F. O. Prado

In this letter we present a protocol to engineer interactions confined to subspaces of the Fock space in trapped ions: we show how to engineer upper-, lower-bounded and sliced Jaynes-Cummings (JC) and anti-Jaynes-Cummings (AJC) Hamiltonians. The upper-bounded (lower-bounded) interaction acting upon Fock subspaces ranging from $\left\vert 0\right\rangle $ to $\left\vert M\right\rangle $ ($\left\vert N\right\rangle $ to$\ \infty$), and the sliced one confined to Fock subspace ranging from $\left\vert M\right\rangle $ to $\left\vert N\right\rangle $, whatever $M<N$. Whereas the upper-bounded JC or AJC interactions is shown to drive any initial state to a steady Fock state $\left\vert N\right\rangle $, the sliced one is shown to produce steady superpositions of Fock states confined to the sliced subspace $\left\{ \left\vert N\right\rangle \text{,}\left\vert N+1\right\rangle \right\} $.

In this paper we extend current perspectives in engineering reservoirs by producing a time-dependent master equation leading to a nonstationary superposition equilibrium state that can be nonadiabatically controlled by the system-reservoir parameters. Working with an ion trapped inside a nonindeal cavity we first engineer effective Hamiltonians that couple the electronic states of the ion with the cavity mode. Subsequently, two classes of decoherence-free evolution of the superposition of the ground and decaying excited levels are achieved: those with time-dependent azimuthal or polar angle. As an application, we generalise the purpose of an earlier study [Phys. Rev. Lett. 96, 150403 (2006)], showing how to observe the geometric phases acquired by the protected nonstationary states even under a nonadiabatic evolution.