  We consider cold Rydberg atoms in a one-dimensional optical lattice in the
Mott regime with a single atom per site at zero temperature. An external laser
drive with Rabi frequency \Omega and laser detuning \Delta, creates Rydberg
excitations whose dynamics is governed by an effective spin-chain model with
(quasi) long-range interactions. This system possesses intrinsically a large
degree of frustration resulting in a ground-state phase diagram in the
(\Delta,\Omega) plane with a rich topology. As a function of \Delta, the
Rydberg blockade effect gives rise to a series of crystalline phases
commensurate with the optical lattice that form a so-called devil's staircase.
The Rabi frequency, \Omega, on the other hand, creates quantum fluctuations
that eventually lead to a quantum melting of the crystalline states. Upon
increasing \Omega, we find that generically a commensurate-incommensurate
transition to a floating Rydberg crystal occurs first, that supports gapless
phonon excitations. For even larger \Omega, dislocations within the floating
Rydberg crystal start to proliferate and a second,
Kosterlitz-Thouless-Nelson-Halperin-Young dislocation-mediated melting
transition finally destroys the crystalline arrangement of Rydberg excitations.
This latter melting transition is generic for one-dimensional Rydberg crystals
and persists even in the absence of an optical lattice. The floating phase and
the concomitant transitions can, in principle, be detected by Bragg scattering
of light.
