With advanced micro- and nanophotonic structures, the vacuum photon-photon coupling rate is anticipated to approach the intrinsic loss rate and lead to unconventional quantum effects. Here, we investigate the classical-to-quantum transition of such photonic nonlinear systems using the quantum cluster-expansion method, which addresses the computational challenge in tracking large photon number states of the fundamental and harmonic optical fields involved in the second-harmonic generation process. Compared to the mean-field approximation used in the weak-coupling limit, the quantum cluster-expansion method solves multimode dynamics efficiently and reveals the quantum behaviors of optical parametric oscillations around the threshold. This paper presents a universal tool to study quantum dynamics of multimode systems and explore the nonlinear photonic devices for continuous-variable quantum information processing.
With advanced micro- and nanophotonic structures, the vacuum photon-photon coupling rate is anticipated to approach the intrinsic loss rate and lead to unconventional quantum effects. Here, we investigate the classical-to-quantum transition of such photonic nonlinear systems using the quantum cluster-expansion method, which addresses the computational challenge in tracking large photon number states of the fundamental and harmonic optical fields involved in the second-harmonic generation process. Compared to the mean-field approximation used in the weak-coupling limit, the quantum cluster-expansion method solves multimode dynamics efficiently and reveals the quantum behaviors of optical parametric oscillations around the threshold. This paper presents a universal tool to study quantum dynamics of multimode systems and explore the nonlinear photonic devices for continuous-variable quantum information processing.
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.
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.
The quantum features of ultrashort-pulse optical parametric oscillators (OPOs) are investigated theoretically in the nonlinear regime near and above threshold. Viewing the pulsed OPO as a multimode open quantum system, we rigorously derive a general input-output model that features nonlinear coupling among many cavity (i.e., system) signal modes and a broadband single-pass (i.e., reservoir) pump field. Under appropriate assumptions, our model produces a Lindblad master equation with multimode nonlinear Lindblad operators describing two-photon dissipation and a multimode four-wave-mixing Hamiltonian describing a broadband, dispersive optical cascade, which we show is required to preserve causality. To simplify the multimode complexity of the model, we employ a supermode decomposition to perform numerical simulations in the regime where the pulsed supermodes experience strong single-photon nonlinearity. We find that the quantum nonlinear dynamics induces pump depletion as well as corrections to the below-threshold squeezing spectrum predicted by linearized models. We also observe the formation of non-Gaussian states with Wigner-function negativity and show that the multimode interactions with the pump, both dissipative and dispersive, can act as effective decoherence channels. Finally, we briefly discuss some experimental considerations for potentially observing such quantum nonlinear phenomena with ultrashort-pulse OPOs on nonlinear nanophotonic platforms.
We numerically investigate the classical limit of quantum trajectories in optical second-harmonic generation. This is a dissipative system of two nonlinearly coupled harmonic oscillators, corresponding to the fundamental and second-harmonic optical cavity modes. Classically it is described by a system of nonlinear differential equations which have a range of attractors: fixed points, limit cycles, and chaotic attractors. We consider the field-amplitude mean values corresponding to the quantum trajectory wave functions. We find that in the classical limit of large photon number the trajectories of these mean values approach the corresponding classical attractors. This is because the mean values of operator products factorize into products of mean values in the classical limit.
Recent advances in nonlinear nanophotonics have enabled optical frequency conversion at energy scales remarkably close to the few-photon regime of quantum optics. A key technique in these efforts is dispersion engineering, allowing dynamical confinement of light into ultrashort pulses with high peak intensities. In this talk, we explore quantum effects expected to emerge in next-generation devices, with a focus on understanding how these phenomena interplay with the multimode physics of femtosecond pulses. Drawing inspiration from recent experiments in thin-film lithium-niobate nanophotonics, we present several numerical studies of broadband frequency conversion in chi(2)-nonlinear waveguides. Our analysis provides a detailed understanding of the dynamics and modal structure of quantum noise and entanglement, e.g., in parametric and supercontinuum generation. This talk also surveys some recent results we obtained for harnessing exotic quantum non-Gaussian states of light in ultrafast nonlinear nanophotonics. Proper engineering of these quantum effects may generate entirely new functionalities for nonlinear optics at and beyond the classical-quantum transition.
The propagation of ultrafast pulses in dispersion-engineered waveguides, exhibiting strong field confinement in both space and time, is a promising avenue towards single-photon nonlinearities in an all-optical platform. However, quantum engineering in such systems requires new numerical tools and physical insights to harness their complicated multimode and nonlinear quantum dynamics. In this work, we use a self-consistent, multimode Gaussian-state model to capture the nonlinear dynamics of broadband quantum fluctuations and correlations, including entanglement. Notably, despite its parametrization by Gaussian states, our model exhibits nonlinear dynamics in both the mean field and the quantum correlations, giving it a marked advantage over conventional linearized treatments of quantum noise, especially for systems exhibiting gain saturation and strong nonlinearities. Numerically, our approach takes the form of a Gaussian split-step Fourier (GSSF) method, naturally generalizing highly efficient SSF methods used in classical ultrafast nonlinear optics; the equations for GSSF evaluate in O(M2logM) time for an M-mode system with O(M2) quantum correlations. To demonstrate the broad applicability of GSSF, we numerically study quantum noise dynamics and multimode entanglement in several ultrafast systems, from canonical soliton propagation in third-order (χ(3)) waveguides to saturated χ(2) broadband parametric generation and supercontinuum generation, e.g., as recently demonstrated in thin-film lithium niobate nanophotonics.