  We investigate a Bose-Einstein condensate trapped in a 2D optical lattice in
the presence of weak disorder within the framework of the Bogoliubov theory. In
particular, we analyze the combined effects of disorder and an optical lattice
on quantum fluctuations and superfluid density of the BEC system. Accordingly,
the analytical expressions of the ground state energy and quantum depletion of
the system are obtained. Our results show that the lattice still induces a
characteristic 3D to 1D crossover in the behavior of quantum fluctuations,
despite the presence of weak disorder. Furthermore, we use the linear response
theory to calculate the normal fluid density of the condensate induced by
disorder. Our results in the 3D regime show that the combined presence of
disorder and lattice induce a normal fluid density that asymptotically
approaches 4/3 of the corresponding condensate depletion. Conditions for
possible experimental realization of our scenario are also proposed.
