Two different schemes are proposed for deterministically generating a pure motion-entangled state for two well-separated ions trapped in a cavity. The first one is that through adjusting the amplitudes and the frequencies of transverse lasers, the two ions trapped in a bimodal cavity can unconditionally evolve into a two-mode squeezed vacuum state with the aid of the cavity dissipation. The second one is based on the interactions of two appropriate series of the driving laser pulses with the two ions trapped in a single-mode cavity, it allows us to produce a two-mod...
