  We theoretically investigate spin pumping into a quantum easy-plane
ferromagnetic spin chain system. This quantum spin chain is effectively
described by the Tomonaga-Luttinger (TL) liquid despite the ferromagnetic
exchange interaction because of the easy-plane magnetic anisotropy. This TL
liquid state has an extremely strong interaction that is hardly realized in
other quantum antiferromagnetic chain systems or weakly interacting electron
systems. We show how the strongly interacting TL liquid affects the
ferromagnetic resonance that occurs in the ferromagnetic insulator. In
particular, we discuss the dependence of the Gilbert damping on the temperature
and the junction length. The Gilbert damping allows us to extract information
about the above-mentioned strong interaction within the quantum ferromagnetic
spin chain. We also point out that a well-known compound CsCuCl$_3$ will be
suitable for the realization of our setup.
