  We analyze time-of-flight absorption images obtained with dilute
Bose-Einstein con-densates released from shaken optical lattices, both
theoretically and experimentally. We argue that weakly interacting, ultracold
quantum gases in kilohertz-driven optical potentials constitute equilibrium
systems characterized by a steady-state distri-bution of Floquet-state
occupation numbers. Our experimental results consistently indicate that a
driven ultracold Bose gas tends to occupy a single Floquet state, just as it
occupies a single energy eigenstate when there is no forcing. When the driving
amplitude is sufficiently high, the Floquet state possessing the lowest mean
energy does not necessarily coincide with the Floquet state connected to the
ground state of the undriven system. We observe strongly driven Bose gases to
condense into the former state under such conditions, thus providing nontrivial
examples of dressed matter waves.
