Sp(4;R) squeezing for Bloch four-hyperboloid via the non-compact Hopf map

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2020-01, Vol.53 (5), p.55303
• Main Author: Hasebe, Kazuki
• Format: Article
• Language: English
• Subjects: geometry of quantum states; Hopf fibration; hyperbolic geometry; quantum optics; squeezed states
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ab3cda

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