  We study analytically and numerically the properties of one-dimensional chain
of cold ions placed in a periodic potential of optical lattice and global
harmonic potential of a trap. In close similarity with the Frenkel-Kontorova
model, a transition from sliding to pinned phase takes place with the increase
of the optical lattice potential for the density of ions incommensurate with
the lattice period. Quantum fluctuations lead to a quantum phase transition and
melting of pinned instanton glass phase at large values of dimensional Planck
constant. The obtained results are also relevant for a Wigner crystal placed in
a periodic potential.
