Exact mappings between quantum relativistic and quantum optical models

We develop a novel quantum optical perspective into a couple of quantum relativistic systems: i) First we show how the two-dimensional extension of the harmonic oscillator, known as the Dirac oscillator, can be exactly mapped onto a chiral Anti-Jaynes-Cummings model of quantum optics. This equivalence allows us to predict a series of novel relativistic phenomena, such as spin-orbit {\it Zitterbewegung}. Furthermore, we also make a realistic experimental proposal, at reach with current technology, for studying the equivalence of both models using a single trapped ion [1]. ii) Second, we show that a relativistic version of Schr\"{o}dinger cat states, here called {\it Dirac cat states}, can be built in relativistic Landau levels when an external magnetic field couples to a relativistic spin $1/2$ charged particle. Under suitable initial conditions, the associated Dirac equation produces unitarily Dirac cat states involving the orbital quanta of the particle in a well defined mesoscopic regime. These states have a purely relativistic origin and cease to exist in the non-relativistic limit [2]. \newline [1] A. Bermudez et. al, Phys. Rev. A.\textbf{76}, 041801(R) (2007). \newline [2] A. Bermudez et. al, Phys. Rev. Lett.\textbf{99}, 123602 (2007).