  We study coherent perfect absorption (CPA) theoretically based on a weakly
coupled atom-cavity system with an optically pumped second-order nonlinear
crystal (SOC) embedded in the cavity. Our system does not require a strong
coupling, which is often needed for CPA in previous studies but is challenging
to implement experimentally in some systems. The role of the SOC is to
introduce a tunable effective decay rate of the cavity, which can lead to CPA
in the weak coupling regime. The proposed system exhibits bistable behaviors,
with bistable patterns switchable between conventional and unconventional
shapes. By varying the properties of the SOC, the operation point of CPA can be
tuned to be inside or outside the bistable regime. It can also be located at
the upper or the lower stable branch or even the unstable branch of the
bistable hysteresis loop. It is however robust against the parameters of the
SOC for any fixed effective decay rate. Our system can potentially be applied
to realize optical devices such as optical switches in the weakly coupled
regime.
