Optomechanical measurement of photon spin angular momentum and optical torque in integrated photonic devices

Photons carry linear momentum, and spin angular momentum when circularly or elliptically polarized. During light-matter interaction, transfer of linear momentum leads to optical forces, while angular momentum transfer induces optical torque. Optical forces including radiation pressure and gradient forces have long been utilized in optical tweezers and laser cooling. In nanophotonic devices optical forces can be significantly enhanced, leading to unprecedented optomechanical effects in both classical and quantum regimes. In contrast, to date, the angular momentum of light and the optical torque effect remain unexplored in integrated photonics. Here, we demonstrate the measurement of the spin angular momentum of photons propagating in a birefringent waveguide and the use of optical torque to actuate rotational motion of an optomechanical device. We show that the sign and magnitude of the optical torque are determined by the photon polarization states that are synthesized on the chip. Our study reveals the mechanical effect of photon's polarization degree of freedom and demonstrates its control in integrated photonic devices. Exploiting optical torque and optomechanical interaction with photon angular momentum can lead to torsional cavity optomechanics and optomechanical photon spin-orbit coupling, as well as applications such as optomechanical gyroscope and torsional magnetometry.Comment: 38 pages, 4 figure

SO(3) invariants of Seifert manifolds and their algebraic integrality

For Seifert manifold M=X({p_1}/_{\f{q_1}},{p_2}/_{\f{q_2}}, ...,{p_n}/_ {\f{q_n}}), \tau^{'}_r(M) is calculated for all $r$ odd $\geq 3$. If $r$ is coprime to at least $n-2$ of $p_k$ (e.g. when $M$ is the Poincare homology sphere), it is proved that $(\sqrt {\dfrac{4}{r}}\sin \dfrac{\pi}{r})^{\nu}\tau^{'}_r(M)$ is an algebraic integer in the r-th cyclotomic field, where $\nu$ is the first Betti number of $M$. For the torus bundle obtained from trefoil knot with framing 0, i.e. X_{tref}(0)=X(-2/_{\f{1}},3/_{\f{1}},6/_{\f{1}}), \tau^{'}_r is obtained in a simple form if $3\mid\llap /r$, which shows in some sense that it is impossible to generalize Ohtsuki's invariant to 3-manifolds being not rational homology spheres.Comment: Late

Darboux transformation and positons of the inhomogeneous Hirota and the Maxwell-Bloch equation

In this paper, we derive Darboux transformation of the inhomogeneous Hirota and the Maxwell-Bloch(IH-MB) equations which is governed by femtosecond pulse propagation through inhomogeneous doped fibre. The determinant representation of Darboux transformation is used to derive soliton solutions, positon solutions of the IH-MB equations.Comment: accepted by SCIENCE CHINA Physics, Mechanics & Astronomy. arXiv admin note: substantial text overlap with arXiv:1205.119

An example concerning Ohtsuki's invariant and the full SO(3) quautum invariant

Two lens spaces are given to show that Ohtsuki's $\tau$ for rational homology spheres does not determine Kirby-Melvin's $\{\tau_r^{'}, r odd\geq3\}$Comment: Latex. To appear in Letters in Math. Physic

The singular set of triholomorphic maps into quartic K3 surface

We prove that any weakly triholomorphic map from a compact hyperk\"ahler surface to an algebraic K3 surface defined by a homogeneous polynomial of degree 4 in $\mathbb{C}P^3$ has only isolated singularities.Comment: 11 page

Synthesizing Filamentary Structured Images with GANs

This paper aims at synthesizing filamentary structured images such as retinal fundus images and neuronal images, as follows: Given a ground-truth, to generate multiple realistic looking phantoms. A ground-truth could be a binary segmentation map containing the filamentary structured morphology, while the synthesized output image is of the same size as the ground-truth and has similar visual appearance to what have been presented in the training set. Our approach is inspired by the recent progresses in generative adversarial nets (GANs) as well as image style transfer. In particular, it is dedicated to our problem context with the following properties: Rather than large-scale dataset, it works well in the presence of as few as 10 training examples, which is common in medical image analysis; It is capable of synthesizing diverse images from the same ground-truth; Last and importantly, the synthetic images produced by our approach are demonstrated to be useful in boosting image analysis performance. Empirical examination over various benchmarks of fundus and neuronal images demonstrate the advantages of the proposed approach

Virasoro symmetry of the constrained multi-component KP hierarchy and its integrable discretization

In this paper, we construct the Virasoro type additional symmetries of a kind of constrained multi-component KP hierarchy and give the Virasoro flow equation on eigenfunctions and adjoint eigenfunctions. It can also be seen that the algebraic structure of the Virasoro symmetry is kept after discretization from the constrained multi-component KP hierarchy to the discrete constrained multi-component KP hierarchy.Comment: 20 Pages, Theoretical and Mathematical Physics, 187(2016), 871-88

Regular solution and lattice miura transformation of bigraded Toda Hierarchy

In this paper, we give finite dimensional exponential solutions of the bigraded Toda Hierarchy(BTH). As an specific example of exponential solutions of the BTH, we consider a regular solution for the $(1,2)$-BTH with $3\times 3$-sized Lax matrix, and discuss some geometric structure of the solution from which the difference between $(1,2)$-BTH and original Toda hierarchy is shown. After this, we construct another kind of Lax representation of $(N,1)$-bigraded Toda hierarchy($(N,1)$-BTH) which does not use the fractional operator of Lax operator. Then we introduce lattice Miura transformation of $(N,1)$-BTH which leads to equations depending on one field, meanwhile we give some specific examples which contains Volterra lattice equation(an useful ecological competition model).Comment: Accepted by Chinese Annals of Mathematics, Series

The extended ZNZ_N-Toda hierarchy

The extended flow equations of a new $Z_N$-Toda hierarchy which takes values in a commutative subalgebra $Z_N$ of $gl(N,\mathbb C)$ is constructed. Meanwhile we give the Hirota bilinear equations and tau function of this new extended $Z_N$-Toda hierarchy(EZTH). Because of logarithm terms, some extended Vertex operators are constructed in generalized Hirota bilinear equations which might be useful in topological field theory and Gromov-Witten theory. Meanwhile the Darboux transformation and bi-hamiltonian structure of this hierarchy are given. From hamiltonian tau symmetry, we give another different tau function of this hierarchy with some unknown mysterious connections with the one defined from the point of Sato theory.Comment: 22 Pages, Theoretical and Mathematical Physics, 185(2015), 1614-163

Solutions of the (2+1)-dimensional KP, SK and KK equations generated by gauge transformations from non-zero seeds

By using gauge transformations, we manage to obtain new solutions of (2+1)-dimensional Kadomtsev-Petviashvili(KP), Kaup-Kuperschmidt(KK) and Sawada-Kotera(SK) equations from non-zero seeds. For each of the preceding equations, a Galilean type transformation between these solutions $u_2$ and the previously known solutions $u_2^{\prime}$ generated from zero seed is given. We present several explicit formulas of the single-soliton solutions for $u_2$ and $u_2^{\prime}$, and further point out the two main differences of them under the same value of parameters, i.e., height and location of peak line, which are demonstrated visibly in three figures.Comment: 18 pages, 6 figures, to appear in Journal of Nonlinear Mathematical Physic