Nonlinear interferometers in quantum optics

We review the main works in the field of nonlinear interferometers, which has been developing ever since the start of nonlinear optics and has seen growing interest in recent years due to the applications in quantum information and quantum metrology. Because the class of schemes that are referred to as nonlinear interferometers is too broad, we restrict the consideration to the cases in which crystals, fibers, or atomic systems producing the nonlinear effect are placed inside the interferometer, and only linear phase shifts are imposed. As a nonlinear effect, we consider four-wave mixing and parametric down-conversion, at both low and high gain.