Nonlinear interactions between photons are fundamentally weak as the photons do not interact directly with each other, and any interaction is mediated by matter. This has motivated researchers over many decades to search for strongly nonlinear materials (by controlling electronic properties) and optical resonators with strong spatial and temporal confinement of light. An extreme form of nonlinear optics is quantum nonlinear optics, where we can realize nonlinear interaction between single photons. Such quantum nonlinear optics is at the heart of any photonic quantum information system including analog quantum simulation and fault-tolerant quantum computing. While engineering light-matter interactions can effectively create photon-photon interactions, the required photon number to observe any nonlinearity are normally very high, where any quantum-mechanical signature disappears. However, with emerging low-dimensional materials and engineered photonic resonators, the photon number can be reduced to reach the quantum nonlinear optical regime. In this review paper, we discuss different mechanisms exploited in solid-state platforms to attain quantum nonlinear optics. We review emerging materials and optical resonator architectures with different dimensionalities. We also present future research directions and open problems in this field.
Nonlinear optical effects are generally limited to the high-intensity regime due to the small value of typical nonlinear coefficients. We demonstrate novel effects where a strong spectator field mediates a large nonlinear interaction between two photons. Such effects open the door to a new field of {\it quantum} nonlinear optics, potentially useful for two-photon all-optical switches (e.g., optical quantum logic gates) as well as other nonlinear effects such as the formation of quantum solitons. I will talk about enhanced upconversion, quantum state preparation, two-photon cross-phase modulation, and potential quantum-information applications to tasks such as Bell state determination.
Photonic integrated circuits with second-order (
χ(2)) nonlinearities are rapidly scaling to remarkably low powers. At this time, state-of-the-art devices achieve saturated nonlinear interactions with thousands of photons when driven by continuous-wave lasers, and further reductions in these energy requirements enabled by the use of ultrafast pulses may soon push nonlinear optics into the realm of single-photon nonlinearities. This tutorial reviews these recent developments in ultrafast nonlinear photonics, discusses design strategies for realizing few-photon nonlinear interactions, and presents a unified treatment of ultrafast quantum nonlinear optics using a framework that smoothly interpolates from classical behaviors to the few-photon scale. These emerging platforms for quantum optics fundamentally differ from typical realizations in cavity quantum electrodynamics due to the large number of coupled optical modes. Classically, multimode behaviors have been well studied in nonlinear optics, with famous examples including soliton formation and supercontinuum generation. In contrast, multimode quantum systems exhibit a far greater variety of behaviors, and yet closed-form solutions are even sparser than their classical counterparts. In developing a framework for ultrafast quantum optics, we will identify what behaviors carry over from classical to quantum devices, what intuition must be abandoned, and what new opportunities exist at the intersection of ultrafast and quantum nonlinear optics. While this article focuses on establishing connections between the classical and quantum behaviors of devices with
χ(2) nonlinearities, the frameworks developed here are general and are readily extended to the description of dynamical processes based on third-order (
χ(3)) nonlinearities.