  The geometric phases for standard coherent states which are widely used in
quantum optics have attracted a large amount of attention. Nevertheless, few
physicists consider about the counterparts of non-linear coherent states, which
are useful in the description of the motion of a trapped ion. In this paper,
the non-unitary and non-cyclic geometric phases for two nonlinear coherent and
one squeezed states are formulated respectively. Moreover, some of their common
properties are discussed respectively, such as gauge invariance, non-locality
and non-linear effects. The non-linear functions have dramatic impacts on the
evolution of the corresponding geometric phases. They speed the evolution up or
down. So this property may have application in controlling or measuring
geometric phase. For the squeezed case, when the squeezed parameter r ->
\infinity, the limiting value of the geometric phase is also determined by
non-linear function at a given time and angular velocity. In addition, the
geometric phases for standard coherent and squeezed states are obtained under a
particular condition. When the time evolution undergoes a period, their
corresponding cyclic geometric phases are achieved as well. And the distinction
between the geometric phases of the two coherent states maybe regarded as a
geometric criterion.

Quantum Physics