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We theoretically propose and investigate a feasible experimental scheme forthe realization of the dynamical Casimir effect (DCE) in a hybridoptomechanical cavity with a moving end mirror containing an interactingcigar-shaped Bose-Einstein condensate (BEC). We show that in the red-detunedregime of cavity optomechanics together with the weak optomechanical couplinglimit by \textit{coherent} modulation of the \textit{s}-wave scatteringfrequency of the BEC and the mechanical spring coefficient of the mechanicaloscillator (MO), the mechanical and atomic quantum vacuum fluctuations areparametrically amplified, which consequently lead to the generation of themechanical/Bogoliubov-type Casimir phonons. Interestingly, in the coherentregime corresponding to the case of largely different optomechanical couplingstrengths of the cavity field to the BEC and the MO, or equivalently largelydifferent cooperativities, one can generate a large number of Casimir photonsdue to the amplification of the intracavity vacuum fluctuations\textit{induced} by the time modulations of the BEC and the MO. The number ofgenerated Casimir particles are externally controllable by the cooperativities,and the modulation amplitudes of the atomic collisions rate and the mechanicalspring coefficient.

Controllable generation of photons and phonons in a coupled BEC-optomechanical-cavity via the parametric dynamical Casimir effect