  We propose a novel realization of Anderson localization in non-equilibrium
states of ultracold atoms trapped in state-dependent optical lattices. The
disorder potential leading to localization is generated with a Rabi pulse
transfering a fraction of the atoms into a different internal state for which
tunneling between lattice sites is suppressed. Atoms with zero tunneling create
a quantum superposition of different random potentials, localizing the mobile
atoms. We investigate the dynamics of the mobile atoms after the Rabi pulse for
non-interacting and weakly interacting bosons, and we show that the evolved
wavefunction attains a quasi-stationary profile with exponentially decaying
tails, characteristic of Anderson localization. The localization length is seen
to increase with increasing disorder and interaction strength, oppositely to
what is expected for equilibrium localization.
