832 research outputs found

### 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

### 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

### 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

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