Extremizing a quadratic form can be computationally straightforward or difficult depending on the feasible domain over which variables are optimized. For example, maximizing E = xTVx for a real-symmetric matrix ๐ with ๐ฅ constrained to a unit ball in ๐ ๐ can be performed simply by finding the maximum (principal) eigenvector of ๐, but can become computationally intractable if the domain of ๐ฅ is limited to corners of the ±1 hypercube in ๐ ๐ (i.e., ๐ฅ is constrained to be a binary vector). Many gain-loss physical systems, such as coherently coupled arrays of lasers or optical parametric oscillators, naturally solve minimum/maximum eigenvector problems (of a matrix of coupling coefficients) in their equilibration dynamics. In this paper we discuss recent case studies on the use of added nonlinear dynamics and real-time feedback to enforce constraints in such systems, making them potentially useful for solving difficult optimization problems. We consider examples in both classical and quantum regimes of operation.
We demonstrate both second harmonic generation (with a normalized efficiency of 0.20 %Wโ1 cmโ2 ) and, to our knowledge, the first degenerate ฯ (2) optical parametric amplifier (with an estimated normalized gain of 0.6 dBWโ1/2 cmโ1 ) using silicon-on-insulator waveguides fabricated in a CMOS-compatible commercial foundry.
Coherent Ising machines (CIMs) are an experimentally promising class of physics-based computational architectures that embed hard combinatorial optimization problems into systems of coupled nonlinear optical oscillators. The solution-finding mechanisms employed by CIMs feature complicated dynamical bifurcations occurring on a network scale, posing significant challenges to the development of theory and models for their underlying principles of operation. These difficulties are especially pronounced in the ultra-low-power or quantum regimes where the benefits in computational efficiency over conventional optimization algorithms are expected to be largest. We discuss some of our recent approaches and results at this intersection of dynamical systems theory and quantum model reduction, which have highlighted some potentially useful architectures and applications on the horizon for CIMs.
Coherent Ising Machines (CIMs) are an emerging class of computational architectures that embed hard combinatorial optimization problems in the continuous dynamics of a physical system with analog degrees of freedom. While crisp theoretical results on the ultimate performance and scaling of such architectures are lacking, large-scale experimental prototypes have begun to exhibit promising results in practice. Our team at Stanford has begun to study the fundamental properties of CIM dynamics using a combination of techniques from statistical physics, random matrices, and dynamical systems theory. Many connections to recent work in neuroscience and deep learning are noted. Our work focuses specifically on CIMs that utilize the nonlinear threshold behavior of optical parametric oscillators to effect a soft (potentially glassy) transition between linear and binary dynamical regimes.
The advent of dispersion-engineered and highly nonlinear nanophotonics is expected to open up an all-optical path towards the strong-interaction regime of quantum optics by combining high transverse field confinement with ultra-short-pulse operation. Obtaining a full understanding of photon dynamics in such broadband devices, however, poses major challenges in the modeling and simulation of multimode non-Gaussian quantum physics, highlighting the need for sophisticated reduced models that facilitate efficient numerical study while providing useful physical insight. In this manuscript, we review our recent efforts in modeling broadband optical systems at varying levels of abstraction and generality, ranging from multimode extensions of quantum input-output theory for sync-pumped oscillators to the development of numerical methods based on a field-theoretic description of nonlinear waveguides. We expect our work not only to guide ongoing theoretical and experimental efforts towards next-generation quantum devices but also to uncover essential physics of broadband quantum photonics.
Due to their strong light con๏ฌnement, waveguides with optical nonlinearities may be a promising platform for energy-e๏ฌcient optical computing. Slow light can enhance a waveguideโs e๏ฌective nonlinearity, which could result in devices that operate in low-power regimes where quantum ๏ฌuctuations are important, and may also have quantum applications including squeezing and entanglement generation. In this manuscript, slow-light structures based on the Kerr (χ(3)) nonlinearity are analyzed using a semi-classical model to account for the quantum noise. We develop a hybrid split-step / Runge-Kutta numerical model to compute the mean ๏ฌeld and squeezing spectrum for pulses propagating down a waveguide, and use this model to study squeezing produced in optical waveguides. Scaling relations are explored, and the bene๏ฌts and limitations of slow light are discussed in the context of squeezing.
Nanoscale integrated photonic devices and circuits offer a path to ultra-low power computation at the few-photon level. Here we propose an optical circuit that performs a ubiquitous operation: the controlled, random-access readout of a collection of stored memory phases or, equivalently, the computation of the inner product of a vector of phases with a binary selector" vector, where the arithmetic is done modulo 2pi and the result is encoded in the phase of a coherent field. This circuit, a collection of cascaded interferometers driven by a coherent input field, demonstrates the use of coherence as a computational resource, and of the use of recently-developed mathematical tools for modeling optical circuits with many coupled parts. The construction extends in a straightforward way to the computation of matrix-vector and matrix-matrix products, and, with the inclusion of an optical feedback loop, to the computation of a weighted" readout of stored memory phases. We note some applications of these circuits for error correction and for computing tasks requiring fast vector inner products, e.g. statistical classification and some machine learning algorithms.
The emerging discipline of coherent-feedback quantum control provides core concepts and methods for nanopho-
tonic circuit theory, which can be assimilated within modern approaches to computer-aided design. Current
research in this area includes the development of software tools to enable a schematic capture workflow for
compilation and analysis of quantum stochastic models for nanophotonic circuits, exploration of elementary
coherent-feedback circuit motifs, and laboratory demonstrations of quantum nonlinear photonic devices.
An experimental demonstration of a quantum-optimal receiver for optical binary signals, developed as a joint effort by the Jet Propulsion Laboratory and the California Institute if Technology, is described in this article. A brief summary of the classical, quantum-optimal, and quantum near optimal solutions to detecting binary signals is first presented. The components and experimental setup used to implement the receivers is then discussed. Experimental performance and results for both optimal and near-optimal receivers are presented and compared to theoretical limits. Finally, experimental shortcomings are discussed along with possible solutions and future direction.
Planar photonic crystals are constructed by combining two-dimensional periodic structures with high refractive index contrast slabs. By suppressing the loss in these structures due to imperfect confinement in the third dimension, one can fully take advantage of their relatively simple fabrication, and achieve the functionality of three-dimensional photonic crystals. One of the greatest challenges in photonic crystal research is a construction of optical nanocavities with small mode volumes and large quality factors, for efficient localization of light. Beside standard applications of these structures (such as lasers or filters), they can potentially be used for cavity QED experiments, or as building blocks for quantum networks. This paper will address our theoretical and experimental results on optical nanocavities based on planar photonic crystals, with mode volumes as small as one half of cubic wavelength of light in material, and with Q factors even larger than 1x104.
Strongly coupled cavity QED systems show great promise for coherent processing of quantum information in the contexts of quantum computing, communication and cryptography. We present here current progress in experiments for which single atoms are strongly coupled to the mode of a high finesse optical resonator.
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