact Hopf Map
- Source :
- J.Phys.A: Math.Theor. 53 (2020) 055303
- Publication Year :
- 2019
-
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.<br />Comment: 1+50 pages, 3 fugures, 1 table; minor corrections, an abridged version to appear in JPA
- Subjects :
- Quantum Physics
High Energy Physics - Theory
Mathematical Physics
Subjects
Details
- Database :
- arXiv
- Journal :
- J.Phys.A: Math.Theor. 53 (2020) 055303
- Publication Type :
- Report
- Accession number :
- edsarx.1904.12259
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1088/1751-8121/ab3cda