Spin of Photons: Nature of Polarisation

Stokes parameters (${\bf S}$) in Poincar\'e sphere are very useful values to describe the polarisation state of photons. However, the fundamental principle of the nature of polarisation is not completely understood, yet, because we have no concrete consensus how to describe spin of photons, quantum-mechanically. Here, we have considered a monochromatic coherent ray of photons, described by a many-body coherent state, and tried to establish a fundamental basis to describe the spin state of photons, in connection with a classical description based on Stokes parameters. We show that a spinor description of the coherent state is equivalent to Jones vector for polarisation states, and obtain the spin operators (${\bf \hat{S}}$) of all components based on rotators in a $SU(2)$ group theory. Polarisation controllers such as phase-shifters and rotators are also obtained as quantum-mechanical operators to change the phase of the wavefunction for polarisation states. We show that the Stokes parameters are quantum-mechanical average of the obtained spin operators, ${\bf S} = \langle {\bf \hat{S}} \rangle$

Quantum Commutation Relationship for Photonic Orbital Angular Momentum

Orbital Angular Momentum (OAM) of photons are already ubiquitously being used for numerous applications. However, there is a fundamental question whether photonic OAM operators satisfy standard quantum mechanical commutation relationship or not; this also poses a serious concern on the interpretation of an optical vortex as a fundamental quantum degree of freedom. Here, we examined canonical angular momentum operators defined in a cylindrical coordinate, and applied them to Laguerre-Gauss (LG) modes in a graded index (GRIN) fibre. We confirmed the validity of commutation relationship for the LG modes and found that ladder operators also work properly with the increment or decrement in units of the Dirac constant ($\hbar$). With those operators, we calculated the quantum-mechanical expectation value of the magnitude of angular momentum, which includes contributions from both intrinsic and extrinsic OAM. The obtained results suggest that OAM characterised by the LG modes exhibits a well-defined quantum degree of freedom

Spin and Orbital Angular Momentum of Coherent Photons in a Waveguide

Spin angular momentum of a photon corresponds to a polarisation degree of freedom of lights, and such that various polarisation properties are coming from macroscopic manifestation of quantum-mechanical properties of lights. An orbital degree of freedom of lights is also manipulated to form a vortex of lights with orbital angular momentum, which is also quantised. However, it is considered that spin and orbital angular momentum of a photon cannot be split from the total orbital angular momentum in a gauge-invariant way. Here, we revisit this issue for a coherent monochromatic ray from a laser source, propagating in a waveguide. We obtained the helical components of spin and orbital angular momentum by the correspondence with the classical Ponyting vector. By applying a standard quantum field theory using a coherent state, we obtained the gauge-independent expressions of spin and orbital angular momentum operators. During the derivations, it was essential to take a finite cross-sectional area into account, which leads the finite longitudinal component along the direction of the propagation, which allows the splitting. Therefore, the finite mode profile was responsible to justify the splitting, which was not possible as far as we are using plane-wave expansions in a standard theory of quantum-electrodynamics (QED). Our results suggest spin and orbital angular momentum are well-defined quantum-mechanical freedoms at least for coherent photons propagating in a waveguide and in a vacuum with a finite mode profile

Macroscopic Single-Qubit Operation for Coherent Photons

Polarisation is described by an $SU(2)$ wavefunction due to macroscopic coherence of photons emitted from a ubiquitous laser source, and thus, a laser pulse is expected to behave as a macroscopic quantum bit (qubit), i.e., a qubit realised by a macroscopic number of photons. Here, we show that an arbitrary single-qubit operation can be carried out for such a macroscopic qubit by employing optical modulators, together with standard optical plates, in a computer-controlled fibre-optic configuration. We named the device as a Poincar\'e rotator, which allows a dynamic control over a polarisation state by executing an arbitrary amount of rotations on the Poincar\'e sphere. The Poincar\'e rotator works as an arbitrary $SU(2)$ operator in a Lie group, by combining a $U(1)$ operation to change the phase and another $U(1)$ operation to change the amplitude of the wavefunction. We have realised various polarisation states, such as $4 \times 4=16$, $8 \times 8=64$, and $10 \times 10=100$ distinguishable states on the sphere. As a locus of the realised polarisation states on the sphere, we have successfully drawn the molecular structure of Buckminsterfullerene (C$_{60}$) and the coastline of the earth

Macroscopic Singlet, Triplet, and Colour-Charged States of Coherent Photons

A ray of photons, emitted from a laser source, is in a coherent state, where macroscopic number of photons are degenerate in the same quantum state. The coherent state has degrees of freedom for spin and orbital angular momentum, which allow an arbitrary superposition state among orthogonal states with varying their amplitudes and phases, described by a representation theory of Lie algebra and Lie group. Here, we experimentally demonstrate that we can construct generators of rotations for the quantum states of coherent photons, simply by combining widely available optical components, such as half- and quarter-wave plates and vortex lenses. We have found that a superposition state between vortexed and no-vortex states is characterised by the motion of the topological charge upon the rotation in the SU(3) states. We also realised singlet and triplet states by combining rays of photons with orthogonal polarisation states and vortexed states. This corresponds to realise an effective SU(4) state and we have confirmed the projection to an SU(2)$\times$SU(2) state upon passing through a polariser

Topological Polarisation States

Polarisation states are described by spin expectation values, known as Stokes parameters, whose trajectories in a rotationally symmetric system form a sphere named after Poincar\'e. Here, we show that the trajectories of broken rotational symmetric systems can exhibit distinct topological structures in polarisation states. We use a phase-shifter to form a polarisation circle (${\mathbb S}^1$), which interferes with the original input due to the phase change of the output state upon the rotation. By rotating the circle using a rotator, the trajectories become a polarisation torus (${\mathbb S}^1 \times {\mathbb S}^1$), which was experimentally confirmed in a simple set-up using passive optical components together with Mach-Zehnder interferometers. We also discuss about realisations of other topological features, such as M\"obius strip, Hopf-links, and topological Dirac bosons with a bulk-edge correspondence

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

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

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

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