  Resonant Raman coupling between internal levels induced by continuous
illumination of non-collinear laser beams can create double-well momentum-space
potentials for multi-level ``periodically-dressed'' atoms. We develop an
approximate many-body formalism for a weakly interacting, trapped
periodically-dressed Bose gas which illustrates how a tunable exchange
interaction yields correlated many-body ground states. In contrast to the case
of a position-space double well, the ground state of stable
periodically-dressed Bose gases with repulsive interactions tends toward a
Schr\"{o}dinger cat state in the regime where interactions dominate the
momentum-space tunnelling induced by the external trapping potential. The
dependence of the momentum-space tunnelling and exchange interaction on
experimental parameters is derived. We discuss how real-time control of
experimental parameters can be used to create Schr\"{o}dinger cat states either
between momentum or internal states, and how these states could be dynamically
controlled towards highly sensitive interferometry and frequency metrology.
