Antiferromagnetism (AF) in AB-stacked centrosymmetric bilayer (BL) CrI$_3$ breaks both spatial inversion ($P$) and time-reversal ($T$) symmetries but maintains the combined $PT$ symmetry, thus inducing novel second-order nonlinear optical (NLO) responses such as second-harmonic generation (SHG), linear electric-optic effect (LEO) and bulk photovoltaic effect (BPVE). In this work, we calculate AF-induced NLO responses of the BL CrI$_3$ based on the density functional theory with the generalized gradient approximation (GGA) plus onsite Coulomb correlation (U), i.e., the GGA+U method. Interestingly, we find that the magnetic SHG, LEO and photocurrent in the AF BL CrI$_3$ are huge, being comparable or even larger than that of the well-known nonmagnetic noncentrosymmetric semiconductors. For example, the calculated SHG coefficients are in the same order of magnitude as that of MoS$_2$ monolayer (ML), the most promising 2D material for NLO devices. The calculated LEO coefficients are almost three times larger than that of MoS$_2$ ML. The calculated NLO photocurrent in the CrI$_3$ BL is among the largest values predicted so far for the BPVE materials. On the other hand, unlike nonmagnetic semiconductors, the NLO responses in the AF BL CrI$_3$ are nonreciprocal and also switchable by rotating magnetization direction. Therefore, our interesting findings indicate that the AF BL CrI$_3$ will not only provide a valuable platform for exploring new physics of low-dimensional magnetism but also have promising applications in magnetic NLO and LEO devices such as frequency conversion, electro-optical switches, and light signal modulators as well as high energy conversion efficiency photovoltaic solar cells.
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