Optical phases and the symplectic group

Arvind, Biswajit Dutta, Narasimhaiengar Mukunda +1 more

Mark H. Kauderer

The n-dimensional symplectic group is discussed and classified in terms of operators for the first time to my knowledge. The symplectic embedding theorem is proved by using Kronecker block multiplication. In addition, the fundamental relation of Fourier optics is proved. It follows that all operator relations have been proved for Fourier optics. In combination with a previous result, the decomposition theorem for nonsingular matrices [ J. Math. Phys.12, 1772 ( 1971)], which is extended here somewhat, it follows that Fourier optics (excluding the phase-conjugate operator) is isomorphic to the symplectic group, the group of linear canonical transforms, and the group of first-order systems. This implies that every mathematical result concerning the symplectic group can be understood simply as a property of ideal cylindrical lenses under the paraxial approximation.

Kay-Michael Voit

M.C.B. Fernandes, Basil J. Hiley

In this paper we discuss a simplified approach to the symplectic Clifford algebra, the symplectic Clifford group and the symplectic spinor by first extending the Heisenberg algebra. We do this by adding a new idempotent element to the algebra. It turns out that this element is the projection operator onto the Dirac standard ket. When this algebra is transformed into the Fock algebra, the corresponding idempotent is the projector onto the vacuum. These additional elements give a very simple way to write down expressions for symplectic spinors as elements of the symplectic Clifford algebra. When this algebra structure is applied to an optical system we find a new way to understand the Gouy effect. It is seen to arise from the covering group, which in this case is the symplectic Clifford group, also known as the metaplectic group. We relate our approach to that of Simon and Mukunda who have shown the relation of the Gouy phase to the Berry phase.

Aleksey K. Fedorov, Evgeniy O. Kiktenko

Quantum tomography for continuous variables is based on the symplectic transformation group acting in the phase space. A particular case of symplectic tomography is optical tomography re-lated to the action of a special orthogonal group. In the tomographic description of spin states, the connection between special unitary and special orthogonal groups is used. We analyze the represen-tation for spin tomography using the Cayley–Klein parameters and discuss an analog of symplectic tomography for discrete variables. We propose a representation for tomograms of discrete variables through quaternions and employ the qubit-state tomogram to illustrate the method elaborated.