  The spin dynamics of all ferromagnetic materials are governed by two types of
collective excitations: spin waves and domain walls. The fundamental processes
underlying these collective modes, such as exchange interactions and magnetic
anisotropy, all originate at the atomic scale; yet, conventional probing
techniques, based on neutron and photon scattering, provide high resolution in
reciprocal space, and thereby poor spatial resolution. Here we present direct
imaging of spin waves in individual chains of ferromagnetically coupled $S=2$
Fe atoms, assembled one by one on a Cu$_2$N surface using a scanning tunnelling
microscope. We are able to map the spin dynamics of these designer nanomagnets
with atomic resolution, in two complementary ways. First, atom to atom
variations of the amplitude of the quantized spin wave excitations, predicted
by theory, are probed using inelastic electron tunnelling spectroscopy. Second,
we observe slow stochastic switching between two opposite magnetisation states,
whose rate varies strongly depending on the location of the tip along the
chain. Our observations, combined with model calculations, reveal that switches
of the chain are initiated by a spin wave excited state which has its antinodes
at the edges of the chain, followed by a domain wall shifting through the chain
from one end to the other. This approach opens the way towards atomic scale
imaging of other types of spin excitations, such as spinons and fractional
end-states, in engineered spin chains.
