  Atoms confined in a three-dimensional dissipative optical lattice oscillate
inside potential wells, occasionally hopping to adjacent wells, thereby
diffusing in all directions. Illumination by a weak probe beam modulates the
lattice, yielding propagating atomic density waves, referred to as Brillouin
modes which travel perpendicular to the direction of travel of the probe. The
probe is made incident at a small angle relative to a lattice symmetry axis,
yielding a driving potential perturbation whose spatial period is not a
multiple of the period of the underlying optical potential, thus enabling
exploration of the regime of space quasiperiodic drive. A theory, based on the
Fourier decomposition of the current into its atomic density wave
contributions, reveals that unlike the previously studied time quasiperiodic
case, wherein a lattice driven by two incommensurate frequencies may exhibit
abrupt suppression in directed current as the driving transitions from
quasiperiodic to periodic, a spatial-quasiperiodically driven lattice exhibits
no such abrupt response. Further, detailed modeling of
spatial-quasiperiodically driven lattices reveals that directed propagation
occurs not only as a consequence of velocity-matching between the propagating
modulation and the average velocity of the atom oscillating inside a well as
was previously reported in the literature, but also as a distinct consequence
of a new mechanism, namely, frequency-matching between the modulation frequency
and the oscillation frequencies. A systematic measurement of the transmitted
probe spectra as a function of off-axis probe angle is presented, which is
consistent with the velocity- and frequency-matching predictions from the
detailed model.
