Nonlinear optical effects in artificial materials

We consider some nonlinear phenomena in metamaterials with negative refractive index properties. Our consideration includes a survey of previously known results as well as identification of the phenomena that are important for applications of this new field. We focus on optical behavior of thin films as well as multi-wave interactions.Comment: 22 pages, no figures. Submitted in book "Nonlinear waves in complex systems: energy flow and geometry

Classification of finite groups with toroidal or projective-planar permutability graphs

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.Comment: 30 pages, 8 figure

Neighboring mapping points theorem

Let f be a continuous map from a metric space X to M. We say that points in a subset N of X are f-neighbors if there exists a sphere S in M such that f(N) lies on S and there are no points of f(X) inside of S. We prove that if X is a unit sphere of any dimension and M is a contractible metric space then there are two f-neighbors in X such that the distance between them is greater than one. This theorem can be derived from the fact that for every non-null-homotopic closed covering C of X there is a set of f-neighbors N in X such that every member of C contains a point of N.Comment: 7 pages, 1 figur

Electrically Controlled Pumping of Spin Currents in Topological Insulators

Pure spin currents are shown to be generated by an electrically controlled quantum pump applied at the edges of a topological insulator. The electric rather than the more conventional magnetic control offers several advantages and avoids, in particular, the necessity of delicate control of magnetization dynamics over tiny regions. The pump is implemented by pinching the sample at two quantum point contacts and phase modulating two external gate voltages between them. The spin current is generated for the full range of parameters. On the other hand, pumping via amplitude modulation of the inter-boundary couplings generates both charge and spin currents, with a pure charge current appearing only for special values of the parameters for which the Bohm-Aharonov flux takes integer values. Our setup can therefore serve to fingerprint the helical nature of the edges states with the zeros of the pumped spin and charge currents occurring at distinct universal locations where the Fabry-Perot or the Aharonov-Bohm phases take integer values.Comment: 5 pages, 5figure

Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations

We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3/2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3/2 for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.Comment: 12 p