  We demonstrate quantum entanglement of two trapped atomic ion qubits using a
sequence of ultrafast laser pulses. Unlike previous demonstrations of
entanglement mediated by the Coulomb interaction, this scheme does not require
confinement to the Lamb-Dicke regime and can be less sensitive to ambient noise
due to its speed. To elucidate the physics of an ultrafast phase gate, we
generate a high entanglement rate using just 10 pulses, each of $\sim20$ ps
duration, and demonstrate an entangled Bell-state with $(76\pm1)$% fidelity.
These results pave the way for entanglement operations within a large
collection of qubits by exciting only local modes of motion.
