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The generally deformed oscillator (GDO) and its multiphoton realization aswell as the coherent and squeezed vacuum states are studied. We discuss, inparticular, the GDO depending on a complex parameter q (therefore we call itq-GDO) together with the finite dimensional cyclic representations. As arealistic physical system of GDO the isospectral oscillator system is studiedand it is found that its coherent and squeezed vacuum states are closelyrelated to those of the oscillator. It is pointed out that starting from theq-GDO with q root of unity one can define the hermitian phase operators inquantum optics consistently and algebraically. The new creation andannihilation operators of the Pegg-Barnett type phase operator theory aredefined by using the cyclic representations and these operators degenerate tothose of the ordinary oscillator in the classical limit q->1.Comment: 21 pages, latex, no figure

Generally deformed oscillator, isospectral oscillator system and Hermitian phase operator