  We propose a new direction in quantum simulation that uses multilevel atoms
in an optical cavity as a toolbox to engineer new types of bosonic models
featuring correlated hopping processes in a synthetic ladder spanned by atomic
ground states. The underlying mechanisms responsible for correlated hopping are
collective cavity-mediated interactions that dress a manifold of excited levels
in the far detuned limit. By weakly coupling the ground state levels to these
dressed states using two laser drives with appropriate detunings, one can
engineer correlated hopping processes while suppressing undesired
single-particle and collective shifts of the ground state levels. We discuss
the rich many-body dynamics that can be realized in the synthetic ladder
including pair production processes, chiral transport and light-cone
correlation spreading. The latter illustrates that an effective notion of
locality can be engineered in a system with fully collective interactions.
