  Quantum mechanical systems with long-range interactions between
quasiparticles provide a promising platform for coherent quantum information
technology. Superconductors are a natural choice for solid-state based quantum
devices, while magnetic impurities inside superconductors give rise to
quasiparticle excitations of broken Cooper pairs that provide characteristic
information about the host superconductor. Here, we reveal that magnetic
impurities embedded below a superconducting La(0001) surface interact via
quasiparticles extending to very large distances, up to several tens of
nanometers. Using low-temperature scanning probe techniques, we observe the
corresponding anisotropic and giant oscillations in the local density of
states. Theoretical calculations indicate that the quasi-two-dimensional
surface states with their strongly anisotropic Fermi surface play a crucial
role for the focusing and long-range extension of the magnetic bound states.
The quasiparticle focusing mechanism should facilitate the design of versatile
magnetic structures with tunable and directed magnetic interactions over large
distances, thereby paving the way toward the design of low-dimensional
magnet-superconductor hybrid systems exhibiting topologically non-trivial
quantum states as possible elements of quantum computation schemes based on
Majorana quasiparticles.
