The birth of SPDC photons

In nonlinear crystals, a special nonlinear process occurs. A pump photon is spontaneously converted into two complementary photons with lower energy. In this process both energy and momentum are conserved, i.e .,  ω_p =ω_s + ω_i  and k_p =k_s + k_i

In PPKTP or APKTP Crystals the momentum mismatch of the process is compensated by the crystal reciprocal vector which is determined by the spatial frequency  k_p = k_i +k_s + k_G

Vacuum energy fluctuations give rise to the spontaneous creation of many random photons.  However, the output photons of the SPDC (spontaneous parametric down-conversion nonlinear process) are the ones that conserve momentum and energy.