Link polynomial calculus and the AENV conjecture

Using the recently proposed differential hierarchy (Z-expansion) technique, we obtain a general expression for the HOMFLY polynomials in two arbitrary symmetric representations of link families, including Whitehead and Borromean links. Among other things, this allows us to check and confirm the recent conjecture of arXiv:1304.5778 that the large representation limit (the same as considered in the knot volume conjecture) of this quantity matches the prediction from mirror symmetry consideration. We also provide, using the evolution method, the HOMFLY polynomial in two arbitrary symmetric representations for an arbitrary member of the one-parametric family of 2-component 3-strand links, which includes the Hopf and Whitehead links.Comment: 20 page

Towards topological quantum computer

One of the principal obstacles on the way to quantum computers is the lack of distinguished basis in the space of unitary evolutions and thus the lack of the commonly accepted set of basic operations (universal gates). A natural choice, however, is at hand: it is provided by the quantum R-matrices, the entangling deformations of non-entangling (classical) permutations, distinguished from the points of view of group theory, integrable systems and modern theory of non-perturbative calculations in quantum field and string theory. Observables in this case are (square modules of) the knot polynomials, and their pronounced integrality properties could provide a key to error correction. We suggest to use R-matrices acting in the space of irreducible representations, which are unitary for the real-valued couplings in Chern-Simons theory, to build a topological version of quantum computing.Comment: 14 page

Photonic and electric control of single photon emission from individual quantum dots

Modern single photon quantum technologies require on-demand and deterministic sources. Single photon emission has been observed from various solid state systems, such as individual molecules, defects and colour centres in diamond and 2D materials, and artificial atoms - quantum dots. However, the quantum emitters suffer from the slow radiation rate and omnidirectional emission, preventing their practical applications. First, this Thesis proposes to exploit the charged excitonic states in individual colloidal quantum dots for the enhancement of radiation rate. Electron injection to a quantum dot increases the number of de-excitation pathways, hence boosting the single photon generation speed. The optical properties of charged excitons depend drastically on the number of injected electrons, which can be controlled by applying a voltage bias in an electrochemical cell, thus providing for an active and deterministic way to manipulate the single photon emission. The charge transfer allows for a deterministic manipulation of quantum dot photodynamics, with an observed 210-fold increase of the radiation rate, accompanied by a 12-fold decrease of the emission intensity, all while preserving single-photon emission characteristics. Secondly, this Thesis proposes a 3D metal-dielectric parabolic antenna with an individual quantum dot in its focal point as a source of collimated single photons which can then be easily extracted and manipulated. Compared to conventional nano-antennas, 3D parabolic antenna design does not require near-field coupling, hence it is very robust against misalignment issues, and minimally affected by absorption in the metal. The parabolic antenna provides one of the largest reported experimental directivity (D=106) and the lowest beam divergence (13.5 deg), a broadband operation over all the visible and near-IR range, together with more than 96% extraction efficiency, offering a practical advantage for quantum technological applications.Open Acces

Cut-and-join structure and integrability for spin Hurwitz numbers

Spin Hurwitz numbers are related to characters of the Sergeev group, which are the expansion coefficients of the Q Schur functions, depending on odd times and on a subset of all Young diagrams. These characters involve two dual subsets: the odd partitions (OP) and the strict partitions (SP). The Q Schur functions Q_R with R\in SP are common eigenfunctions of cut-and-join operators W_\Delta with \Delta\in OP. The eigenvalues of these operators are the generalized Sergeev characters, their algebra is isomorphic to the algebra of Q Schur functions. Similarly to the case of the ordinary Hurwitz numbers, the generating function of spin Hurwitz numbers is a \tau-function of an integrable hierarchy, that is, of the BKP type. At last, we discuss relations of the Sergeev characters with matrix models.Comment: 22 page

A Hurwitz theory avatar of open-closed strings

We review and explain an infinite-dimensional counterpart of the Hurwitz theory realization of algebraic open-closed string model a la Moore and Lizaroiu, where the closed and open sectors are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphes respectively. An intriguing feature of this Hurwitz string model is coexistence of two different multiplications, reflecting the deep interrelation between the theory of symmetric and linear groups S_\infty and GL(\infty).Comment: 11 page