$Sp(4; \mathbb{R})$ 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, 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; R) analysis to Sp(4; R) -the isometry of an split-signature four-hyperboloid.We explicitly construct the Sp(4; 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; R) squeezed one-photon state is equal to an entangled superposition state of two Sp(2; 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; R) squeezed coherent states are also investigated. In particular, we show that the Sp(4; R) squeezed vacuum naturally realizes a generalized squeezing in a 4D manner. Sp(2; R) group and squeezingThe isomorphism Sp(2; R) ≃ Spin(2, 1) suggests that the Sp(2; R) one-and two-mode operators are equivalent to the Majorana and the Dirac spinor operators of SO(2, 1). Based on the identification of the squeeze operator with the SU (1, 1) ≃ Spin(2, 1) "rotation" operator, we introduce two types of squeeze operators, the (usual) Dirac-and Schwinger-types. We discuss how the non-compact 1st Hopf map is embedded in the geometry of the Sp(2; R) squeezed state. 7 U Sp(2n) = U (n; H) and π 1 (U Sp(2n)) = 1. For instance, U Sp(2) = SU (2) = Spin(3), U Sp(4) = Spin(5). 8 The anti-de Sitter, de Sitter and Euclidean anti-de Sitter spaces are realized as the special cases of the ultra-hyperboloids:

Wigner function for SU(1,1)

“…We can safely take |α| > |β|, for the opposite case can be obtained by just a relabelling of modes, with no physical consequences. The parameters χ and τ can be interpreted as azimuthal and "polar" angles on a two-sheeted hyperboloid H 2 [64]. A similar parametrization as in (15), wherein the hyperbolic functions are replaced with trigonometric ones, maps two complex modes into the Bloch sphere S 2 .…”

Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

“…Refs. [29][30][31][32][33]. As a consequence, it may be skipped by those readers already familiar with the use of symplectic groups in quantum mechanics.…”

The Quantum Dynamics of Two-component Bose-Einstein Condensate: an $Sp(4,R)$ Symmetry Approach

“…By diagonalizing gH k , where g = diag(1, 1, −1, −1), we can get the quasiparticles, whose annihilation operators are given by the Bogoliubov transformation α i,k = 2 j=1 u ij a j,k + v ij a † j,−k , where u, v are general 2 × 2 complex matrices, if the eigen-energies of these two excited states both are real. And the Bogoliubov transformation has the property [16,17]…”

“…where i, j, k, l = {1, 2}, with ten of them independent, they satisfy the commutation relations of the Lie algebra of the real symplectic group Sp(4, R) [16,17], i.e.…”

“…UU † −VV † = I, UV T = VU T }, where U, V are 2×2 complex matrices [16,17]. The ground state can be parameterized as a point in a more complicated 6-dimensional space Sp(4, R)/U (2), which can be shown to reduce to the direct product of two Poincare disk in certain limit.…”