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SPIE Quantum Nanophotonics

Quantum localization issues in nonlinear frequency conversion and harmonic generation

Davıd L. Andrews, Kayn A. Forbes, Jack Ford

Published Aug 29, 2017• 0 citations
Publisher

Abstract

Issues of a fundamental quantum origin exert a significant effect on the output mode structures in optically parametric processes. An assumption that each frequency conversion event occurs in an infinitesimal volume produces uncertainty in the output wave-vector, but a rigorous, photon-based theory can provide for a finite conversion volume. It identifies the electrodynamic mechanisms operating within the corresponding region of space and time, on an optical wavelength and cycle timescale. Based on quantum electrodynamics, this theory identifies specific material parameters that determine the extent and measure of delocalized frequency conversion, and its equations deliver information on the output mode structures. The results also indicate that a system of optimally sized nanoparticles can display a substantially enhanced efficiency of frequency conversion.

DOI
10.1117/12.2269517

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Issues of a fundamental quantum origin exert a significant effect on the output mode structures in optically parametric processes. An assumption that each frequency conversion event occurs in an infinitesimal volume produces uncertainty in the output wave-vector, but a rigorous, photon-based theory can provide for a finite conversion volume. It identifies the electrodynamic mechanisms operating within the corresponding region of space and time, on an optical wavelength and cycle timescale. Based on quantum electrodynamics, this theory identifies specific material parameters that determine the extent and measure of delocalized frequency conversion, and its equations deliver information on the output mode structures. The results also indicate that a system of optimally sized nanoparticles can display a substantially enhanced efficiency of frequency conversion.
The generation of correlated photon pairs is a key to the production of entangled quantum states, which have a variety of applications within the area of quantum information. In spontaneous parametric down-conversion---the primary method of generating correlated photon pairs---the associated photon annihilation and creation events are generally thought of as being colocated: The correlated pair of photons is localized with regards to the pump photon and its positional origin. A detailed quantum electrodynamical analysis highlights a mechanism exhibiting the possibility of a delocalized origin for paired output photons: The spatial extent of the region from which the pair is generated can be much larger than previously thought. The theory of both localized and nonlocalized degenerate down-conversion is presented, followed by a quantitative analysis using discrete-volume computational methods. The results may have significant implications for quantum information and imaging applications, and the design of nonlinear optical metamaterials.
This thesis presents a study of photon generation and conversion processes in nonlinear optics. Our results extend beyond the first order perturbative regime, in which only pairs of photons are generated in spontaneous parametric down-conversion or four wave mixing. They also allow us to identify the limiting factors for achieving unit efficiency frequency conversion, and to correctly account for the propagation of bright nonclassical photon states generated in the nonlinear material. This is done using the Magnus expansion, which allows us to incorporate corrections due to time-ordering effects in the time evolution of the state, while also respecting the quantum statistics associated with squeezed states (in photon generation) and single photons or coherent states (in photon conversion). We show more generally that this expansion should be the preferred strategy when dealing with any type of time-dependent Hamiltonian that is a quadratic form in the creation and annihilation operators of the fields involved. Using the Magnus expansion, simple figures of merit to estimate the relevance of these time-ordering corrections are obtained. These quantities depend on the group velocities of the modes involved in the nonlinear process and provide a very simple physical picture of the interactions between the photons at different times. These time-ordering effects are shown to be important only when the photons and pump beam overlap within the nonlinear material for a significant amount of time. We also discuss the possibility of preparing Fock states with many photons using parametric down-conversion sources and realistic photon-number-resolving detectors. After discussing photon generation processes, we study photon conversion processes in which a nonlinear material and a classical strong pump field are used modify the frequency of a second incoming low-intensity field. We show that time-ordering effects limit the efficiency of this conversion process, re-examine strategies to get around this problem, and show that it should be possible to have a clear observation of modified upconversion probabilities due to these effects, and thus have an experimental demonstration of time-ordering effects in nonlinear optics.
A quantum optical model for high-order harmonic generation is presented in which both the exciting field and the high-order harmonic modes are quantized, while the target material appears only via parameters. As a consequence, the model is independent of the excited material system to a large extent and allows us to focus on the properties of the electromagnetic fields. Technically, the Hamiltonian known for parametric down-conversion is adopted, where photons in the nthn\mathrm{th}nth harmonic mode are created during the annihilation of nnn photons from the fundamental mode. In our treatment, initially, the fundamental mode is in a coherent state corresponding to large photon numbers, while the high-order harmonic modes are in vacuum state. Due to the interaction, the latter modes get populated while the fundamental one loses photons. Analytical approximations are presented for the time evolution that are verified by numerically exact calculations. For multimode, finite-bandwidth excitation, the time dependence of the high-order harmonic radiation is also given.
The existing widely-accepted theory of photon-pair generation via spontaneous down-conversion (SPDC) in nonlinear optical crystals and waveguides is incomplete, as it fails to account for the important physical phenomenon of parametric resonances. We demonstrate that exponential gain of classical fields in the regime of parametric resonance corresponds to resonant delocalization in the Glauber-Fock model of quantum SPDC. We propose a quantitative measure of localisation of Floquet eigen-modes as an analogue of classical gain to identify regimes of resonant delocalization. Using this method, we are able to reconstruct the classical "Arnold tongues" map of domains of instabilities for SPDC. We also predict novel regimes of resonant delocalization in the two-level model describing quantum frequency conversion processes.
Frequency generation in highly multimode nonlinear optical systems is inherently a complex process, giving rise to an exceedingly convoluted landscape of evolution dynamics. While predicting and controlling the global conversion efficiencies in such nonlinear environments has long been considered impossible, here, we formally address this challenge even in scenarios involving a very large number of spatial modes. By utilizing fundamental notions from optical statistical mechanics, we develop a universal theoretical framework that effectively treats all frequency components as chemical reactants/products, capable of undergoing optical thermodynamic reactions facilitated by a variety of multi-wave mixing effects. These photon-photon reactions are governed by conservation laws that directly determine the optical temperatures and chemical potentials of the ensued chemical equilibria for each frequency species. In this context, we develop a comprehensive stoichiometric model and formally derive an expression that relates the chemical potentials to the optical stoichiometric coefficients, in a manner akin to atomic/molecular chemical reactions. This advancement unlocks new predictive capabilities that can facilitate the optimization of frequency generation in highly multimode photonic arrangements, surpassing the limitations of conventional schemes that rely exclusively on nonlinear optical dynamics. Notably, we identify a universal regime of Rayleigh-Jeans thermalization where an optical reaction at near-zero optical temperatures can promote the complete and entropically irreversible conversion of light to the fundamental mode at a target frequency. Our theoretical results are corroborated by numerical simulations in settings where second-harmonic generation, sum-frequency generation and four-wave mixing processes can manifest.
The achievement of optimum conversion efficiency in conventional spontaneous parametric down-conversion requires consideration of quantum processes that entail multisite electrodynamic coupling, actively taking place within the conversion material. The physical mechanism, which operates through virtual photon propagation, provides for photon pairs to be emitted from spatially separated sites of photon interaction; occasionally pairs are produced in which each photon emerges from a different point in space. The extent of such nonlocalized generation is influenced by individual variations in both distance and phase correlation. Mathematical analysis of the global contributions from this mechanism provides a quantitative measure for a degree of positional uncertainty in the origin of down-converted emission.