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 $leftvert 0rightrangle $ to $leftvert Mrightrangle $ ($leftvert Nrightrangle $ to$ infty$), and the sliced one confined to Fock subspace ranging from $leftvert Mrightrangle $ to $leftvert Nrightrangle $, whatever $M<N$. Whereas the upper-bounded JC or AJC interactions is shown to drive any initial state to a steady Fock state $leftvert Nrightrangle $, the sliced one is shown to produce steady superpositions of Fock states confined to the sliced subspace $left{ leftvert Nrightrangle text{,}leftvert N+1rightrangle right} $.