Parity-deformed $sl(2,R)$, $su(2)$ and $so(3)$ Algebras: a Basis for Quantum Optics and Quantum Communications Applications

Having in mind the significance of parity (reflection) in various areas of physics, the single-mode and two-mode Wigner algebras are considered adding to them a reflection operator. The associated deformed $sl(2, R)$ algebra, $sl_{\nu}(2,R)$ and the deformed $so(3)$ algebra, $so_{\nu}(3)$, are constructed for the widely used Jordan-Schwinger and Holstein-Primakoff realizations, commenting on various aspects and ingredients of the formalism for both single-mode and two-mode cases. Finally, due to its potential application in the study of qubit and qutrit systems, the parity-deformed $so_{\nu}(3)$ representation is analyzed based on the isomorphy of $so(3)$ and $su(2)$. Related applications are discussed as well.