Slicing the Fock space for state production and protection

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 $\left\{ \left\vert N\right\rangle \text{,}\left\vert N+1\right\rangle \right\}$.