There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

BACKGROUND Singularities are critical points for which the behavior of a mathematical model governing a physical system is of a fundamentally different nature compared to the neighboring points. Exceptional points are spectral singularities in the parameter space of a system in which two or more eigenvalues, and their corresponding eigenvectors, simultaneously coalesce. Such degeneracies are peculiar features of nonconservative systems that exchange energy with their surrounding environment. In the past two decades, there has been a growing interest in investigating such nonconservative systems, particularly in connection with the quantum mechanics notions of parity-time symmetry, after the realization that some non-Hermitian Hamiltonians exhibit entirely real spectra. Lately, non-Hermitian systems have raised considerable attention in photonics, given that optical gain and loss can be integrated as nonconservative ingredients to create artificial materials and structures with altogether new optical properties. ADVANCES As we introduce gain and loss in a nanophotonic system, the emergence of exceptional point singularities dramatically alters the overall response, leading to a range of exotic functionalities associated with abrupt phase transitions in the eigenvalue spectrum. Even though such a peculiar effect has been known theoretically for several years, its controllable realization has not been made possible until recently and with advances in exploiting gain and loss in guided-wave photonic systems. As shown in a range of recent theoretical and experimental works, this property creates opportunities for ultrasensitive measurements and for manipulating the modal content of multimode lasers. In addition, adiabatic parametric evolution around exceptional points provides interesting schemes for topological energy transfer and designing mode and polarization converters in photonics. Lately, non-Hermitian degeneracies have also been exploited for the design of laser systems, new nonlinear optics phenomena, and exotic scattering features in open systems. OUTLOOK Thus far, non-Hermitian systems have been largely disregarded owing to the dominance of the Hermitian theories in most areas of physics. Recent advances in the theory of non-Hermitian systems in connection with exceptional point singularities has revolutionized our understanding of such complex systems. In the context of optics and photonics, in particular, this topic is highly important because of the ubiquity of nonconservative elements of gain and loss. In this regard, the theoretical developments in the field of non-Hermitian physics have allowed us to revisit some of the well-established platforms with a new angle of utilizing gain and loss as new degrees of freedom, in stark contrast with the traditional approach of avoiding these elements. On the experimental front, progress in fabrication technologies has allowed for harnessing gain and loss in chip-scale photonic systems. These theoretical and experimental developments have put forward new schemes for controlling the functionality of micro- and nanophotonic devices. This is mainly based on the anomalous parameter dependence in the response of non-Hermitian systems when operating around exceptional point singularities. Such effects can have important ramifications in controlling light in new nanophotonic device designs, which are fundamentally based on engineering the interplay of coupling and dissipation and amplification mechanisms in multimode systems. Potential applications of such designs reside in coupled-cavity laser sources with better coherence properties, coupled nonlinear resonators with engineered dispersion, compact polarization and spatial mode converters, and highly efficient reconfigurable diffraction surfaces. In addition, the notion of the exceptional point provides opportunities to take advantage of the inevitable dissipation in environments such as plasmonic and semiconductor materials, which play a key role in optoelectronics. Finally, emerging platforms such as optomechanical cavities provide opportunities to investigate exceptional points and their associated phenomena in multiphysics systems.

Exceptional points in optics and photonics.

Recent advances in shape–memory polymers: Structure, mechanism, functionality, modeling and applications

Mathematical Handbook for Scientists and Engineers

A review of stimuli-responsive shape memory polymer composites

We studied both global and local effects of cylindrical confinement on the diffusive behavior of hard sphere (HS) colloids. Using confocal scanning laser microscopy (CSLM) and particle tracking, we measured the mean squared displacement (MSD) of 1 micron sized silica particles in water–glycerol. This combination of fluid and setup allowed us to measure MSDs in a 4-dimensional parameter space, defined by the HS volume fraction (Φ: 0.05–0.39), cylinder radius (R: 2.5–20 micron), distance to the wall (z) and lagtime (τ: 0.03–60 s). MSDs for the entire cylinder confirm earlier findings that both narrowing the cylinder and populating it cause a slower dynamics. Additionally a decrease in R was found to cause a stronger ordering of the fluid. The effect of confinement on dynamics was further examined as a function of (z) location. For the largest cylinder (with minor curvature), we found that the strong decrease in MSD near the wall, becomes much less pronounced for higher Φ. Analyzing the radial (r) and azimuthal (θ) components, we found pronounced differences in the z-dependence that were ‘hidden’ in the total MSD. Near the wall, the r-MSD shows a much steeper z-dependence while at larger z, it shows a remarkable anti-correlation with the (peaked) density n(z). Also the dependence of the r-MSD on lagtime correlates with n(z): diffusive in between layers, but subdiffusive inside layers. These observations bring earlier findings together, while also shedding new light on the diffusive dynamics of concentrated colloids in narrow capillaries.

Dynamics of colloids confined in microcylinders

Nitrate reductases (NRs) are molybdoenzymes that reduce nitrate (NO3−) to nitrite (NO2−) in both mammals and plants. In mammals, the salival microbes take part in the generation of the NO2− from NO3−, which further produces nitric oxide (NO) either in acid-induced NO2− reduction or in the presence of nitrite reductases (NiRs). Here, we report a new approach of VCl3 (V3+ ion source) induced step-wise reduction of NO3− in a CoII-nitrato complex, [(12-TMC)CoII(NO3−)]+ (2,{CoII–NO3−}), to a CoIII–nitrosyl complex, [(12-TMC)CoIII(NO)]2+ (4,{CoNO}8), bearing an N-tetramethylated cyclam (TMC) ligand. The VCl3 inspired reduction of NO3− to NO is believed to occur in two consecutive oxygen atom transfer (OAT) reactions, i.e., OAT-1 = NO3− → NO2− (r1) and OAT-2 = NO2− → NO (r2). In these OAT reactions, VCl3 functions as an O-atom abstracting species, and the reaction of 2 with VCl3 produces a CoIII-nitrosyl ({CoNO}8) with VV-Oxo ({VVO}3+) species, via a proposed CoII-nitrito (3, {CoII–NO2−}) intermediate species. Further, in a separate experiment, we explored the reaction of isolated complex 3 with VCl3, which showed the generation of 4 with VV-Oxo, validating our proposed reaction sequences of OAT reactions. We ensured and characterized 3 using VCl3 as a limiting reagent, as the second-order rate constant of OAT-2 (k2/) is found to be ∼1420 times faster than that of the OAT-1 (k2) reaction. Binding constant (Kb) calculations also support our proposition of NO3− to NO transformation in two successive OAT reactions, as Kb(CoII–NO2−) is higher than Kb(CoII–NO3−), hence the reaction moves in the forward direction (OAT-1). However, Kb(CoII–NO2−) is comparable to Kb{CoNO}8, and therefore sequenced the second OAT reaction (OAT-2). Mechanistic investigations of these reactions using 15N-labeled-15NO3− and 15NO2− revealed that the N-atom in the {CoNO}8 is derived from NO3− ligand. This work highlights the first-ever report of VCl3 induced step-wise NO3− reduction (NRs activity) followed by the OAT induced NO2− reduction and then the generation of Co-nitrosyl species {CoNO}8.

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Oxygen atom transfer promoted nitrate to nitric oxide transformation: a step-wise reduction of nitrate → nitrite → nitric oxide

An expression is derived for the diffusivity-mobility ratio in degenerate narrow-gap ternary semiconductors in the presence of a quantizing magnetic field. With the help of this expression, the diffusivity-mobility ratio is shown, taking n-Hg1−xCdxTe as an example, to have an oscillatory dependence on a quantizing magnetic field. The influence of alloy composition on such dependence is also investigated.

Influence of alloy composition on the dependence of the diffusivity-mobility ratio in ternary semiconductors on a quantizing magnetic field

Here, we report the nitric oxide monooxygenation (NOM) reactions of a CoIII-nitrosyl complex (1, {Co-NO}8) in the presence of mono-oxygen reactive species, i.e., a base (OH−, tetrabutylammonium hydroxide (TBAOH) or NaOH/15-crown-5), an oxide (O2− or Na2O/15-crown-5) and water (H2O). The reaction of 1 with OH− produces a CoII-nitrito complex {3, (CoII-NO2−)} and hydrogen gas (H2), via the formation of a putative N-bound Co-nitrous acid intermediate (2, {Co-NOOH}+). The homolytic cleavage of the O–H bond of proposed [Co-NOOH]+ releases H2via a presumed CoIII-H intermediate. In another reaction, 1 generates CoII-NO2− when reacted with O2−via an expected CoI-nitro (4) intermediate. However, complex 1 is found to be unreactive towards H2O. Mechanistic investigations using 15N-labeled-15NO and 2H-labeled-NaO2H (NaOD) evidently revealed that the N-atom in CoII-NO2− and the H-atom in H2 gas are derived from the nitrosyl ligand and OH− moiety, respectively.

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Somnath Ghosh

Papers

Dynamics of colloids confined in microcylinders

Oxygen atom transfer promoted nitrate to nitric oxide transformation: a step-wise reduction of nitrate → nitrite → nitric oxide

Influence of alloy composition on the dependence of the diffusivity-mobility ratio in ternary semiconductors on a quantizing magnetic field

Nitric oxide monooxygenation (NOM) reaction of cobalt-nitrosyl {Co(NO)}8 to CoII-nitrito {CoII(NO2-)}: base induced hydrogen gas (H2) evolution.

On the Modification of the Einstein Relation for Heavily Doped Semiconductors Having Gaussian Band Tails