  Gaussian unitaries are specified by a second order polynomial in the bosonic
operators, that is, by a quadratic polynomial and a linear term. From the
Hamiltonian other equivalent representations of the Gaussian unitaries are
obtained, such as Bogoliubov and real symplectic transformations. The paper
develops an alternative representation, called complex symplectic
transformation, which is more compact and is comprehensive of both Bogoliubov
and real symplectic transformations. Moreover, it has other advantages. One of
the main results of the theory, not available in the literature, is that the
final displacement is not simply given by the linear part of the Hamiltonian,
but depends also on the quadratic part. In particular, it is shown that by
combining squeezing and rotation, it is possible to achieve a final
displacement with an arbitrary amount.
