  Quantum Zeno and anti-Zeno effects are studied in a symmetric nonlinear
optical coupler, which is composed of two nonlinear ($\chi^{\left(2\right)}$)
waveguides that are interacting with each other via the evanescent waves. Both
the waveguides operate under second harmonic generation. However, to study
quantum Zeno and anti-Zeno effects one of them is considered as the system and
the other one is considered as the probe. Considering all the fields involved
as weak, a completely quantum mechanical description is provided, and the
analytic solutions of Heisenberg's equations of motion for all the field modes
are obtained using a perturbative technique. Photon number statistics of the
second harmonic mode of the system is shown to depend on the presence of the
probe, and this dependence is considered as quantum Zeno and anti-Zeno effects.
Further, it is established that as a special case of the momentum operator for
$\chi^{\left(2\right)}-\chi^{\left(2\right)}$ symmetric coupler we can obtain
momentum operator of $\chi^{\left(2\right)}-\chi^{\left(1\right)}$ asymmetric
coupler with linear ($\chi^{\left(1\right)}$) waveguide as the probe, and in
such a particular case, the expressions obtained for Zeno and anti-Zeno effects
with nonlinear probe (which we referred to as nonlinear quantum Zeno and
anti-Zeno effects) may be reduced to the corresponding expressions with linear
probe (which we referred to as the linear quantum Zeno and anti-Zeno effects).
Linear and nonlinear quantum Zeno and anti-Zeno effects are rigorously
investigated, and it is established that in the stimulated case, we may switch
between quantum Zeno and anti-Zeno effects just by controlling the phase of the
second harmonic mode of the system or probe.
