  We consider the problem of time optimal control of a continuous bosonic
quantum system subject to the action of a Markovian dissipation. In particular,
we consider the case of a one mode Gaussian quantum system prepared in an
arbitrary initial state and which relaxes to the steady state due to the action
of the dissipative channel. We assume that the unitary part of the dynamics is
represented by Gaussian operations which preserve the Gaussian nature of the
quantum state, i.e. arbitrary phase rotations, bounded squeezing and unlimited
displacements. In the ideal ansatz of unconstrained quantum control (i.e. when
the unitary phase rotations, squeezing and displacement of the mode can be
performed instantaneously), we study how control can be optimized for speeding
up the relaxation towards the fixed point of the dynamics and we analytically
derive the optimal relaxation time. Our model has potential and interesting
applications to the control of modes of electromagnetic radiation and of
trapped levitated nanospheres.
