  Dynamical processes induced by the external time-dependent fields can provide
valuable insight into the characteristic energy scales of a given physical
system. We investigate them here in a nanoscopic heterostructure, consisting of
the double quantum dot coupled in series to the superconducting and the
metallic reservoirs, analyzing its response to (i)~abrupt bias voltage applied
across the junction, (ii) sudden change of the energy levels, and imposed by
(iii)~their periodic driving. We explore subgap properties of this setup which
are strictly related to the in-gap quasiparticles and discuss their signatures
manifested in the time-dependent charge currents. The characteristic multi-mode
oscillations, their beating patters and photon-assisted harmonics reveal a rich
spectrum of dynamical features that might be important for designing the
superconducting qubits.
