Lattice Quantum Algorithm for the Schrodinger Wave Equation in 2+1 Dimensions With a Demonstration by Modeling Soliton Instabilities

A lattice-based quantum algorithm is presented to model the non-linear Schrödinger-like equations in 2 + 1 dimensions. In this lattice-based model, using only 2 qubits per node, a sequence of unitary collide (qubit-qubit interaction) and stream (qubit translation) operators locally evolve a discrete field of probability amplitudes that in the long-wavelength limit accurately approximates a non-relativistic scalar wave function. The collision operator locally entangles pairs of qubits followed by a streaming operator that spreads the entanglement throughout the two dimensional lattice. The quantum algorithmic scheme employs a non-linear potential that is proportional to the moduli square of the wave function. The model is tested on the transverse modulation instability of a one dimensional soliton wave train, both in its linear and non-linear stages. In the integrable cases where analytical solutions are available, the numerical predictions are in excellent agreement with the theory

Experiments on synthetic dimensions in photonics

The first and introductory section of the dissertation presents the working principle of a one- and two-dimensional photonic mesh lattice based on the time-multiplexing technique. The basis of a random walk interrelated to the corresponding light and quantum walk is comprehensively discussed as well. The second part of the dissertation consists of three experiments on a one-dimensional photonic mesh lattice. Firstly, the Kapitza-based guiding light project models the Kapitza potential as a continuous Pauli-Schrödinger-like equation and presents an experimental observation of light localization when the transverse modulation is bell-shaped but with a vanishing average along the propagation direction. Secondly, the optical thermodynamics project experimentally demonstrates for the first time that any given initial modal occupancy reaches thermal equilibrium by following a Rayleigh-Jeans distribution when propagates through a multimodal photonic mesh lattice with weak nonlinearity. Remarkably, the final modal occupancy possesses a unique temperature and chemical potential that have nothing to do with the actual thermal environment. Finally, the quantum interference project discusses an experimental all-optical architecture based on a coupled-fiber loop for generating and processing time-bin entangled single-photon pairs. Besides, it shows coincidence-to-accidental ratio and quantum interference measurements relying on the phase modulation of those time bins. The third part of the dissertation comprises two experiments on a two-dimensional photonic mesh lattice. The first project discusses the experimental realization of a two-dimensional mesh lattice employing short- and long-range interaction. To some extent, the second project presents a nonconservative system based on a two-dimensional photonic mesh lattice exploiting parity-time (PT) symmetry

Soliton generation and control in engineered materials

Optical solitons provide unique opportunities for the control of light‐bylight. Today, the field of soliton formation in natural materials is mature, as the main properties of the possible soliton states are well understood. In particular, optical solitons have been observed experimentally in a variety of materials and physical settings, including media with cubic, quadratic, photorefractive, saturable, nonlocal and thermal nonlinearities. New opportunities for soliton generation, stability and control may become accessible in complex engineered, artificial materials, whose properties can be modified at will by, e.g., modulations of the material parameters or the application gain and absorption landscapes. In this way one may construct different types of linear and nonlinear optical lattices by transverse shallow modulations of the linear refractive index and the nonlinearity coefficient or complex amplifying structures in dissipative nonlinear media. The exploration of the existence, stability and dynamical properties of conservative and dissipative solitons in settings with spatially inhomogeneous linear refractive index, nonlinearity, gain or absorption, is the subject of this PhD Thesis. We address stable conservative fundamental and multipole solitons in complex engineered materials with an inhomogeneous linear refractive index and nonlinearity. We show that stable two‐dimensional solitons may exist in nonlinear lattices with transversally alternating domains with cubic and saturable nonlinearities. We consider multicomponent solitons in engineered materials, where one field component feels the modulation of the refractive index or nonlinearity while the other component propagates as in a uniform nonlinear medium. We study whether the cross‐phase‐modulation between two components allows the stabilization of the whole soliton state. Media with defocusing nonlinearity growing rapidly from the center to the periphery is another example of a complex engineered material. We study such systems and, in contrast to the common belief, we have found that stable bright solitons do exist when defocusing nonlinearity grows towards the periphery rapidly enough. We consider different nonlinearity landscapes and analyze the types of soliton solution available in each case. Nonlinear materials with complex spatial distributions of gain and losses also provide important opportunities for the generation of stable one‐ and multidimensional fundamental, multipole, and vortex solitons. We study onedimensional solitons in focusing and defocusing nonlinear dissipative materials with single‐ and double‐well absorption landscapes. In two‐dimensional geometries, stable vortex solitons and complexes of vortices could be observed. We not only address stationary vortex structures, but also steadily rotating vortex solitons with azimuthally modulated intensity distributions in radially symmetric gain landscapes. Finally, we study the possibility of forming stable topological light bullets in focusing nonlinear media with inhomogeneous gain landscapes and uniform twophoton absorption

Complex extreme nonlinear waves: classical and quantum theory for new computing models

The historical role of nonlinear waves in developing the science of complexity, and also their physical feature of being a widespread paradigm in optics, establishes a bridge between two diverse and fundamental fields that can open an immeasurable number of new routes. In what follows, we present our most important results on nonlinear waves in classical and quantum nonlinear optics. About classical phenomenology, we lay the groundwork for establishing one uniform theory of dispersive shock waves, and for controlling complex nonlinear regimes through simple integer topological invariants. The second quantized field theory of optical propagation in nonlinear dispersive media allows us to perform numerical simulations of quantum solitons and the quantum nonlinear box problem. The complexity of light propagation in nonlinear media is here examined from all the main points of view: extreme phenomena, recurrence, control, modulation instability, and so forth. Such an analysis has a major, significant goal: answering the question can nonlinear waves do computation? For this purpose, our study towards the realization of an all-optical computer, able to do computation by implementing machine learning algorithms, is illustrated. The first all-optical realization of the Ising machine and the theoretical foundations of the random optical machine are here reported. We believe that this treatise is a fundamental study for the application of nonlinear waves to new computational techniques, disclosing new procedures to the control of extreme waves, and to the design of new quantum sources and non-classical state generators for future quantum technologies, also giving incredible insights about all-optical reservoir computing. Can nonlinear waves do computation? Our random optical machine draws the route for a positive answer to this question, substituting the randomness either with the uncertainty of quantum noise effects on light propagation or with the arbitrariness of classical, extremely nonlinear regimes, as similarly done by random projection methods and extreme learning machines

Fiber Laser Based Nonlinear Spectroscopy

To date, nonlinear spectroscopy has been considered an expensive technique and confined mostly to experimental laboratory settings. Over recent years, optical-fiber lasers that are highly reliable, simple to operate and relatively inexpensive have become commercially available, removing one of the major obstacles to widespread utilization of nonlinear optical measurement in biochemistry. However, fiber lasers generally offer relatively low output power compared to lasers traditionally used for nonlinear spectroscopy, and much more careful design is necessary to meet the excitation power thresholds for nonlinear signal generation. On the other hand, reducing the excitation intensity provides a much more suitable level of user-safety, minimizes damage to biological samples and reduces interference with intrinsic chemical processes. Compared to traditional spectroscopy systems, the complexity of nonlinear spectroscopy and imaging instruments must be drastically reduced for them to become practical. A nonlinear spectroscopy tool based on a single fiber laser, with electrically controlled wavelength-tuning and spectral resolution enhanced by a pulse shaping technique, will efficiently produce optical excitation that allows quantitative measurement of important nonlinear optical properties of materials. The work represented here encompasses the theory and design of a nonlinear spectroscopy and imaging system of the simplest architecture possible, while solving the difficult underlying design challenges. With this goal, the following report introduces the theories of nonlinear optical propagation relevant to the design of a wavelength tunable system for nonlinear spectroscopy applications, specifically Coherent Anti-Stokes Spectroscopy (CARS) and Förster Resonance Energy Transfer (FRET). It includes a detailed study of nonlinear propagation of optical solitons using various analysis techniques. A solution of the generalized nonlinear Schrödinger equation using the split-step Fourier method is demonstrated and investigation of optical soliton propagation in fibers is carried out. Other numerical methods, such as the finite difference time domain approach and spectral-split step Fourier methods are also described and compared. Numerical results are contrasted with various measurements of wavelength shifted solitons. Both CARS and FRET test-bed designs and experiments are presented, representing two valuable biochemical measurement applications. Two-photon excitation experiments with a simplified calibration process for quantitative FRET measurement were conducted on calmodulin proteins modified with fluorescent dyes, as well as modified enhanced green fluorescent protein. The resulting new FRET efficiency measurements showed agreement with those of alternative techniques which are slower and can involve destruction of the sample. In the second major application of the nonlinear spectroscopy system, CARS measurement with enhanced spectral resolution was conducted on cyclohexane as well as on samples of mouse brain tissue containing lipids with Raman resonances. The measurements of cyclohexane verified the ability of the system to precisely determine its Raman resonances, thus providing a benchmark within a similar spectral range for biological materials which have weaker Raman signal responses. The improvement of spectral resolution (resonance frequency selectivity), was also demonstrated by measuring the closely-spaced resonances of cyclohexane. Finally, CARS measurements were also made on samples of mouse brain tissue which has a lipids-based Raman signature. The CARS spectrum of the lipid resonances matched well with other cited studies. The imaging of mouse brain tissue with Raman resonance contrast was also partially achieved, but it was hindered by low signal to noise ratio and limitations of the control hardware that led to some dropout of the CARS signal due to power coupling fluctuations. Nevertheless, these difficulties can be straightforwardly addressed by refinement of the wavelength tuning electronics. In conclusion, it is hoped that these efforts will lead to greater accessibility and use of CARS, FRET and other nonlinear spectral measurement instruments, in line with the promising advances in optics and laser technology

Applications of microstructured fibers : supercontinua and novel components

Microstructured fibers are a special class of pure-silica optical fibers. They consist of a silica core, surrounded by a periodic array of air-holes running along the entire length of the fiber. These air-holes permit guidance of light through total-internal reflection. Diameter and spacing of the air-holes determines the optical properties of the fiber, therefore allowing for tailoring of the fiber according to the intended application. This thesis contains novel results on supercontinuum generation in microstructured fibers. Several critical advances have been made in tailoring of the fiber properties in order to further reduce power requirements hindering miniaturization of supercontinuum sources. In particular, the influence of a second zero-dispersion wavelength of the fiber and the input polarization of highly-birefringent fibers have been studied. Furthermore, a novel two-pump scheme allows for efficient generation of broadband blue-light. The generated supercontinua are applied to characterization of absorption and transmission spectra of novel optical components. The high spectral power density of supercontinuum allows for observation of several new excited-state absorption lines of Erbium-doped fibers and characterization of optical components with strong variations in the transmission spectrum. The second part of the thesis deals with applications developed for microstructured fibers. A tapered microstructured fiber is designed for coupling between standard fibers and photonic-crystal waveguides. An elliptical-core microstructured fiber is proposed as an efficient adapter between standard fibers and highly asymmetric waveguides. In addition, a microstructured fiber based optically bistable fiber cavity is applied to all-optical switching. In particular, an optical flip-flop is numerically studied.reviewe

Discrete Wave Propagation In Quadratically Nonlinear Media

Discrete models are used in describing various microscopic phenomena in many branches of science, ranging from biology through chemistry to physics. Arrays of evanescently coupled, equally spaced, identical waveguides are prime examples of optical structures in which discrete dynamics can be easily observed and investigated. As a result of discretization, these structures exhibit unique diffraction properties with no analogy in continuous systems. Recently nonlinear discrete optics has attracted a growing interest, triggered by the observation of discrete solitons in AlGaAs waveguide arrays reported by Eisenberg et al. in 1998. So far, the following experiments involved systems with third order nonlinearities. In this work, an experimental investigation of discrete nonlinear wave propagation in a second order nonlinear medium is presented. This system deserves particular attention because the nonlinear process involves two or three components at different frequencies mutually locked by a quadratic nonlinearity, and new degrees of freedom enter the dynamical process. In the first part of dissertation, observation of the discrete Talbot effect is reported. In contrast to continuous systems, where Talbot self-imaging effect occurs irrespective of the pattern period, in discrete configurations this process is only possible for a specific set of periodicities. The major part of the dissertation is devoted to the investigation of soliton formation in lithium niobate waveguide arrays with a tunable cascaded quadratic nonlinearity. Soliton species with different topology (unstaggered all channels in-phase, and staggered neighboring channels with a pi relative phase difference) are identified in the same array. The stability of the discrete solitons and plane waves (modulational instability) are experimentally investigated. In the last part of the dissertation, a phase-insensitive, ultrafast, all-optical spatial switching and frequency conversion device based on quadratic waveguide array is demonstrated. Spatial routing and wavelength conversion of milliwatt signals is achieved without pulse distortions