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

TL;DR

This work introduces parity-deformed Wigner algebras by incorporating a reflection operator $R$ with $R = (-1)^N$ into the oscillator algebra, producing the deformed commutator $[a, a^{\dagger}] = 1 + 2 \nu R$. It develops both single- and two-mode Wigner algebras and constructs two realizations of the deformed $sl(2,\mathbb{R})$ via Jordan–Schwinger and Holstein–Primakoff, including the necessary extra generators to close the algebra, with explicit actions on basis states. It also derives a parity-deformed $so_{\nu}(3)$ algebra through the isomorphism with $su_{\nu}(2)$ and provides odd/even representations for $2j$, presenting matrix realizations for several low-dimensional cases. The results offer a new mathematical framework for quantum optics and quantum communications, with potential applications to qubit/qutrit systems, entanglement transfer, and generalized light–matter Hamiltonians, and they point to future work on dynamical models and photonic networks in the parity-deformed setting.

Abstract

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, $sl_ν(2,R)$ and the deformed $so(3)$ algebra, $so_ν(3)$, 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 $so_ν(3)$ representation is analyzed based on the isomorphy of $so(3)$ and $su(2)$. Related applications are discussed as well.