  For finite-dimensional quantum systems, such as qubits, a well established
strategy to protect such systems from decoherence is dynamical decoupling.
However many promising quantum devices, such as oscillators, are infinite
dimensional, for which the question if dynamical decoupling could be applied
remained open. Here we first show that not every infinite-dimensional system
can be protected from decoherence through dynamical decoupling. Then we develop
dynamical decoupling for continuous variable systems which are described by
quadratic Hamiltonians. We identify a condition and a set of operations that
allow us to map a set of interacting harmonic oscillators onto a set of
non-interacting oscillators rotating with an averaged frequency, a procedure we
call homogenization. Furthermore we show that every quadratic
system-environment interaction can be suppressed with two simple operations
acting only on the system. Using a random dynamical decoupling or
homogenization scheme, we develop bounds that characterize how fast we have to
work in order to achieve the desired uncoupled dynamics. This allows us to
identify how well homogenization can be achieved and decoherence can be
suppressed in continuous variable systems.
