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The relationships between quantum entangled states and braid matrices have been well studied in recent years. However, most of the results are based on qubits. In this paper, we investigate the applications of 2-qutrit entanglement in the braiding associated with Z 3 parafermion. The 2-qutrit entangled state | Ψ ( θ ) 〉 , generated by the action of the localized unitary solution R ˘ ( θ ) of YBE on 2-qutrit natural basis, achieves its maximal l 1 -norm and maximal von Neumann entropy simultaneously at θ = π / 3 . Meanwhile, at θ = π / 3 , the solutions of YBE reduces braid matrices, which implies the role of l 1 -norm and entropy plays in determining real physical quantities. On the other hand, we give a new realization of 4-anyon topological basis by qutrit entangled states, then the 9 × 9 localized braid representation in 4-qutrit tensor product space ( C 3 ) ⊗ 4 is reduced to Jones representation of braiding in the 4-anyon topological basis. Hence, we conclude that the entangled states are powerful tools in analysing the characteristics of braiding and R ˘ -matrix.

In this paper, we demonstrate that there are vortex beam solutions for the photon in the rotating medium. By constructing the photon wave function with Riemann–Silberstein vector, we derive the dynamic equation of the photon in moving medium from the Maxwell equations and the non-relativistic Minkowski relations. In case of the stationary state, the dynamic equation of the photon can be written as a Dirac-like equation, where the velocity of the medium plays the role of a vector potential. By giving the medium different forms of rotating velocity fields, we obtain different vortex beam solutions of the photon, such as the diffracting and non-diffracting Laguerre–Gaussian (LG) beam solutions via proper approximations. For the diffracting LG beam solution, we acquire a new term arising from the medium rotation that can change the Gouy phase, and then accordingly predict the rotation behavior of the photon interference pattern. In addition, the rotation of the medium can lead to the change of the relative intensity distribution of the interference pattern. Furthermore, our theory predicts the existence of the Landau levels of transverse photon energy in the nondiffracting LG beam solution.

We discuss a new type of solutions of the Yang-Baxter equation, called type-II solutions. They are related to quantum entanglements. The action of the corresponding braiding operator on the topological basis associated with a topological quantum field theory generates a (2J+1)-dimensional matrix form of the R-matrix for spin J, i.e., the Wigner function D with the spectral parameter θ denoting the entanglement degree. We present concrete examples for J = 1/2 and J = 1 in an explicit form. We show that the Hamiltonian related to the type-II R-matrix is Kitaev’s toy model.

This paper investigates the physical effects of Yang-Baxter equation (YBE) to quantum entanglements through the 3-body S-matrix in entangling parameter space. The explicit form of 3-body S-matrix $\breve{R}_{123}(\theta,\varphi)$ based on the 2-body S-matrices is given due to the factorization condition of YBE. The corresponding chain Hamiltonian has been obtained and diagonalized, also the Berry phase for 3-body system is given. It turns out that by choosing different spectral parameters the $\breve{R}(\theta,\varphi)$-matrix gives GHZ and W state respectively. The extended 1-D Kitaev toy model has been derived. Examples of the role of the model in entanglement transfer are discussed., Comment: 13 pages, 2 figures

In this paper some results associated with a new type of Yang---Baxter equation (YBE) are reviewed. The braiding matrix of Kauffman---Lomonaco has been extended to the solution (called type-II) of Yang---Baxter equation (YBE) and the related chain Hamiltonian is given. The Lorentz additivity for spectral parameters is found, rather than the Galilean rule for the familiar solutions (called type-I) of YBE associated with the usually exact solvable models. Based on the topological basis, the N-dimensional solution of YBE is found to be the Wigner D-functions. The explicit examples for spin-$$\frac{1}{2}$$12 and spin-1 have been shown. The extremes of $$\ell _1$$l1-norm of $$D$$D-functions are introduced to distinguish the type-I from type-II of braiding matrices that also correspond to those of von Neumann entropy for quantum information.

Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced $\ell_1$ -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigner's D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to $\mathbb{Z}_3$ parafermion model based on $SU(3)$ principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity $\frac{u+v}{1+uv}$ rather than Galilean rule $(u+v)$. Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for $\mathbb{Z}_3$ parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces $\{1,\frac{1}{2},\frac{1}{2}\}$., 20 pages, 1 figure; Conference paper for Group32 in Prague, 2018. arXiv admin note: text overlap with arXiv:1704.00252 by other authors

Since Dirac developed the theory of monopoles consistent with both quantum mechanics and the gauge invariance of the electromagnetic field in 1931, it led to many discussions about the generalized spherical harmonics for the wave function of charged particle around magnetic monopole. In recent years, a lot of studies about monopole are presented. Some analogues of magnetic monopole have been found in exotic spin ices and other systems. Several studies on magnetic monopole are also carried out in BECs and superfluid. Especially the Dirac monopole in the synthetic magnetic field not only has been produced theoretically by a spinor Bose-Einstein condensate, and but also its observation has been also reported. As it is well-known that the Dirac monopole is described by string which causes the discontinuities for generalized spherical function in depicting an electrical charge particle around magnetic monopole. To overcome the difficulty Wu and Yang presented the monopole harmonics to describe Dirac monopole without string based on the manifold with two cups. In our work, we notice that the fluid field and electromagnetic field have similar features, by identifying the Schrodinger equation for ψ(r, t) = □(r, t) = eiS(r,t) and v = (ħ/μ)∇S(r, t) with the hydrodynamic equations with the quantum pressure, the effective potential is introduced in Schrodinger equation. Due to the existence of the pressure in the hydrodynamic equations, it turns out that the effective potential exactly corresponds to the potential of Wu-Yang monopole for the pure gauge solution of the hydrodynamic equations. The incompressibility will limit the physical models, such as hard core and harmonic oscillator model for Bose-Einstein condensate (BEC). The results provide a new idea for the simulation of the Wu-Yang monopole.

We construct the 1D "Equation missing" parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the "Equation missing" parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the "Equation missing" parafermionic model is a direct generalization of 1D "Equation missing" Kitaev model. Both the "Equation missing" and "Equation missing" model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian "Equation missing" based on Yang-Baxter equation. Different from the Majorana doubling, the "Equation missing" holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ω-parity P"Equation missing" and emergent parafermionic operator Γ, which are the generalizations of parity PM and emergent Majorana operator in Lee-Wilczek model, respectively. Both the "Equation missing" parafermionic model and "Equation missing" can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.

We propose a scheme of acoustic spherical cloaking by means of background irrotational flow in compressible fluid. The background flow forms a virtual curved spacetime and guides the sound waves bypass the cloaked objects. To satisfy the laws of real fluid, we show that spatially distributed mass source and momentum source are necessary to supply. The propagation of sound waves in this system is studied via both geometric acoustics approximation and full wave approach. The analytic solution of sound fields is obtained for plane wave incidence. The results reveal the effect of phase retardation (or lead) in comparison with the ordinary transformation-acoustic cloak. In addition, the ability of cloaking is also evaluated for unideal background flows by analyzing the scattering cross section., Comment: 15 pages, 6 figures