  Controlling the non-Markovian dynamics of open quantum systems is essential
in quantum information technology since it plays a crucial role in preserving
quantum memory. Albeit in many realistic scenarios the quantum system can
simultaneously interact with composite environments, this condition remains
little understood, particularly regarding the effect of the coupling between
environmental parts. We analyze the non-Markovian behavior of a qubit
interacting at the same time with two coupled single-mode cavities which in
turn dissipate into memoryless or memory-keeping reservoirs. We show that
increasing the control parameter, that is the two-mode coupling, allows for
triggering and enhancing a non-Markovian dynamics for the qubit starting from a
Markovian one in absence of coupling. Surprisingly, if the qubit dynamics is
non-Markovian for zero control parameter, increasing the latter enables
multiple transitions from non-Markovian to Markovian regimes. These results
hold independently on the nature of the reservoirs. This work highlights that
suitably engineering the coupling between parts of a compound environment can
efficiently harness the quantum memory, stored in a qubit, based on
non-Markovianity.
