The Cationminus signpi Interaction.

(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

This article reviews the electronic and transport properties of carbon nanotubes. The focus is mainly theoretical, but when appropriate the relation with experimental results is mentioned. While simple band-folding arguments will be invoked to rationalize how the metallic or semiconducting character of nanotubes is inferred from their topological structure, more sophisticated tight-binding and ab initio treatments will be introduced to discuss more subtle physical effects, such as those induced by curvature, tube-tube interactions, or topological defects. The same approach will be followed for transport properties. The fundamental aspects of conduction regimes and transport length scales will be presented using simple models of disorder, with the derivation of a few analytic results concerning specific situations of shortand long-range static perturbations. Further, the latest developments in semiempirical or ab initio simulations aimed at exploring the effect of realistic static scatterers chemical impurities, adsorbed molecules, etc. or inelastic electron-phonon interactions will be emphasized. Finally, specific issues, going beyond the noninteracting electron model, will be addressed, including excitonic effects in optical experiments, the Coulomb-blockade regime, and the Luttinger liquid, charge density waves, or superconducting transition.

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Electronic and transport properties of nanotubes

Metamaterials are composed of periodic subwavelength metal/dielectric structures that resonantly couple to the electric and/or magnetic components of the incident electromagnetic fields, exhibiting properties that are not found in nature. Planar metamaterials with subwavelength thickness, or metasurfaces, consisting of single-layer or few-layer stacks of planar structures, can be readily fabricated using lithography and nanoprinting methods, and the ultrathin thickness in the wave propagation direction can greatly suppress the undesirable losses. Metasurfaces enable a spatially varying optical response, mold optical wavefronts into shapes that can be designed at will, and facilitate the integration of functional materials to accomplish active control and greatly enhanced nonlinear response. This paper reviews recent progress in the physics of metasurfaces operating at wavelengths ranging from microwave to visible. We provide an overview of key metasurface concepts such as anomalous reflection and refraction, and introduce metasurfaces based on the Pancharatnam-Berry phase and Huygens' metasurfaces, as well as their use in wavefront shaping and beam forming applications, followed by a discussion of polarization conversion in few-layer metasurfaces and their related properties. An overview of dielectric metasurfaces reveals their ability to realize unique functionalities coupled with Mie resonances and their low ohmic losses. We also describe metasurfaces for wave guidance and radiation control, as well as active and nonlinear metasurfaces. Finally, we conclude by providing our opinions of opportunities and challenges in this rapidly developing research field.

A review of metasurfaces: physics and applications

Sources and systems for far-infrared or terahertz (1 THz = 10(12) Hz) radiation have received extensive attention in recent years, with applications in sensing, imaging and spectroscopy. Terahertz radiation bridges the gap between the microwave and optical regimes, and offers significant scientific and technological potential in many fields. However, waveguiding in this intermediate spectral region still remains a challenge. Neither conventional metal waveguides for microwave radiation, nor dielectric fibres for visible and near-infrared radiation can be used to guide terahertz waves over a long distance, owing to the high loss from the finite conductivity of metals or the high absorption coefficient of dielectric materials in this spectral range. Furthermore, the extensive use of broadband pulses in the terahertz regime imposes an additional constraint of low dispersion, which is necessary for compatibility with spectroscopic applications. Here we show how a simple waveguide, namely a bare metal wire, can be used to transport terahertz pulses with virtually no dispersion, low attenuation, and with remarkable structural simplicity. As an example of this new waveguiding structure, we demonstrate an endoscope for terahertz pulses.

Metal wires for terahertz wave guiding

Thanks to the recent progresses in epitaxy techniques, it is now possible to grow alternating layers of semiconductors with different band gap energies with a thickness control down to one atomic layer. It is now well established that these systems behave, as far as the electron motion is concerned, as a succession of potential barriers and wells. If the width of the potential well is less than the deBroglie wavelength of the electrons in the material (e.g. less than ≈15 nm in GaAs), the motion of the electrons may be considered, at sufficiently low temperature, as quantized in the direction normal to the growth axis1. The electrons are quantized into subbands where their wavefunctions in the growth direction have the form of envelope functions with an extension equal to the well width, i.e. in the few nanometer range. Electromagnetic waves may induce electronic transitions between these subbands. The dipole matrix elements associated to these intersubband transitions (ISBT) have the same order of magnitude as the quantum well width leading to extremely large absorption. These large absorptions have been observed for the first time in GaAs/AlGaAs multiquantum wells (MQW) by West and Eglash2.

Model system for optical nonlinearities: Asymmetric quantum wells.

Frequency conversion in nonlinear optical crystals1,2 is an effective means of generating coherent light at frequencies where lasers perform poorly or are unavailable. For efficient conversion, it is necessary to compensate for optical dispersion, which results in different phase velocities for light of different frequencies. In anisotropic birefringent crystals such as LiNbO3 or KH2PO4 (‘KDP’), phase matching can be achieved between electromagnetic waves having different polarizations. But this is not possible for optically isotropic materials, and as a result, cubic materials such as GaAs (which otherwise have attractive nonlinear optical properties) have been little exploited for frequency conversion applications. Quasi-phase-matching schemes1,3, which have achieved considerable success in LiNbO3 (ref. 4), provide a route to circumventing this problem5,6, but the difficulty of producing the required pattern of nonlinear properties in isotropic materials, particularly semiconductors, has limited the practical utility of such approaches. Here we demonstrate a different route to phase matching — based on a concept proposed by Van der Ziel 22 years ago7 — which exploits the artificial birefringence of multilayer composites of GaAs and oxidised AlAs. As GaAs is the material of choice for semiconductor lasers, such optical sources could be integrated in the core of frequency converters based on these composite structures.

Phase matching using an isotropic nonlinear optical material

Second-order optical nonlinearities in materials are of paramount importance for optical wavelength conversion techniques, which are the basis of new high-resolution spectroscopic tools. Semiconductor technology now makes it possible to design and fabricate artificially asymmetric quantum structures in which optical nonlinearities can be calculated and optimized from first principles. Extremely large second-order susceptibilities can be obtained in these asymmetric quantum wells. Moreover, properties such as double resonance enhancement or electric field control will open the way to new devices, such as fully solid-state optical parametric oscillators.

https://science.sciencemag.org/content/sci/271/5246/168.full.pdf

Quantum Engineering of Optical Nonlinearities

The potential energy profile in semiconductor heterostructures can now be controlled in a fascinating way that could barely be dreamed of twenty years ago 1. When dealing with interband optical transitions, additional features related to electron-hole interactions (see for instance exciton descriptions in this book) are coming into play and the one-electron wavefunctions and energy levels may fail to describe or predict experimental results. Moreover, the quantization energy is usually small compared to the forbidden band gap, so that typical interband transitions always occur in the same energy range for a given materials pair. On the contrary, intersubband transitions (ISBT) are very sensitive to the exact potential profile and transitions have been observed at wavelengths between lμm and 100μm. In addition, they can be quantitatively described by a simple formalism based on one-electron approaches and many-body effects usually appear as small corrections only. Since 19852, many devices have been designed according to this quantum engineering and have shown unsurpassed properties3. Various materials have been successfully used for these quantum well (QW) heterostructures: GaAs/AlGaAs, InP/InGaAs/InAlAs, Si/SiGe, D/VI compounds…We will focus here on the GaAs/AlGaAs system which has been the most widely studied. First, the calculation of the ISBT matrix element will evidence two major characteristic properties: the optical transitions take advantage of giant dipoles but must verify in the same time a rather drastic selection rule. Then, examples will be given in different fields of application: detection, modulation and emission. Some interesting aspects of coupling and propagation in these structures involve a photon mode density alteration. Finally, a detailed study of second order non linearities will exemplify the beauty of quantum engineering for improving optical properties.

Intersubband transitions in quantum wells

Optical transitions in single-wall boron nitride nanotubes are investigated by means of optical absorption spectroscopy. Three absorption lines are observed. Two of them (at 4.45 and 5.5 eV) result from the quantification involved by the rolling up of the hexagonal boron nitride (h-BN) sheet. The nature of these lines is discussed, and two interpretations are proposed. A comparison with single-wall carbon nanotubes leads one to interpret these lines as transitions between pairs of van Hove singularities in the one-dimensional density of states of boron nitride single-wall nanotubes. But the confinement energy due to the rolling up of the h-BN sheet cannot explain a gap width of the boron nitride nanotubes below the h-BN gap. The low energy line is then attributed to the existence of a Frenkel exciton with a binding energy in the 1 eV range.

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Emmanuel Rosencher

Papers

Model system for optical nonlinearities: Asymmetric quantum wells.

Phase matching using an isotropic nonlinear optical material

Quantum Engineering of Optical Nonlinearities

Intersubband transitions in quantum wells

Optical Transitions in Single-Wall Boron Nitride Nanotubes