A photon is an elementary particle and it must follow the principle of quantum mechanics [6–9]. A photon can be created in a laser source, or it can be annihilated in a detector. The creation and annihilation are described by operators a ̂ σ † and a ̂ σ , respectively, which satisfy the commutation relationships for Bose particles [9,16,41,42] as [ a ̂ σ , a ̂ σ ′ ] = 0 , [ a ̂ σ † , a ̂ σ ′ † ] = 0 , and [ a ̂ σ , a ̂ σ ′ † ] = δ σ , σ ′ , where σ stands for the polarisation states such as σ = H for horizontally polarised state and σ = V for vertically polarised state. δ _σ,σ′ is the Kronecker delta, which becomes 1 if the polarisation states of σ and σ′ coincide, and 0 if the polarisation states are orthogonal. We can also choose another orthogonal base such as linearly polarised states along diagonal (σ = D) and anti-diagonal (σ = A) directions, or left (σ = L) and right (σ = R) circularly-polarised states (Supplementary Material).

We are interested in a monochromatic coherent ray of photons propagating in a material such as a waveguide or an optical fibre or in a vacuum, emitted from a laser source [5], because lasers are ubiquitously available. The coherent states [16,41,42] are described as |α _H, α _V⟩ = |α _H⟩|α _V⟩, where. | α H 〉 = e − | α H | 2 2 e α H a ̂ H † | 0 〉 (1) | α V 〉 = e − | α V | 2 2 e α V a ̂ V † | 0 〉 , (2) for which we assign α H = N H = N cos ⁡ α , α V = N V e i δ = N sin ⁡ α e i δ with the average number of photons for each polarisation given by N _H and N _V, and the total number of photons in the system is N = N _H + N _V. The auxiliary angle of α is the angle to split the electric field between horizontal and vertical directions (Supplementary Material) and the relative phase of δ = δ _y − δ _x is the amount of the phase shift for the vertical direction (δ _y ), measured from that for the horizontal direction (δ _x ). The coherent states have important characteristics. a ̂ H | α H 〉 = α H | α H 〉 (3) a ̂ V | α V 〉 = α V | α V 〉 , (4) which mean these are eigenstates of annihilation operators.

We consider the following complex electric field operator, defined as, E ̂ z , t = 2 ℏ ω ϵ V e i β a ̂ H x ̂ + a ̂ V y ̂ , (5) where β = kz − ωt + δ _x , x ̂ and y ̂ are unit vectors along x and y directions, respectively, ϵ is the dielectric constant of the material, and V is the volume of the system. In a uniform material of the refractive index n, the dispersion is simply obtained as ω = vk = ck/n by solving the Maxwell equations (4) and (5). By applying E ̂ ( z , t ) to |α _H, α _V⟩ from the left, we obtain E ̂ z , t | α H , α V 〉 = E z , t | α H , α V 〉 , (6) which means the coherent state is an eigenstate of this operator and the operation did not change the state except for the factor, E ( z , t ) , which gives the complex amplitude of the electric field as E z , t = E x E y = E 0 e i β cos ⁡ α sin ⁡ α e i δ , (7) with the amplitude of E 0 = 2 ℏ ω N / ( ϵ V ) . Therefore, E ̂ ( z , t ) | α H , α V 〉 also works as a wavefunction to describe the coherent state of photons. Then, we recognise that E ( z , t ) is actually a spinor representation of the wavefunction, and it is indeed rewritten as E z , t = E 0 Ψ z , t | J o n e s 〉 , (8) where Ψ(z, t) = e ^iβ is the orbital part of the wavefunction, and |Jones⟩ it the Jones vector to describe the polarisation states (Supplementary Material). Therefore, the Jones vector is actually the wavefunction itself to describe the spin state of the coherent photons, quantum mechanically. It is interesting to note that the many-body coherent state is described simply by a single mode of Ψ(z, t) with the spin state as inherent internal degrees of freedom. This is coming from the nature of the Bose-Einstein condensation for the coherent laser beam, in which macroscopic number of photons are degenerate to occupy the single mode.

In a real material with a specific geometrical structure, patterned into a form of a waveguide or a fibre, we must solve the Helmholtz equation ∇ 2 Ψ r = μ 0 ϵ r ∂ 2 ∂ t 2 Ψ r , (9) because the dielectric constant has a profile in the form of ϵ = ϵ(r), due to the spacial distribution of material compositions. We are aware that this is very important to take into account for the more realistic considerations. By respecting the symmetry of the waveguide [43], we can also consider various forms of the orbital wavefunction, including the vortex nature of the beam with orbital angular momentum [4,5,12–17]. Here, we consider only the plane wave solution of the Helmholtz equation as Ψ(z, t) = e ^iβ , which makes a lot of serious problems for separating the spin and orbital parts of the angular momentum [11,13–15,18], as we shall see briefly below. Nevertheless, the plane wave solution makes calculations easy, such that it is still useful for a theoretical perspective.

We also note that E ̂ ( z , t ) is not observable, since all physical observables must be real. In order to observe the electric field, which is observable, we must define the real electric field operator given by E ̂ = 1 2 E ̂ + E ̂ † . (10)

If we take the quantum-mechanical average over the coherent state, we obtain. E z , t = 〈 E ̂ 〉 (11) = 〈 α H , α V | E ̂ | α H , α V 〉 (12) = E 0 cos ⁡ α ⁡ cos ⁡ β x ̂ + sin ⁡ α ⁡ cos β + δ y ̂ (13) = R E z , t , (14) which is indeed real. Please also note that the application of E ̂ to |α _H, α _V⟩ changes the original state, because the coherent state is not the eigenstate of the creation operator [16,41,42]. Therefore, it is essential to treat the electric field by using a complex value rather than real value, so that the use of a complex value is not a mere mathematical convention. The wavefunction is intrinsically described by a complex value for a photon, just like the other quantum-mechanical systems [6–9].

We can also obtain the quantum-mechanical average of the energy for the photons. U ̄ QM = V 2 E ̂ ⋅ D ̂ ̄ + B ̂ ⋅ H ̂ ̄ = ϵ V E ̂ ⋅ E ̂ ̄ (15) = ϵ V 4 E ̂ ⋅ E † ̂ ̄ + E † ̂ ⋅ E ̂ ̄ + E ̂ ⋅ E ̂ ̄ + E † ̂ ⋅ E † ̂ ̄ (16) = ϵ V 4 E ̂ ⋅ E † ̂ ̄ + E † ̂ ⋅ E ̂ ̄ (17) = ϵ V 4 2 ℏ ω ϵ V a ̂ H † a ̂ H + a ̂ H a ̂ H † + a ̂ V † a ̂ V + a ̂ V a ̂ V † (18) = ℏ ω a ̂ H † a ̂ H + 1 2 + a ̂ V † a ̂ V + 1 2 (19) = ℏ ω N H + 1 2 + N V + 1 2 (20) = ℏ ω N + 1 , (21) where the bar stands for the t average, and 1/2 is coming from the zero-point oscillations.

We will obtain various electro-magnetic field operators to describe a coherent ray of photons emitted from a laser. We use a Coulomb gauge, which satisfy ∇ ⋅ A ̂ = 0 , (22) where A ̂ is the vector potential operator. The magnetic induction operator, B ̂ , and E ̂ are obtained from A ̂ , as. B ̂ = ∇ × A ̂ = k × E ̂ ω (23) E ̂ = − ∂ t A ̂ , (24) respectively. Alternatively, we have already obtained E ̂ as, E ̂ = ℏ ω 2 ϵ V a ̂ H e i β + a ̂ H † e − i β x ̂ + a ̂ V e i β + a ̂ V † e − i β y ̂ , (25) we can obtain A ̂ , instead, as A ̂ = − i ω ℏ ω 2 ϵ V a ̂ H e i β − a ̂ H † e − i β x ̂ + a ̂ V e i β − a ̂ V † e − i β y ̂ . (26)

Consequently, we obtain B ̂ = 1 v ℏ ω 2 ϵ V a ̂ H e i β + a ̂ H † e − i β y ̂ − a ̂ V e i β + a ̂ V † e − i β x ̂ . (27)

Here, we assumed that the ray is described by a single mode, which is remarkably different from a standard description of the E ̂ and B ̂ in a Quantum Electro-Dynamics (QED) theory [6,9,16,41,42], for which the sum over all possible modes with different wavelengths are included. For the application of a coherent ray from lasers, the single mode are dominated over other modes [5]. We can easily extend the theory to include a few modes for describing propagation of multiple modes just by summing up these contributions based on a superposition principle. But, the main point here is the Bose-Einstein condensation character of the coherent ray, and we do not have to consider infinite number of electromagnetic fields in a vacuum. In that sense, our system, considering here for a standard optical laser lab, is remarkably different from situations in high energy physics, where the Lorentz invariance is inevitable [11,14,18]. In a waveguide, the spatial symmetry is broken a priori, such that the orbital is not uniform, reflecting the shape and the profile of the material compositions [43].

The momentum operator of the electro-magnetic field, p ̂ field , is given from the Poynting vector operator, S ̂ = E ̂ × H ̂ , with the magnetic field operator of H ̂ , as p ̂ field = ϵ E ̂ × B ̂ = ϵ μ 0 E ̂ × H ̂ = 1 v 2 S ̂ , (28) where μ ₀ is the permeability of a material, which is usually almost the same as that in a vacuum [5] for most of the optical materials, except for optical isolators. By integrating over V, we obtain P ̂ field = ℏ k z ̂ a ̂ H † a ̂ H + 1 2 + a ̂ V † a ̂ V + 1 2 , (29) for which we have used ℏ ω 2 ϵ V 1 v ℏ ω 2 ϵ V 1 v 2 μ 0 V = 1 v ℏ ω 2 1 v 2 ϵ μ 0 = ℏ k 2 (30) for the factor, x ̂ × x ̂ = y ̂ × y ̂ = 0 , x ̂ × y ̂ = − y ̂ × x ̂ = z ̂ , and for the boundary condition ∫ d z L e ± 2 i β = 0 , (31) for the length of L along z. The contribution of 1/2 for each polarisation states are coming from the zero-point fluctuations, which will cancel among contributions between modes propagating opposite directions (ℏk and −ℏk).

Then, it is natural to expect that the total angular momentum operator for the ray should be given by [4,5,11–18,44]. J ̂ z = ∫ d 3 r r × p ̂ field = ϵ ∫ d 3 r r × E ̂ × B ̂ . (32)

By using the identities. r × E ̂ × ∇ × A ̂ = E ̂ r ⋅ ∇ × A ̂ − r ⋅ E ̂ ∇ × A ̂ (33) r ⋅ ∇ × A ̂ = r i ϵ ijk ∂ j A ̂ k = ϵ ijk r i ∂ j A ̂ k = r × ∇ ⋅ A ̂ , (34)

J ̂ z = L ̂ z + S ̂ z is split into its orbital angular momentum part, L ̂ z = ϵ ∫ d 3 r E ̂ r × ∇ ⋅ A ̂ , (35) and the spin angular momentum part, S ̂ z = − ϵ ∫ d 3 r r ⋅ E ̂ ∇ × A ̂ . (36) Furthermore, using the identity [4,5], r ⋅ E ̂ ∇ × A ̂ i = r j E ̂ j ϵ ilm ∂ l A ̂ m = ϵ ilm r j E ̂ j ∂ l A ̂ m , (37) we obtain for the ith component, S ̂ i = − ϵ ∫ d 3 r r ⋅ E ̂ ∇ × A ̂ i (38) = − ϵ ∫ d 3 r ϵ ilm r j E ̂ j ∂ l A ̂ m (39) = − ϵ r j E ̂ j A ̂ m − ∞ ∞ + ϵ ∫ d 3 r ϵ ilm A ̂ m ∂ l r j E ̂ j , (40) whose first term vanishes [11] for the finite mode size in a waveguide. After the integration only i = z component survive, and we obtain. S ̂ z = ϵ ∫ d 3 r E ̂ × A ̂ (41) = − i ℏ z ̂ a ̂ H † a ̂ V − a ̂ V † a ̂ H , (42) for which we have used ϵ ℏ ω 2 ϵ V − i ω ℏ ω 2 ϵ V V = − i ℏ 2 . (43)

If we change the basis states for describing the polarisation states from horizontal/vertical linear polarised states to left/right circular polarised states by the transformations [5,21,22]. a ̂ L † = 1 2 a ̂ H † + i a ̂ V † (44) a ̂ R † = 1 2 a ̂ H † − i a ̂ V † , (45) and their conjugate. a ̂ L = 1 2 a ̂ H − i a ̂ V (46) a ̂ R = 1 2 a ̂ H + i a ̂ V , (47) S ̂ z = S ̂ z z ̂ is further simplified to be S ̂ z = ℏ a ̂ L † a ̂ L − a ̂ R † a ̂ R . (48)

We are aware that there are significant criticisms [4,5,11–19] on this derivation such as the intentional choice of the Coulomb gauge, the artificial choice of the boundary condition, the apparent gauge dependent expression in the form of E ̂ × A ̂ , the disappearance of the x and y components, and so on. We are not happy, either, and we will address some of these issues in a separate paper [20]. Nevertheless, we think the last expression is quite intuitive, and we could diagonalise the spin component in the chiral bases, which implies the spin is deeply linked to the polarisation. Moreover, this expression is not gauge dependent, since the number of photons should not depend on the choice of the gauge. Here, we accept this form as an expected expression derived from the correspondence from classical expectation, although we do not know why only z component appeared for spin operator [20]. In the next section, we will apply standard quantum-mechanical technique to consider the spin operators and their expectation values. We naturally obtained Stokes operators simply from the SU(2) consideration of the spin states, and established the expectation values of spin operators are Stokes parameters. Therefore, we show that the Poincaré sphere is essential the same as the Bloch sphere.

Here, we describe the polarisation state of photons by a chiral representation using left and right circular-polarised states, | L 〉 = | ↺ 〉 = 1 0 (49) | R 〉 = | ↻ 〉 = 0 1 . (50)

We will call this basis as the LR-basis. According to the result of the previous section, the spin operator along z direction can be written as. S ̂ z = ℏ a ̂ L † , a ̂ R † 1 0 0 − 1 a ̂ L a ̂ R (51) = ℏ ψ ̂ LR † σ 3 ψ ̂ LR , (52) where ψ ̂ LR † = ( a ̂ L † , a ̂ R † ) and ψ ̂ LR are the chiral representation of the creation and the annihilation operator, respectively, and the Pauli matrices are defined as σ 1 = 0 1 1 0 , σ 2 = 0 − i i 0 , σ 3 = 1 0 0 − 1 . (53)

Now, it is clear that the photon in the left-circular-polarised state has a spin of ℏ along the direction of propagation (σ = 1), and right-circular-polarised state has a spin of −ℏ along the same direction (σ = −1). Spin states pointing the other directions such as x and y would be realised by the superposition states of a ̂ L and a ̂ R in the chiral representation, because the 2 level systems are described by an SU(2) theory [6–9,20,45–47]. It is thus straightforward to expect the spin operators for x and y components as S ̂ x = ℏ a ̂ L † , a ̂ R † 0 1 1 0 a ̂ L a ̂ R (54) = ℏ ψ ̂ LR † σ 1 ψ ̂ LR , (55) S ̂ y = ℏ a ̂ L † , a ̂ R † 0 − i i 0 a ̂ L a ̂ R (56) = ℏ ψ ̂ LR † σ 2 ψ ̂ LR , (57) respectively

The general spin state, pointing to the (θ, ϕ) direction, is obtained by the Bloch state [6–9]. | B l o c h 〉 = | θ , ϕ 〉 (58) = e − i ϕ 2 ⁡ cos θ 2 e + i ϕ 2 ⁡ sin θ 2 , (59) where θ is the polar angle and ϕ is the azimuthal angle (Figure 1).

Thus, the corresponding coherent state with the spin state (θ, ϕ) is obtained as |α _L α _R⟩ = |α _L⟩|α _R⟩, where. | α L 〉 = e − | α L | 2 2 e α L a ̂ L † | 0 〉 (60) | α R 〉 = e − | α R | 2 2 e α R a ̂ R † | 0 〉 , (61) for which we assign α L = N e − i ϕ 2 ⁡ cos θ 2 , α R = N e + i ϕ 2 ⁡ sin θ 2 . The complex electric field operator in the chiral representation is given by E ̂ z , t = 2 ℏ ω ϵ V e i β a ̂ L l ̂ + a ̂ R r ̂ , (62) where l ̂ = ( x ̂ + i y ̂ ) / 2 and r ̂ = ( x ̂ − i y ̂ ) / 2 are complex unit vectors to describe directions of left and right polarisation states with phases. By applying this to the coherent state, we obtain the complex wavefunction in the chiral representation as E z , t = E L E R (63) = E 0 e i β e − i ϕ 2 ⁡ cos θ 2 e + i ϕ 2 ⁡ sin θ 2 (64) = E 0 Ψ z , t | B l o c h 〉 . (65)

By calculating the expectation values of spin components by the coherent state, we obtain 〈 S ̂ 〉 = 〈 S ̂ x 〉 〈 S ̂ y 〉 〈 S ̂ z 〉 (66) = ℏ N sin ⁡ θ ⁡ cos ⁡ ϕ sin ⁡ θ ⁡ sin ⁡ ϕ cos ⁡ θ . (67)

By realising the correspondences between angles, θ = π 2 − 2 χ (68) ϕ = 2 Ψ . (69)

We realised ⟨ S ̂ ⟩ = ℏ N cos 2 χ cos 2 Ψ cos 2 χ sin 2 Ψ sin 2 χ (70) = ℏ N S 1 S 2 S 3 , (71) showing that the expectation values of the spin operators are essentially equivalent to the Stokes parameters. Thus, the Poincaré sphere is equivalent to the Bloch sphere.

It is also useful to define the spin operator to represent the magnitude of the spin, S ̂ 0 = ℏ a ̂ L † a ̂ L + a ̂ R † a ̂ R , (72) such that its expectation value is ⟨ S ̂ 0 ⟩ = ℏ N . (73)

This actually shows the order parameter of the coherent states. The onset of lasing is similar to the second order phase transition, which show the continuous increase of the macroscopic order parameter upon changing the control parameter such as temperature [48–55]. In the case for lasing, the control parameter is the pumping power, provided by injecting electrons and holes for a laser diode, or by optical populating of electrons to higher energy levels to realise a population inversion state. Above the lasing threshold, the macroscopic number of photons are degenerate to occupy the single mode, such that ⟨ S ̂ 0 ⟩ can posses a non-zero value, and ⟨ S ̂ 0 ⟩ increases gradually upon the increase of the pumping power.

The theory of the order parameter description of the phase transition was first developed for the theory of superconductivity, as the Ginzburg–Landau theory [48–55], for which the order parameter was the energy gap, |Δ|, and the U (1) gauge symmetry of the phase (e ^iϕ ) was broken. Therefore, the order parameter is described by a scalar.

In the case of lasing, two phases of the wave, such as (θ, ϕ) in chiral representation and (α, δ) in Jones representation, are fixed, and the order parameters are described by a vector, not by a scalar. This is why 3D vectorial representation using the Poincaré sphere is so useful [5,21,22].

The 3D description of the order parameter similar to the Poincaré sphere is not restricted to the photonic systems, and actually they are ubiquitously available for describing various order parameters. For example, magnetic Heisenberg model was used to describe the superfluid-solid phase transition for a liquid He [56]. Another example is the SO(5) (special orthogonal) theory, which was developed for describing antiferromagnetic-superconducting phase transition for high-critical-temperature superconducting cuprates [55,57].

These spin operators are previously known as Stokes operators [29–33,38,58,59], and their commutation relationships are S ̂ x , S ̂ y = 2 i ℏ S ̂ z , S ̂ y , S ̂ z = 2 i ℏ S ̂ x , S ̂ z , S ̂ x = 2 i ℏ S ̂ y , (74) which are directly obtained by the commutation relationships of a ̂ σ † and a ̂ σ . The factor of 2 is unusual, because we have just 2 degrees of freedom, regardless of the spin 1 nature of a photon, which normally allow 3 states (1, 0, −1) along the principle quantisation axis [6–9]. This restriction is coming from the transverse nature of the ray of photons.

We also obtain S ̂ 0 , S ̂ x = S ̂ 0 , S ̂ y = S ̂ 0 , S ̂ z = 0 , (75) which are commutable, such that the magnitude can be a simultaneous eigenstate with the spin vector, S ̂ = S ̂ x , S ̂ y , S ̂ z .

It is also intuitive to evaluate the quantum fluctuations [31–33] of the spin of photons, by calculating S ̂ ⋅ S ̂ / ℏ 2 = n ̂ L + n ̂ R n ̂ L + n ̂ R + 2 (76) = n ̂ n ̂ + 2 , (77) where n ̂ = n ̂ L + n ̂ R is the total number operator, and then we obtain the expectation value of the quantum-mechanical fluctuation as [31–33]. δ S = ⟨ S ̂ ⋅ S ̂ ⟩ − ⟨ S ̂ 0 ⟩ 2 ⟨ S ̂ 0 ⟩ 2 (78) = 2 N , (79) Which means that the quantum-mechanical fluctuation decreases significantly upon increasing the order parameter. This is quite typical among other macroscopically quantum ordered systems [48,52–54].

Here, we will develop a similar description of spin of a photon, using Jones vector representation (Supplementary Material). Our starting point is S ̂ z = − i ℏ a ̂ H † a ̂ V − a ̂ V † a ̂ H , (80) = ℏ a ̂ H † , a ̂ V † 0 − i i 0 a ̂ H a ̂ V (81) = ℏ ψ ̂ HV † σ 2 ψ ̂ HV , (82) where ψ ̂ HV † = ( a ̂ H † , a ̂ V † ) and ψ ̂ HV are the creation and annihilation operators in Jones vector representation. We also call this basis as the HV-basis. In the HV-basis, it is expected that the spin state is diagonalised along the horizontal and vertical directions, such that we can expect the spin operator along x as S ̂ x = ℏ ψ ̂ HV † σ 3 ψ ̂ HV , (83) and consequently, S ̂ y = ℏ ψ ̂ HV † σ 1 ψ ̂ HV , (84) for the y component, describing diagonal/anti-diagonal linear polarisation.

By taking the quantum-mechanical average over the coherent state, |α _H, α _V⟩, we obtain ⟨ S ̂ ⟩ = ℏ N cos 2 α sin 2 α cos ⁡ δ sin 2 α sin ⁡ δ , (85) which is shown on a Poincaré sphere of Figure 2.

The expectation value must be independent on the choice of the fundamental basis. By comparing ⟨ S ̂ ⟩ , obtained for both LR- and HV-bases, we obtain the identities, tan 2 Ψ = tan 2 α cos ⁡ δ (86) sin 2 χ = sin 2 α sin ⁡ δ , (87) for the transformations of angles. These are exactly the same ones as those obtained classically, by rotating the horizontal axis to the principal axis of the polarisation ellipse (Supplementary Material). Therefore, the rotation of the axes in the real space to change from (α, δ) to (χ, Ψ) is equivalent to transforming from Jones vector representation to chiral representation. The comparison between chiral and Jones representations are summarised in Table 1 for Poincaré sphere.

The reason why the factor of 2 appeared in front of angles such as 2Ψ, 2χ, and 2α, is the quantum-mechanical average. By taking the complex conjugate and applying it to the original phase factor, we obtain this factor of 2, compared with the actual angle in the real space for E . This difference could be very important similar to geometrical Pancharatnam-Berry’s phase [61, 62], since the adiabatic rotation in Bloch/Poincaré sphere would not change the expectation value, but nevertheless, it can change the sign of the electric field, which leads the non-trivial interference [60].

We can also consider another representation, using diagonal |D⟩ and anti-diagonal |A⟩ basis states. In this basis, we will diagonalise the spin operator along y, and we obtain S ̂ x = ℏ ψ ̂ DA † σ 2 ψ ̂ DA (88) S ̂ y = ℏ ψ ̂ DA † σ 3 ψ ̂ DA (89) S ̂ z = ℏ ψ ̂ DA † σ 1 ψ ̂ DA , (90) where ψ ̂ DA † = ( a ̂ D † , a ̂ A † ) and ψ ̂ DA are the creation and the annihilation operator in the diagonal representation. For this DA-representation, the coherent state and the average of the spin operators are best described by the polar angle θ′ measured from the S ₂ axis and the azimuthal angle ϕ′ measured from the S ₃ axis (Figure 3). The expectation value is given by ⟨ S ̂ ⟩ = ℏ N 1 sin θ ′ ⁡ sin ϕ ′ cos θ ′ sin θ ′ ⁡ cos ϕ ′ . (91)

As far as we are aware, this representation is barely used.

Now, we realise the Jones vector treatments of the polarisation states are fully consistent with the quantum-mechanical treatment in SU(2). Here, we will double check this equivalence by transforming the Jones vector state to the corresponding representation in chiral state, which is made by the unitary transformation | H 〉 | V 〉 = e i γ 2 1 1 − i i | L 〉 | R 〉 , (92) where γ is the uncertainty of the global U (1) phase.

The original Jones vector is prepared as. | α , δ 〉 = e i β cos ⁡ α sin ⁡ α e i δ (93) = e i β ⁡ cos ⁡ α | H 〉 + e i β ⁡ sin ⁡ α e i δ | V 〉 , (94) and after the unitary transformation, we obtain in the form of | α , δ 〉 = C L | L 〉 + C R | R 〉 = | θ , ϕ 〉 . (95)

Therefore, we need to determine C _L and C _R, which are. C L = e i β + γ cos ⁡ α − i e i δ ⁡ sin ⁡ α 2 (96) C R = e i β + γ cos ⁡ α + i e i δ ⁡ sin ⁡ α 2 . (97)

We still need to express these as a function of (θ, ϕ).

To this aid, we assume the expectation values of ⟨ S ̂ ⟩ are independent on the choice of the basis states.

The ratio of the coefficient becomes C R C L = tan θ 2 e i ϕ . (98)

In addition, we can confirm that the wavefunction is normalised | C L | 2 + | C R | 2 = 1 . (99)

These equations for 2 complex values of C _L and C _R correspond to 3 equations for real values. Therefore, we cannot determine the global phase degree of freedom, e ^iγ .

Assuming C L = l ∈ R , we obtain C R = l ⁡ tan θ / 2 e i ϕ . Inserting this into the normalisation condition, we obtain l = ± cos θ / 2 . Thus, we obtain 2 states, C L C R = ± cos θ 2 e i ϕ ⁡ sin θ 2 , (100) which yield the same expectation value but the overall sign is opposite each other. This phase is different from the global phase, and this is coming from the two-fold coverage of SU(2) as SU(2)/SO(3) ≈ Z₂ = {−1, 1}, according to the mathematical theory for isomorphism [64, 65, 66? ]. We know that the wavefunction of the polarisation state is the spinor representation of the complex electric field. Therefore, the factor of −1 means that the change between ( E x , E y ) and ( − E x , − E y ) , which cannot change the polarisation state, but the phase is observable in the interference experiments [60–62]. We can express these states together, by shifting the global phase, while keeping the relative phase, as C L C R = e − i ϕ / 2 ⁡ cos θ 2 e + i ϕ / 2 ⁡ sin θ 2 , (101) which is indeed the Bloch vector. Here, we should consider the range of ϕ should be (0, 4π) to account for the change of the sign. Thus, the Jones vector is equivalent to the Bloch vector.

It is less obvious of this sign change in the above-defined Jones vector, if we describe the phase dependence as e ^iδ . This could be improved by shifting the global phase with the amount of e ^iδ/2, and then Jones vector can be rewritten as | J o n e s 〉 = e i β e − i δ / 2 ⁡ cos γ / 2 e i δ / 2 ⁡ sin γ / 2 , (102) where γ = 2α is the azimuthal angle measured from S ₁ on the Poincaré sphere (Figure 2). In this spinor representation, it is clear that the state will change the sign after 1 rotation on the Poincaré sphere, irrespective to whether the adiabatic rotation is the azimuthal rotation along the equator or the polar rotation along the meridian. This is exactly the same form of the Bloch state in chiral state, such that the change of basis from LR to HV simply corresponds to change from (θ, ϕ) to (γ, δ) in polar coordinates. This corresponds to the cyclic exchange of Pauli matrices from (σ ₁, σ ₂, σ ₃) to (σ ₃, σ ₁, σ ₂) (Table 2).

Now, we understand the spin state of a photon is described by an SU(2) group theory [6–9]. By using a standard Lie algebra, using Pauli matrices, we can obtain the many-body spin operators for the coherent monochromatic ray for photons, in a more elegant way. The general rotation operator [6–9] along the direction n ̂ with the amount of δϕ is defined as D ̂ n ̂ , δ ϕ = exp − i σ ⋅ n ̂ δ ϕ 2 , (103) where | n ̂ ̂ | = 1 and σ = (σ ₁, σ ₂, σ ₃). We have chosen the direction of rotation in a standard mathematical way. Specifically, the positive rotation along z is equivalent to the left-hand rotation (anti-clock-wise) rotation in xy-plane, seen from the top of the z-axis, which is equivalent to see from the observer in the detector side. By expanding the exponential and using the formulas, σ i , σ j = 2 δ i j 1 , and σ i , σ j = 2 i ϵ ijk σ k , we obtain [6–9]. D ̂ n ̂ , δ ϕ = 1 ⁡ cos δ ϕ 2 − i σ ⋅ n ̂ sin δ ϕ 2 . (104)

In particular, we describe the rotation around x, y, and z, axes as D ̂ 1 ( δ ϕ ) = D ̂ x ( δ ϕ ) = D ̂ ( x ̂ , δ ϕ ) , D ̂ 2 ( δ ϕ ) = D ̂ y ( δ ϕ ) = D ̂ ( y ̂ , δ ϕ ) , and D ̂ 3 ( δ ϕ ) = D ̂ z ( δ ϕ ) = D ̂ ( z ̂ , δ ϕ ) , for simplicity.

Previously, as outlined above, the spin operator along the direction of propagation ( S ̂ z ) was obtained by using the Poynting vector and considerations of angular momentum [4,5,11–18]. Then, we can obtain the spin operator along x, S ̂ x , by rotating S ̂ z with the amount of π/2 along y, and therefore, S ̂ x = D ̂ y π 2 S ̂ z D ̂ y † π 2 (105) = D ̂ y π 2 ψ LR † ℏ σ 3 ψ LR D ̂ y † π 2 (106) = ψ LR † D ̂ y π 2 ℏ σ 3 D ̂ y † π 2 ψ LR (107) = ℏ ψ LR † σ 1 ψ LR . (108)

Similarly, we obtain S ̂ y from S ̂ z by rotating along x with the amount of −π/2 as S ̂ y = D ̂ x − π 2 S ̂ z D ̂ x † − π 2 (109) = D ̂ x − π 2 ψ LR † ℏ σ 3 ψ LR D ̂ x † − π 2 (110) = ψ LR † D ̂ x − π 2 ℏ σ 3 D ̂ x † − π 2 ψ LR (111) = ℏ ψ LR † σ 2 ψ LR . (112)

Alternatively, we can also rotate 2π/3 along ( 1,1,1 ) / 3 direction, for cyclic permutation of axes. The rotation operator becomes D ̂ 1,1,1 3 , 2 π 3 = 1 2 − i 2 σ 1 + σ 2 + σ 3 , (113) which yields D ̂ 1,1,1 3 , 2 π 3 σ 3 D ̂ 1,1,1 3 , 2 π 3 † = σ 1 . (114)

Therefore, we successfully obtain S ̂ x , and the opposite rotation yield S ̂ y .

For the expressions using the HV-basis, we can use a unitary transformation a ̂ H † a ̂ V † = 1 2 1 1 − i i a ̂ L † a ̂ R † , (115) and its conjugate a ̂ H a ̂ V = 1 2 1 1 i − i a ̂ L a ̂ R . (116)

We can, of course, come back to LR-basis from HV-basis by the inverse unitary transformation. The transfer to the DA-basis is also straightforward by using the unitary transformation a ̂ D † a ̂ A † = 1 2 1 1 1 − 1 a ̂ H † a ̂ V † , (117) and its conjugate a ̂ D a ̂ A = 1 2 1 1 1 − 1 a ̂ H a ̂ V . (118)

The summary of the assignments of Pauli matrices to spin operator components for each representation is given by Table 2.

Here, we consider the rotation in real space rather than Hilbert space for spin. We define the rotation operators for the amount of the rotation of δϕ along x, y, and z-axes as R x ( δ ϕ ) , R y ( δ ϕ ) , R z ( δ ϕ ) , respectively. These are rotations in a SO(3) (Special Orthogonal) group theory. By applying 2 successive rotations along y and x, R x π 2 R y π 2 x ̂ = y ̂ (119) R x π 2 R y π 2 y ̂ = z ̂ (120) R x π 2 R y π 2 z ̂ = x ̂ . (121) We can perform cyclic exchange of axes from (x, y, z) to (y, z, x) as R x π 2 R y π 2 = 0 0 1 1 0 0 0 1 0 . (122)