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CRC Press eBooks

Nonlinear Dynamics in Quantum Photonic Structures

Gabriela M. Slavcheva, Mirella Koleva

Published Oct 12, 2017• 1 citation
Publisher

Abstract

Quantum optics unites physical optics and the quantum field theory of light. A fundamental consequence of light-field quantization is the appearance of vacuum fluctuations, or quantum noise, which has no classical analogue and stems from the Heisenberg uncertainty relations. Nonlinear optical interactions "on a chip" offer new, unexplored functionalities, and may play a key role in future technologies. In this respect, a major theoretical and computational challenge is the development of new, tractable models of the nonlinear quantum dynamics in realistic devices with complex geometries in multiple dimensions. Description of nonlinear pulse propagation, interaction, and localization phenomena in multiple dimensions requires dipole coupling to at least a 2D resonant medium, whereby the electric field of a single-photon excitation is coupled to at least two distinct dipole optical transitions. The chapter applies the vector Maxwell-pseudospin formalism to the problem of nonlinear optical pulse propagation in planar waveguides and microcavities with embedded quantum systems.

DOI
10.4324/9781315152318-24

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Quantum optics unites physical optics and the quantum field theory of light. A fundamental consequence of light-field quantization is the appearance of vacuum fluctuations, or quantum noise, which has no classical analogue and stems from the Heisenberg uncertainty relations. Nonlinear optical interactions "on a chip" offer new, unexplored functionalities, and may play a key role in future technologies. In this respect, a major theoretical and computational challenge is the development of new, tractable models of the nonlinear quantum dynamics in realistic devices with complex geometries in multiple dimensions. Description of nonlinear pulse propagation, interaction, and localization phenomena in multiple dimensions requires dipole coupling to at least a 2D resonant medium, whereby the electric field of a single-photon excitation is coupled to at least two distinct dipole optical transitions. The chapter applies the vector Maxwell-pseudospin formalism to the problem of nonlinear optical pulse propagation in planar waveguides and microcavities with embedded quantum systems.
The interaction of light with matter can generate different types of scattering that can be applied to quantum optics, photonics and integrated optics appropriately. The generation of certain pulses in certain waveguides can produce certain nonlinear optical pulses with important properties such as low energy and information loss applied, for example, to communication systems and quantum information. The method we describe is based on the generation and propagation of nonlinear optical pulses in multidimensional material structures: waveguides in 1 dimension, planar systems in 2 dimensions and crystalline systems in 3 dimensions. These structures can be appropriately designed to generate nonlinear and quantum optical pulses depending on the crystal structure and electrical susceptibility of the material. These optical pulses can be appropriately characterized whose respective scattering signals can be identified and processed providing an application basis for emerging technologies for transmission and processing systems based on quantum optics and quantum computing.
Optical fibers are the backbone of the world's communication systems. They also represent the simplest example of a photonic device, where photons in waveguides replace electrons in wires. More sophisticated devices than this can be made, in which all the optical components – including multiple waveguided modes – are integrated on a chip. It is important to understand the quantum noise properties of these systems. While in some respects these systems can be treated as one-dimensional, there are additional features. Dispersion is due not just to material properties, but also to the waveguide geometry. The nonlinearity comes from a combination of material and geometric properties as well. There is, of course, a quantum aspect to these photonic devices, in which quantum noise occurs due to random effects caused by nonlinearity and gain. Optical devices based on waveguides are, of course, not just simple fibers. One can have multiple waveguides with various types of linear coupling. The intrinsic nonlinearity of the dielectric leads to four-wave mixing, which can be an important source of correlated photons, both for fundamental tests of quantum mechanics, and for applications in quantum information. It is often essential to include effects due to anisotropies of various types, as well as material dispersion. In dielectrics, another effect that should be included is the coupling of vibrational modes of the (solid) dielectric to the propagating field, which gives rise to Raman and Brillouin scattering effects. This allows us to obtain a more detailed and correct theory of quantum noise than the simplified case treated in Chapter 3.
Nonlinear optical phenomena associated with high-order soliton breakup in photonic crystal fibres and squeezed state generation in three dimensional photonic crystal microcavities are investigated. In both cases, the properties of periodically patterned, high-index contrast dielectric structures are engineered to control the dispersion and local field enhancements of the electromagnetic field. Ultra-short pulse propagation in a polarization-maintaining microstructured fibre (with 1 um core diameter and 1.1 m length) is investigated experimentally and theoretically. For an 80 MHz train of 130 fs pulses with average propagating powers in the fibre up to 13.8 mW, the output spectra consist of multiple discrete solitons that shift continuously to lower energies as they propagate in the lowest transverse mode of the fibre. The number of solitons and the amount that they shift both increase with the launched power. All of the data is quantitatively consistent with solutions of the nonlinear Schrodinger equation, but only when the Raman nonlinearity is treated without approximation, and self-steepening is included. The feasibility of using a parametric down-conversion process to generate squeezed electromagnetic states in 3D photonic crystal microcavity structures is investigated for the first time. The spectrum of the squeezed light is theoretically calculated by using an open cavity quantum mechanical formalism. The cavity communicates with two main channels, which model vertical radiation losses and coupling into a single-mode waveguide respectively. The amount of squeezing is determined by the correlation functions relating the field quadratures of light coupled into the waveguide. All of the relevant model parameters are realistically estimated using 3D finite-difference time-domain (FDTD) simulations. Squeezing up to ~20% below the shot noise level is predicted for reasonable optical excitation levels. To preserve the squeezed nature of the light generated in the microcavity, a unidirectional coupling geometry from the microcavity to a ridge waveguide in a slab photonic crystal structure is studied. The structure was successfully fabricated in a silicon membrane, and experimental measurements of the efficiency for the signal coupled out of the structure are in good agreement with the result of FDTD simulations. The coupling efficiency of the cavity mode to the output channel is ~60%.
In a realistic treatment of a three-dimensional nonlinear optical experiment, the complete Maxwell equations in (3 + 1) space-time dimensions should be employed. It is then necessary to utilize a multi-mode Hamiltonian that correctly describes the propagating modes. There is an important difference between these experiments and traditional particle scattering. Quantum field dynamics in nonlinear media is dominated by multiple scattering, which is the reason why perturbation theory is less useful. Nevertheless, it is interesting to make a link to conventional perturbation theory. Accordingly, we start by considering a perturbative theory of propagation in a one-dimensional nonlinear optical system in a χ (3) medium. While this calculation cannot treat long interaction times, it does give a qualitative understanding of the important features. This problem is the ‘hydrogen atom’ of quantum field theory: it has fully interacting fields with exact solutions for their energy levels. This is because a photon in a waveguide is an elementary boson in one dimension. The interactions between these bosons are mediated by the Kerr effect, which in quantum field theory is a quartic potential, equivalent to a delta-function interaction. The quantum field theory involved is the simplest model of a quantum field that has an exact solution. This elementary model is still nontrivial in terms of its dynamics, as the calculation of quantum dynamics using standard eigenfunction techniques would require exponentially complex sums over multi-dimensional overlap integrals. This is not practicable, and accordingly we use other methods including quantum phase-space representations to solve this problem.
A detectable nonlinear response between light and matter depends on one of two key ingredients: Either there is a resonance between the light wave and some natural oscillation mode of the material, or the light is sufficiently intense. In this chapter, the authors let the matter variables themselves play active roles and derive coupled nonlinear differential equations for the electric and matter fields. They outline the use of Schrodinger's equation to derive the Bloch equations, which describe the dynamics of the oscillators of matter and help to derive the Maxwell equation for the envelope of an almost monochromatic electromagnetic wave. The authors analyse the case of two distinct co- or counter-propagating waves interacting with a three-level medium where both waves excite separate quantum transitions sharing a common level. They also help to derive an effective two-level atom model that describes two-photon absorption and stimulated Raman scattering.
At the quantum limit of nonlinear optics, where single photons interact with individual emitters, new phenomena are found. One-dimensional, subwavelength waveguides have recently emerged as a promising quantum optical platform capable of both guiding and confining photons, leading to efficient light-matter interactions and enhancing quantum optical nonlinearities. We explore this regime using both single organic molecules and epitaxially grown quantum dots coupled to nanoscale waveguides. In the first case, two-color experiments allow us to observe distinctly quantum nonlinearities such as Stark shifts of the molecular resonances or amplification without population inversion, although we are limited by relatively weak molecule-waveguide coupling. In contrast, quantum dots can be nearly perfectly coupled to waveguides, allowing us to demonstrate optical nonlinearities at the true single photon level. These, as we discuss, can be used as a basis for novel on-chip quantum technologies.