Photons are robust carriers of quantum information as they can propagate long distances without losing quantum entanglement and coherence. Compared to quantum information in matter-based carriers, such as superconducting oscillators, trapped ions and atoms, quantum dots, and vacancy centers in crystals, the photonic quantum states are robust against perturbations from the environment, such as parasitic electromagnetic fields and thermal fluctuations (phonons), making it an ideal candidate for room-temperature-based quantum metrology and information processing applications. Such robustness is due to photon-photon scattering in the vacuum being extremely improbable and photon-atom interactions being in the linear regime for most materials. Nevertheless, photon-photon or photon-atom nonlinear interactions are also critical for all quantum photonic applications as nonlinearity is required for generating non-classical states of light. Furthermore, nonlinear interactions greatly expand the variety of Hamiltonian that can be engineered for a given system or subsystem, which is a direct measure of the system's functionality. Thus, the ability to engineer nonlinear interactions has been one of the primary research focuses in quantum photonics. This thesis presents research on using nonlinear photonic chips to harness the unique properties offered by quantum mechanics, with applications in precision metrology and information procession. Atoms possess a rich set of quantum properties that have no counterparts in the classical world. Even in warm vapor form, atomic gases maintain sufficient coherence for tasks, including time keeping, electric field sensing and quantum memories. We develop chip-based light sources that can interact with narrow-band atomic transitions in order to miniaturize these applications. Typical Alkali atoms have transition around the visible light regime, where photonic materials exhibit strong normal group-velocity dispersion (GVD) which inhibits light generation via nonlinear interactions. We offer a systematic solution by re-examining the dispersion engineer techniques, which revealed that higher-order waveguide modes can have stronger anomalous GVD. With this technique, we demonstrate on-chip mode-locked pulses (Kerr combs) at a record-low wavelength, which can be used for high-precision atomic clocks. We also develop chip-based narrow-band high-brightness photon sources at the visible regime using nonlinear interactions. Such photons can interact with atom-based quantum memories and gates, which can find applications in both quantum communication and computation. Squeezed state is also an important class of non-classical states with key applications in quantum metrology, quantum simulation, and continuous-variable quantum information processing. Typically, squeezed states are generated using χ² processes, which are not readily available on most photonic platforms. For the first time, we demonstrate squeezed state generation using a dual-pumped four-wave-mixing process, which we implement on a silicon-nitride chip. To perform quantum simulation or computation with squeezed states, we need programmable interferometer arrays and photon-number resolving (PNR) detectors. Current PNR detectors rely on superconducting effects which require Kelvin level temperatures. We propose a room-temperature PNR scheme based on optical nonlinearity. We show that using cascaded χ² interactions, a single photon can impart an observable phase on a probe beam, which can be implemented within the current fabrication capabilities. Our squeezed-state-generation and PNR-detection devices lay a practical path towards room-temperature quantum simulation and computing.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Meteorology and Physical Oceanography, 1982. / Microfiche copy available in Archives and Science / Vita. / Bibliography: leaves 321-326. / by Teresa Kathleen Chereskin. / Ph.D.

It is well established that many physical and chemical phenomena such as those in chemical reaction kinetics, laser cavities, rotating fluids, and in plasmas and in solid state physics are governed by nonlinear differential equations whose solutions are of variable character and even may lack regularities. Such systems are usually first studied qualitatively by examining their temporal behavior near singular points of their phase portrait. In this work we will be concerned with systems governed by the time evolution equations [see PDF for mathematical formulas] The xi may generally be considered to be concentrations of species in a chemical reaction, in which case the k's are rate constants. In some cases the xi may be considered to be position and momentum variables in a mechanical system. We will divide the equations into two classes: those in which the evolution can be carried out by the action of one of Lie's transformation groups of the plane, and those for which this is not possible. Members of the first class can be integrated by quadrature either directly or by use of an integrating factor; those in the second class cannot. Of those in the first class the most interesting evolve by transformations of the projective group, and these, as well as the equations that cannot be integrated by quadrature, we study in some detail. We seek a qualitative analysis of systems which have no linear terms in their evolution equations when the origin from which the xi are measured is a critical point. The standard, linear, phase plane analysis is of course not adequate for our purposes.

No description available.

No description available.

No description available.

No description available.

The method of multiple scales is used to study the nonlinear response of a relief valve under a constant static pressure and a sinusoidal dynamic pressure that can be generated by variations in the pneumatic transmission lines. Under certain conditions, large vibrations may occur, causing unsatisfactory performance as well as undesirable noise. In this theoretical analysis, the valve is modeled by a continuous spring fixed at one end and attached to a valve ball at the other end. The ball rests on a valve seat having nonlinear spring characteristics. The method of multiple scales is used to determine the non-linear response of the valve for the cases of harmonic, higher-harmonic, subharmonic and combination resonances. Conditions are determined for the onset and stability of large-amplitude oscillations. / M.S.

The work described in this dissertation can be divided into two parts. The first part is an investigation of the transient behavior and stability property of a phase conjugate resonator (PCR) below threshold. The second part is an experimental and theoretical study of the PCR's spatiotemporal dynamics above threshold. The time-dependent coupled wave equations for four-wave mixing (FWM) in a photorefractive crystal, with two distinct interaction regions caused by feedback from an ordinary mirror, was used to model the transient dynamics of a PCR below threshold. Analytical expressions of the steady state cavity's fields for the case of nondepleted pumps and an absorption free medium were derived and used to determine the self-oscillation conditions. The solutions, through simple frequency domain transformation techniques, were used to define the PCR's transfer function and analyse its stability. Taking into account pump depletion and medium absorption, the transient buildup and decay times of the cavity's fields as well as the specularly reflected and phase conjugate reflected intensities were numerically calculated as functions of a number of system parameters such as the coupling parameter and the pump and probe ratios. General trends were unveiled and discussed in view of the possible use of the PCR in image storage or processing architectures. Experimental results for the buildup and decay times confirmed qualitatively the predicted behavior. / Ph. D.

There are algorithms for finding zeros or fixed points of nonlinear systems of (algebraic) equations that are globally convergent for almost all starting points, i.e., with probability one. The essence of all such algorithms is the construction of an appropriate homotopy map and then tracking some smooth curve in the zero set of this homotopy map. The augmented Jacobian matrix algorithm is part of the software package HOMPACK, and is based on an algorithm developed by W.C. Rheinboldt. The algorithm exists in two forms-one for dense Jacobian matrices, and the other for sparse Jacobian matrices. / M.S.