  In magnetic trilayer systems, spin pumping is generally addressed as a
reciprocal mechanism characterized by one unique spin mixing conductance common
to both interfaces. However, this assumption is questionable in cases where
different types of interfaces are present in the material. Here, we present a
general theory for analyzing spin pumping in cases with more than one unique
interface. The theory is applied to analyze layer-resolved ferromagnetic
resonance experiments on the trilayer system
Ni$_{20}$Fe$_{80}$/Ru/Fe$_{49}$Co$_{49}$V$_2$ where the Ru spacer thickness is
varied to tune the indirect exchange coupling. The results show that the spin
pumping in trilayer systems with dissimilar magnetic layers is non-reciprocal,
with a surprisingly large difference between spin-pumping induced damping of
different interfaces. Our findings have importance on dynamics of spintronic
devices based on magnetic multilayer materials.
