  By moving the pivot of a pendulum rapidly up and down one can create a stable
position with the pendulum's bob above the pivot rather than below it. This
surprising and counterintuitive phenomenon is a widespread feature of driven
systems and carries over into the quantum world. Even when the static
properties of a quantum system are known, its response to an explicitly
time-dependent variation of its parameters may be highly nontrivial, and
qualitatively new states can appear that were absent in the original system. In
quantum mechanics the archetype for this kind of behaviour is an atom in a
radiation field, which exhibits a number of fundamental phenomena such as the
modification of its g-factor in a radio-frequency field and the dipole force
acting on an atom moving in a spatially varying light field. These effects can
be successfully described in the so-called dressed atom picture. Here we show
that the concept of dressing can also be applied to macroscopic matter waves,
and that the quantum states of "dressed matter waves" can be coherently
controlled. In our experiments we use Bose-Einstein condensates in driven
optical lattices and demonstrate that the many-body state of this system can be
adiabatically and reversibly changed between a superfluid and a Mott insulating
state by varying the amplitude of the driving. Our setup represents a versatile
testing ground for driven quantum systems, and our results indicate the
direction towards new quantum control schemes for matter waves.
