Deformações quânticas de sistemas relativísticos e suas conexões com sistemas ópticos.

Abstract

In this thesis, we aim to analyze the effects of quantum deformation on the dynamics of collapses and revivals of the spin inversion of an optical system. This deformation occurs in the structures of spacetime, called the κ-deformation. To carry out this analysis, we will map the Hamiltonian of the κ-deformed Dirac oscillator onto the Jaynes-Cummings Hamiltonian. Firstly, we show how to map the Dirac oscillator in (2+1) dimensions in the Jaynes-Cummings and anti-Jaynes-Cummings system simultaneously, without the deformation. We show that using a suitable basis of Dirac matrices to write the system in (2+1) dimensions, we find a chirality symmetry of the system. From this, we use the κ-deformed Dirac equation obtained from the κ-deformed Poincar ́e generators to write the

Hamiltonian of the κ-deformed Dirac oscillator. We show that the Hamiltonian of the κ- deformed Dirac oscillator is non-Hermitian, but its eigenvalues are real, which characterizes

the Hamiltonian as being quasi-Hermitian. Since the Hamiltonian of the κ-deformed Dirac oscillator is quasi-Hermitian, we can apply a similarity transformation, called Dyson map, which allows us to obtain its Hermitian counterpart, leaving its eigenvalues and expected values invariant. Finally, we show that the deformation adds new entangled states to the system which causes a change in the frequency of Zitterbewegung and, as a consequence of this, the collapses and revivals of the spin inversion of system are also affected. We also show that the total angular momentum in the z direction is not conserved anymore, as a direct consequence of the deformation.