Effects of quantum deformation on the Jaynes-Cummings and anti-Jaynes-Cummings models

The theory of non-Hermitian systems and the theory of quantum deformations have attracted a great deal of attention in the past decades. In general, non-Hermitian Hamiltonians are constructed by an ad hoc manner. Here, we study the (2+1) Dirac oscillator and show that in the context of the $\kappa$-deformed Poincar\'e-Hopf algebra its Hamiltonian is non-Hermitian but has real eigenvalues. The non-Hermiticity stems from the $\kappa$-deformed algebra. From the mapping in Bermudez et al., Phys. Rev. A 76, 041801(R) (2007), we propose the $\kappa$-Jaynes-Cummings and $\kappa$-anti-Jaynes-Cummings models, which describe an interaction between a two-level system with a quantized mode of an optical cavity in the $\kappa$-deformed context. We find that the $\kappa$-deformation modifies the Zitterbewegung frequencies and the collapses and revivals of quantum oscillations. In particular, the total angular momentum in the $z$ direction is not conserved anymore, as a direct consequence of the deformation.