We generate and characterise entangled states of a register of 20
individually controlled qubits, where each qubit is encoded into the electronic
state of a trapped atomic ion. Entanglement is generated amongst the qubits
during the out-of-equilibrium dynamics of an Ising-type Hamiltonian, engineered
via laser fields. Since the qubit-qubit interactions decay with distance,
entanglement is generated at early times predominantly between neighbouring
groups of qubits. We characterise entanglement between these groups by
designing and applying witnesses for genuine multipartite entanglement. Our
results show that, during the dynamical evolution, all neighbouring qubit
pairs, triplets, most quadruplets, and some quintuplets simultaneously develop
genuine multipartite entanglement. Witnessing genuine multipartite entanglement
in larger groups of qubits in our system remains an open challenge.Comment: 20 pages, 4 figure