  We elucidate the fate of neighboring two and three-$\alpha$ particles in cold
neutron matter by focusing on an analogy between such $\alpha$ systems and
Fermi polarons realized in ultracold atoms. We describe in-medium excitation
properties of an $\alpha$ particle and neutron-mediated two- and three-$\alpha$
interactions using theoretical approaches developed for studies of cold atomic
systems. We numerically solve the few-body Schr\"odinger equation of $\alpha$
particles within standard $\alpha$ cluster models combined with in-medium
properties of $\alpha$ particles. We point out that the resultant two-$\alpha$
ground state and three-$\alpha$ first excited state, which correspond to $^8$Be
and the Hoyle state, respectively, known as main components in the
triple-$\alpha$ reaction, can become bound states in such a many-neutron
background although these states are unstable in vacuum. Our results suggest a
significance of these in-medium cluster states not only in astrophysical
environments such as core-collapsed supernova explosions and neutron star
mergers but also in neutron-rich nuclei.
