  We present the results of experiments performed on cold caesium in a pulsed
sinusoidal optical potential created by counter-propagating laser beams having
a small frequency difference in the laboratory frame. Since the atoms, which
have average velocity close to zero in the laboratory frame, have non-zero
average velocity in the co-moving frame of the optical potential, we are able
to centre the initial velocity distribution of the cloud at an arbitrary point
in phase-space. In particular, we demonstrate the use of this technique to
place the initial velocity distribution in a region of phase-space not
accessible to previous experiments, namely beyond the momentum boundaries
arising from the finite pulse duration of the potential. We further use the
technique to explore the kicked rotor dynamics starting from a region of
phase-space where there is a strong velocity dependence of the diffusion
constant and quantum break time and demonstrate that this results in a marked
asymmetry in the chaotic evolution of the atomic momentum distribution.
