Ultrafast second-order nonlinear photonics—from classical physics to non-Gaussian quantum dynamics: a tutorial

Abstract

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 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. Although 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.

Strong Coupling and Single-Photon Nonlinearity in Free-Electron Quantum Optics

“…Figure 5b shows the corresponding supermode populations for different values of |g Q |, and Figure 5c shows the time evolution of the expectation value of the total excitation number, for the same values of |g Q | as in Figure 5a,b. The total excitation number experiences a vacuum Rabi oscillation, though with limited visibility, owing to the excitation of other supermodes (interestingly, a similar behavior is obtained in the deepquantum regime of multimode parametric downconversion 69 ). Finally, Figure 5d shows the value of the second-order c o r r e l a t i o n f u n c t i o n a t z e r o d e l a y ,…”

“…The realization of single-photon nonlinearities with free electrons opens up many exciting prospects in free-electron quantum optics, such as the efficient and deterministic (or heralded) generation of non-Gaussian light. In the proposed system, owing to the availability of a large coupling coefficient g Q , such effects can in principle be observed even after a single pass of a single free electron, instead of slower, multielectron consecutive interactions previously considered in the literature. , As shown in Figure , the strong nonlinearity induced by free electrons also results in complex multimode dynamics that are reminiscent of broadband parametric down conversion in the deep quantum regime . Extending these ideas to correlated many-electron experiments − can reveal further nonlinear and strong coupling effects.…”

“…4,9 As shown in Figure 5, the strong nonlinearity induced by free electrons also results in complex multimode dynamics that are reminiscent of broadband parametric down conversion in the deep quantum regime. 69 Extending these ideas to correlated many-electron experiments 73−76 can reveal further nonlinear and strong coupling effects.…”