Topological carbon allotropes: paradigm shift for materials innovation

Topology is a central concept of mathematics, which allows us to distinguish two isolated rings with linked ones. In material science, researchers discovered topologically different carbon allotropes in a form of a cage, a tube, and a sheet, which have unique translational and rotational symmetries, described by a crystallographic group theory, and the atoms are arranged at specific rigid positions in 3-dimensional ($D$) space. However, topological orders must be robust against deformations, so that we can make completely different families of topological materials. Here we propose various topological structures such as knots and links using covalent $\sigma$ bonds of carbon atoms, while allowing various topologically equivalent arrangements using weak $\pi$ bonds. By extending this idea, we invented a new 3D carbon allotrope, Hopfene, which has periodic arrays of Hopf-links to knit horizontal Graphene sheets into vertical ones without connecting by $\sigma$ bonds.Comment: 19 pages, 14 figure

Intrinsic optical gain of ultrathin silicon quantum wells from first-principles calculations

Optical gains of ultrathin Si(001) quantum wells are calculated from first principles, and found to be positive because of an intrinsic quantum confinement effect. The gain of the ultrathin silicon film is comparable to that of the bulk GaAs if the carrier density is large enough. The impact of surface structure of the silicon film on the efficiency of light emission is also investigated and we found that SiO2 crystal that forms a strainless connection with a Si(001) surface such as quartz enhances optical gain

Special Theory of Relativity for a Graded Index Fibre

The speed of light ($c$) in a vacuum is independent on a choice of frames to describe the propagation, according to the theory of relativity. We consider how light is characterised in a material, where the speed of light is different from that in a vacuum due to the finite dielectric constant. The phase velocity in a material is smaller than $c$, such that the speed of a moving frame can be larger than the phase velocity, such that the frame can move faster than the speed of light in a material. Consequently, an unusual Doppler effect is expected, and the wavelength in the moving frame changes from the red-shift to the blue-shift upon increasing the speed of the frame. The corresponding energy of the light also changes sign from positive to negative, while momentum is always positive, leading to the changes of sings for the phase velocity and the helicity. In a graded index fibre, where the exact solution is available, even more complicated phenomena are expected, due to the finite effective mass of photons. Upon the increase of the energy gap, generated by optical confinements and optical orbital angular momentum, the effective mass of photons increases. If the gap is large enough, momentum starts to change the sign upon increasing the frame velocity, while the energy of photons is always positive. In this case, the phase velocity diverges if momentum is in agreement with the fame velocity. Contrary to the unusual behaviours of the phase velocity, the group velocity is always below $c$. This thought-experiment might be useful to consider the insight for the polarisation sate of light

Representation Theory and Topology of Coherent Photons with Angular Momentum

Photons are elementary particles of lights, which have both spin and orbital angular momentum as internal degrees of freedom. Nature of spin is known as polarisation, which is widely used for sunglasses, liquid-crystal displays, digital-coherent communications, while orbital angular momentum is useful for optical tweezers, laser-patterning, and quantum optics. However, spin and orbital angular momentum of photons are considered to be impossible for splitting into two independent degrees of freedom in a proper gauge invariant way, proved by plane wave expansions in a free space. Here, we show these degrees of freedom are well-defined quantum observables in a waveguide and a free space as far as the propagation mode is sufficiently confined in the core. We found Stokes parameters are spin expectation values of coherent photons, which exhibit non-trivial topological features like a torus, a M\"obius strip, and a bosonic Dirac cone. We have applied an SU(N) representation theory to describe both spin and orbital angular momentum of photons, and experimentally demonstrated their controls over a full Poincar\'e sphere to show a fullerene C$_{60}$ and the earth by qubits. We have also ascribed topological colour charge to photonic orbital angular momentum, whose SU(3) states are shown on a proposed Gell-Mann hypersphere in SO(8), whose parameters could be embedded in SO(5). We have also realised photonic SU(4) states of singlet and triplet states, which were successfully projected into SU(2)$\times$SU(2) states by a rotated polariser. Our results indicate that our platform of manipulating spin and orbital angular momentum is useful for exploring a photonic quantum chromodynamics and a higher order macroscopic quantum state

Dirac Equation for Photons: Origin of Polarisation

Spin is a fundamental degree of freedom, whose existence was proven by Dirac for an electron by imposing the relativity to quantum mechanics, leading to the triumph to derive the Dirac equation. Spin of a photon should be linked to polarisation, however, the similar argument for an electron was not applicable to Maxwell equations, which are already Lorentz invariant. Therefore, the origin of polarisation and its relationship with spin are not completely elucidated, yet. Here, we discuss propagation of coherent rays of photons in a graded-index optical fibre, which can be solved exactly using the Laguerre-Gauss or Hermite-Gauss modes in a cylindrical or a Cartesian coordinate. We found that the energy spectrum is massive with the effective mass as a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional ($1D$) non-relativistic Schr\"odinger equation, which is equivalent to the $2D$ space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the $2D$ Dirac equation with spin. We applied the Bardeen-Cooper-Schrieffer (BCS)-Bogoliubov theory of superconductivity to a coherent ray from a laser and identified a radiative Nambu-Anderson-Higgs-Goldstone mode for recovering the broken symmetry. The spin expectation value of a photon corresponds to the polarisation state in the Poincar\'e sphere, which is characterised by fixed phases after the onset of lasing due to the broken $SU(2)$ symmetry, and it is shown that its azimuthal angle is coming from the phase of the energy gap

SU(2) Symmetry of Coherent Photons and Application to Poincar\'e Rotator

Lie algebra is a hidden mathematical structure behind various quantum systems realised in nature. Here, we consider $SU(2)$ wavefunctions for polarisation states of coherent photons emitted from a laser source, and discuss the relationship to spin expectation values with SO(3) symmetry based on isomorphism theorems. In particular, we found rotated half-wave-plates correspond to mirror reflections in the Poincar\'e sphere, which do not form a subgroup in the projected O(2) plane due to anti-hermitian property. This could be overcome experimentally by preparing another half-wave-plate to realise a pristine rotator in $SU(2)$, which allows arbitrary rotation angles determined by the physical rotation. By combining another 2 quarter-wave-plates, we could also construct a genuine phase-shifter, thus, realising passive control over the full Poincar\'e sphere