  In the recent publication [Phys. Rev. B 87, 235319 (2013)], the conventional
photon blockade(CPB) was studied for the low frequency mode in a second-order
nonlinear system. In this paper, we will study the CPB for the high frequency
mode in a second-order nonlinear system with the Kerr nonlinearity filling in
the low-frequency cavity. By solving the master equation and calculating the
zero-delay-time second order correlation function $g^{(2)}(0)$, strong photon
antibunching can be obtained in the high frequency cavity. The optimal
condition for strong antibunching is found by analyticcal culations and
discussions of the optimal condition are presented. We find that the
Kerr-nonlinearities can enhanced the CPB effect. In addition, this scheme is
not sensitive to the reservoir temperature, which make the current system
easier to implement experimentally.
