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We study the transient phenomena appearing in a subgap region of the doublequantum dot coupled in series between the superconducting and normal metallicleads, focusing on the development of the superconducting proximity effect. Forthe uncorrelated nanostructure we derive explicit expressions of thetime-dependent occupancies in both quantum dots, charge currents, and electronpairing induced on individual dots and between them. We show that the initialconfigurations substantially affect the dynamical processes, in which thein-gap bound states emerge upon coupling the double quantum dot tosuperconducting reservoir. In particular, the superconducting proximity effectwould be temporarily blocked whenever the quantum dots are initially singlyoccupied. Such {\it triplet}/{\it Andreev blockade} has been recently reportedexperimentally for double quantum dots embedded in the Josephson [D. Bouman etal., Phys. Rev. B 102, 220505 (2020)] and Andreev [P. Zhang et al.,arXiv:2102.03283 (2021)] junctions. We also address the role of correlationeffects within the lowest-order decoupling scheme and by the time-dependentnumerical renormalization group calculations. Competition of the repulsiveCoulomb interactions with the superconducting proximity effect leads torenormalization of the in-gap quasiparticles, speeding up the quantumoscillations and narrowing a region of transient phenomena, whereas thedynamical Andreev blockade is well pronounced in the weak inter-dot couplinglimit. We propose feasible methods for detecting the characteristic time-scalesthat could be observable by the Andreev spectroscopy.

Transient effects in double quantum dot sandwiched laterally between superconducting and metallic leads