  We study the subradiant collective states of a periodic chain of two-level
atoms with either transversal or longitudinal transition dipole moments with
respect to the chain axis. We show that long-lived subradiant states can be
obtained for the transversal polarization by properly choosing the chain period
for a given number of atoms in the case of no open diffraction channels. These
highly subradiant states have a linewidth that decreases with the number of
atoms much faster than it was shown previously. In addition, our study shows
that such states are present even if there are additional interaction channels
between the atoms, i.e. they interact via waveguide mode of an optical
nanofiber for instance. We develop a theoretical framework allowing to describe
the spectral properties of the system in terms of contributions from each
collective eigenstate and we show that subradiant states manifest themselves in
the transmission and reflection spectra, allowing to observe
interaction-induced transparency in a very narrow spectral range. Such
long-lived collective states may find potential applications in nanophotonics
and quantum optics.
