  We introduce a formalism of nonlinear canonical transformations for general
systems of multiphoton quantum optics. For single-mode systems the
transformations depend on a tunable free parameter, the homodyne local
oscillator angle; for n-mode systems they depend on n heterodyne mixing angles.
The canonical formalism realizes nontrivial mixings of pairs of conjugate
quadratures of the electromagnetic field in terms of homodyne variables for
single-mode systems; and in terms of heterodyne variables for multimode
systems. In the first instance the transformations yield nonquadratic model
Hamiltonians of degenerate multiphoton processes and define a class of non
Gaussian, nonclassical multiphoton states that exhibit properties of coherence
and squeezing. We show that such homodyne multiphoton squeezed states are
generated by unitary operators with a nonlinear time evolution that realizes
the homodyne mixing of a pair of conjugate quadratures. Tuning of the local
oscillator angle allows to vary at will the statistical properties of such
states. We discuss the relevance of the formalism for the study of degenerate
(up-)down-conversion processes. In a companion paper, ``Structure of
multiphoton quantum optics. II. Bipartite systems, physical processes, and
heterodyne squeezed states'', we provide the extension of the nonlinear
canonical formalism to multimode systems, we introduce the associated
heterodyne multiphoton squeezed states, and we discuss their possible
experimental realization.
