Description
We study the localization transition of an atom confined by
an external optical lattice in a high-finesse cavity. The
atom-cavity coupling yields an effective secondary lattice
potential, whose wavelength is incommensurate with the
periodicity of the optical lattice. The cavity lattice can
induce localization of the atomic wave function analogously
to the Aubry-André localization transition. Starting from
the master equation for the cavity and the atom we perform a
mapping of the system dynamics to a Hubbard Hamiltonian,
which can be reduced to the Harper's Hamiltonian in
appropriate limits. We evaluate the phase diagram for the
atom ground state and show that the transition between
extended and localized wavefunction is controlled by the
strength of the cavity nonlinearity, which determines the
size of the localized region and the behaviour of the
Lyapunov exponent. The Lyapunov exponent, in particular,
exhibits resonance-like behaviour in correspondence with the
optomechanical resonances. Finally we discuss the
experimental feasibility of these predictions.
