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

Unfolding of Orbifold LG B-Models: A Case Study

In this note we explore the variation of Hodge structures associated to the orbifold Landau-Ginzburg B-model whose superpotential has two variables. We extend the Getzler-Gauss-Manin connection to Hochschild chains twisted by group action. As an application, we provide explicit computations for the Getzler-Gauss-Manin connection on the universal (noncommutative) unfolding of $\mathbb{Z}_2$-orbifolding of A-type singularities. The result verifies an example of deformed version of Mckay correspondence.Comment: 19 page

Symplectic genus, minimal genus and diffeomorphisms

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least -1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least -1 in any 4-manifold admitting a symplectic structure.Comment: 28 page

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

Color degree and color neighborhood union conditions for long heterochromatic paths in edge-colored graphs

Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let $d^c(v)$ denote the color degree and $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. In a previous paper, we showed that if $d^c(v)\geq k$ (color degree condition) for every vertex $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil\frac{k+1}{2}\rceil$, and if $|CN(u)\cup CN(v)|\geq s$ (color neighborhood union condition) for every pair of vertices $u$ and $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil\frac{s}{3}\rceil+1$. Later, in another paper we first showed that if $k\leq 7$, $G$ has a heterochromatic path of length at least $k-1$, and then, based on this we use induction on $k$ and showed that if $k\geq 8$, then $G$ has a heterochromatic path of length at least $\lceil\frac{3k}{5}\rceil+1$. In the present paper, by using a simpler approach we further improve the result by showing that if $k\geq 8$, $G$ has a heterochromatic path of length at least $\lceil\frac{2k}{3}\rceil+1$, which confirms a conjecture by Saito. We also improve a previous result by showing that under the color neighborhood union condition, $G$ has a heterochromatic path of length at least $\lfloor\frac{2s+4}{5}\rfloor$.Comment: 12 page

Ring frustration and factorizable correlation functions of critical spin rings

Basing on the exactly solvable prototypical model, the critical transverse Ising ring with or without ring frustration, we establish the concept of nonlocality in a many-body system in the thermodynamic limit by defining the nonlocal factors embedded in its factorizable correlation functions. In the context of nonlocality, the valuable traditional finite-size scaling analysis is reappraised. The factorizable correlation functions of the isotropic $XY$ and the spin-1/2 Heisenberg models are also demonstrated with the emphasis on the effect of ring frustration.Comment: 15 pages, 4 figure

Long rainbow path in properly edge-colored complete graphs

Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the edges incident to $v$, and denote it to be $d^c(v)$. It was shown that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a rainbow path of length at least $\min\{\lceil\frac{2k+1}{3}\rceil,k-1\}$. In the present paper, we consider the properly edge-colored complete graph $K_n$ only and improve the lower bound of the length of the longest rainbow path by showing that if $n\geq 20$, there must have a rainbow path of length no less than $\displaystyle \frac{3}{4}n-\frac{1}{4}\sqrt{\frac{n}{2}-\frac{39}{11}}-\frac{11}{16}$.Comment: 12 page

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

Quantum Torus symmetry of the KP, KdV and BKP hierarchies

In this paper, we construct the quantum Torus symmetry of the KP hierarchy and further derive the quantum torus constraint on the tau function of the KP hierarchy. That means we give a nice representation of the quantum Torus Lie algebra in the KP system by acting on its tau function. Comparing to the $W_{\infty}$ symmetry, this quantum Torus symmetry has a nice algebraic structure with double indices. Further by reduction, we also construct the quantum Torus symmetries of the KdV and BKP hierarchies and further derive the quantum Torus constraints on their tau functions. These quantum Torus constraints might have applications in the quantum field theory, supersymmetric gauge theory and so on.Comment: published in Lett. Math. Phys. online ahead of print 15 August 201

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