  The role of disorder on physical systems has been widely studied in the
macroscopic and microscopic world. While static disorder is well understood in
many cases, the impact of time-dependent disorder on quantum gases is still
poorly investigated. In our experimental setup, we introduce and characterize a
method capable of producing time-controlled optical-speckle disorder.
Experimentally, coherent light illuminates a combination of a static and a
rotating diffuser, thereby collecting a spatially varying phase due to the
diffusers' structure and a temporally variable phase due to the relative
rotation. Controlling the rotation of the diffuser allows changing the speckle
realization or, for future work, the characteristic time scale of the change of
the speckle pattern, i.e. the correlation time, matching typical time scales of
the quantum gases investigated. We characterize the speckle pattern ex-situ by
measuring its intensity distribution cross-correlating different intensity
patterns. In-situ, we observe its impact on a molecular Bose-Einstein
condensate (BEC) and cross-correlate the density distributions of BECs probed
in different speckle realizations. As one diffuser rotates relative to the
other around the common optical axis, we trace the optical speckle's intensity
cross-correlations and the quantum gas' density cross-correlations. Our results
show comparable outcomes for both measurement methods. The setup allows us to
tune the disorder potential adapted to the characteristics of the quantum gas.
These studies pave the way for investigating nonequilibrium physics in
interacting quantum gases using controlled dynamical-disorder potentials.
