  We demonstrate the rise-and-fall of multiple pseudogaps in the
Bardeen-Cooper-Schrieffer-Bose-Einstein-condensation (BCS-BEC) crossover in
two-band fermionic systems having different pairing strengths in the deep band
and in the shallow band. The striking features of this phenomenon are an
unusual many-body screening of pseudogap state and the importance of
pair-exchange couplings, which induces multiple pseudogap formation in the two
bands. The multi-band configuration suppresses pairing fluctuations and the
pseudogap opening in the strongly-interacting shallow band at small
pair-exchange couplings by screening effects, with possible connection to the
pseudogap phenomenology in iron based superconductors. On the other hand, the
multiple pseudogap mechanism accompanies with the emergence of binary preformed
Cooper pairs originating from interplay between intra-band and pair-exchange
couplings.
