  The generally deformed oscillator (GDO) and its multiphoton realization as
well as the coherent and squeezed vacuum states are studied. We discuss, in
particular, the GDO depending on a complex parameter q (therefore we call it
q-GDO) together with the finite dimensional cyclic representations. As a
realistic physical system of GDO the isospectral oscillator system is studied
and it is found that its coherent and squeezed vacuum states are closely
related to those of the oscillator. It is pointed out that starting from the
q-GDO with q root of unity one can define the hermitian phase operators in
quantum optics consistently and algebraically. The new creation and
annihilation operators of the Pegg-Barnett type phase operator theory are
defined by using the cyclic representations and these operators degenerate to
those of the ordinary oscillator in the classical limit q->1.
