Deformed Clifford algebra and supersymmetric quantum mechanics on a phase space with applications in quantum optics

TL;DR

This paper develops a novel approach combining deformed Clifford algebra and supersymmetric quantum mechanics on phase space, with applications to quantum optics models like Jaynes-Cummings, providing explicit spectra, eigenstates, and constants of motion.

Contribution

It introduces a new method of extending Clifford algebra with Moyal star-product for supersymmetric quantum mechanics on phase space, applied to quantum optics models.

Findings

Derived isospectral matrix Hamiltonians with common bosonic parts

Obtained spectra, eigen-spinors, and Wigner functions for models

Presented constants of motion within deformation quantization framework

Abstract

In order to realize supersymmetric quantum mechanics methods on a four dimensional classical phase-space, the complexified Clifford algebra of this space is extended by deforming it with the Moyal star-product in composing the components of Clifford forms. Two isospectral matrix Hamiltonians having a common bosonic part but different fermionic parts depending on four real-valued phase space functions are obtained. The Hamiltonians are doubly intertwined via matrix-valued functions which are divisors of zero in the resulting Moyal-Clifford algebra. Two illustrative examples corresponding to Jaynes-Cummings-type models of quantum optics are presented as special cases of the method. Their spectra, eigen-spinors and Wigner functions as well as their constants of motion are also obtained within the autonomous framework of deformation quantization.