We consider three orthogonal quantum states, namely, left- and right-twisted states and the no-vortex state, which are described in the Hilbert space with three complex numbers, C 3 . We allow arbitrary mixing of these three states, realized by the superposition principle, and the wavefunction could be considered to be normalized to 1 or to the fixed number of photons, N, such that the radius of the complex sphere is fixed. Consequently, the number of freedom is 2 × 3–1 = 5, and the Hilbert space is equivalent to the sphere of five dimensions, S ⁵. In order to describe arbitrary rotational operations of the wavefunction in the Hilbert space, we need complex matrices of 3 × 3 for the SU(3) Lie group, which is realized by the exponential mapping form of the s u ( 3 ) Lie algebra. The SU(3) forms a group, whose determinant must be unity, which corresponds to the traceless condition for the Lie algebra. Therefore, we need 3 × 3–1 = 8 bases, defined by Gell-Mann [4, 5, 10, 28–31] as λ ̂ 1 = 0 1 0 1 0 0 0 0 0 , (1) λ ̂ 2 = 0 − i 0 i 0 0 0 0 0 , (2) λ ̂ 3 = 1 0 0 0 − 1 0 0 0 0 , (3) λ ̂ 4 = 0 0 1 0 0 0 1 0 0 , (4) λ ̂ 5 = 0 0 − i 0 0 0 i 0 0 , (5) λ ̂ 6 = 0 0 0 0 0 1 0 1 0 , (6) λ ̂ 7 = 0 0 0 0 0 − i 0 i 0 , (7) λ ̂ 8 = 1 3 1 0 0 0 1 0 0 0 − 2 , (8) which are all Hermite matrices, λ ̂ i † = λ ̂ i (i = 1, …, 8), implying that their expectation values must be real and observable. We can also use the bases, e ̂ i = λ ̂ i / 2 , reflecting the underlying s u ( 2 ) symmetry between two orthogonal states. The bases satisfy the normalization relationship for the trace, Tr λ ̂ i ⋅ λ ̂ j = 2 δ i j , where δ _ij is the Kronecker delta function.

The commutation relationship is obtained by the straightforward calculation of basis matrices, which is derived as: λ ̂ i , λ ̂ j = 2 i ∑ k C ijk λ ̂ k , (9) where the structure constants, C _ijk , are listed in Table 1. C _ijk is an asymmetric tensor, such that odd permutation of indices changes its sign. Most commutation relationships involve only one term in the summation on the right-hand side of the equation, similar to the s u ( 2 ) commutation relationship for spin. On the other hand, we must account for two terms involved in equations λ ̂ 4 , λ ̂ 5 = 2 i 1 2 λ ̂ 3 + 3 2 λ ̂ 8 , (10) λ ̂ 6 , λ ̂ 7 = 2 i − 1 2 λ ̂ 3 + 3 2 λ ̂ 8 . (11)

We also confirm that there are two mutually commutable operators, λ ̂ 1 , λ ̂ 8 = λ ̂ 2 , λ ̂ 8 = λ ̂ 3 , λ ̂ 8 = 0 (12) while we see λ ̂ 1 , λ ̂ 2 ≠ 0 , λ ̂ 1 , λ ̂ 3 ≠ 0 , λ ̂ 2 , λ ̂ 3 ≠ 0 . (13)

Therefore, the rank-2 character of the s u ( 3 ) Lie algebra is confirmed. It is also evident that λ ̂ 3 and λ ̂ 8 are already diagonalized in our representation for the basis.

We consider three orthogonal states for the s u ( 3 ) Lie algebra, and we select two states among three available states. There are three ways to choose two states, and each of the pairs of states will form the s u ( 2 ) Lie algebra.

For example, if we choose the first and second states, corresponding to the left- and right-twisted states, respectively, we use bases e ̂ 1 t = 1 2 λ ̂ 1 , (14) e ̂ 2 t = 1 2 λ ̂ 2 , (15) e ̂ 3 t = 1 2 λ ̂ 3 , (16) for describing the SU(2) states since they are equivalent to Pauli matrices, σ ̂ 1 = 0 1 1 0 , (17) σ ̂ 2 = 0 − i i 0 , (18) σ ̂ 3 = 1 0 0 − 1 , (19) if we neglect the third quantum state for the no-vortex state. These operators, e ̂ i ( t ) , were originally used for describing isospin for quarks [5]. For our applications, they will be useful to describe the rotation between the left- and right-twisted states. The rotation corresponds to mixing the left- and right-twisted states, which are described in the Poincaré sphere for vortices.

The introduction of the coupling between the left- and right-twisted states corresponds to allowing the direct SU(2) coupling between states with ℓ = −1 and ℓ = +1, as shown in Figure 2B. In standard quantum mechanics for orbital angular momentum [5, 8, 31], this coupling is not considered since it changes to the topological charge of ± 2, such that the ladder operations of SU(2) cannot directly transfer states to each other. On the other hand, we allow this coupling for photons simply by mixing states with certain amplitudes and phases [23, 25, 35, 47, 48, 50, 51, 70], which allows us to extend photonic states to obtain an SU(3) symmetry.

Another pair of states is made of the right-twisted state and the no-vortex state, whose bases are e ̂ 1 u = 1 2 λ ̂ 6 , (20) e ̂ 2 u = 1 2 λ ̂ 7 , (21) e ̂ 3 u = 1 2 − 1 2 λ ̂ 3 + 3 2 λ ̂ 8 , (22) = 1 2 0 0 0 0 1 0 0 0 − 1 . (23)

Here, it is noteworthy that we can define a new vector operator of e ̂ 3 ( u ) , for example, from λ ̂ 3 and λ ̂ 8 , since they are basis vector operators in the s u ( 3 ) Lie algebra, which forms a vector space. We cannot simply add components in the SU(3) Lie group since the SU(3) Lie group is not a vector space. We observe that e ̂ 3 ( u ) is normalized to be similar to e ̂ 3 ( t ) , such that it is useful to consider SU(2) rotations by e ̂ 1 ( u ) , e ̂ 2 ( u ) , and e ̂ 3 ( u ) .

Similarly, we consider the pair of states consisting of left-vortex state and the no-vortex state, whose bases are e ̂ 1 v = 1 2 λ ̂ 4 , (24) e ̂ 2 v = 1 2 λ ̂ 5 , (25) e ̂ 3 v = 1 2 1 2 λ ̂ 3 + 3 2 λ ̂ 8 , (26) = 1 2 1 0 0 0 0 0 0 0 − 1 . (27)

These s u ( 2 ) commutation relationships are summarized as e ̂ 1 x , e ̂ 2 x = i e ̂ 3 x , (28) where x = t, u, or v. We used lower-case letters (x = t, u, or v) for operators describing a single quanta such as a quark or a photon, and we used upper-case letters for coherent states of photons (X = T, U, or V) under Bose–Einstein condensation, where a macroscopic number, N, of photons are occupying the same state, as discussed further in this paper.

The s u ( 3 ) algebra does not contain a non-trivial invariant group. For example, we see that e ̂ 1 t , e ̂ 4 = − 1 4 λ ̂ 1 , λ ̂ 4 = − 1 4 i λ ̂ 7 = − 1 2 e ̂ 7 , (29) such that the internal SU(2) groups are connected by commutation relationships, and the s u ( 2 ) Lie algebra is not closed.

We utilize the Cartan–Dynkin formulation [5] for describing the SU(3) states. In the formalism, we consider to fix the quantization axis, rather than isotropic to all directions in the s u ( 2 ) Lie algebra, and consider ladder operators for increasing and decreasing the quantum number along the quantization axis [5, 9]. More specifically, we define t ̂ ± = 1 2 λ ̂ 1 ± i λ ̂ 2 , (30) t ̂ 3 = 1 2 λ ̂ 3 , (31) where t ̂ 3 stands for the operator of the z component of the rotationally symmetric s u ( 2 ) operator t ̂ = ( t ̂ 1 , t ̂ 2 , t ̂ 3 ) and t ̂ + and t ̂ − represent the rising and the lowering operators, respectively, to increase and decrease the quantum number for t ̂ 3 . We use t ̂ 3 and t ̂ ± instead of e ̂ i ( t ) (i = 1, 2, 3), and their commutation relationships become t ̂ 3 , t ̂ ± = ± t ̂ ± , (32) t ̂ + , t ̂ − = 2 t ̂ 3 . (33)

For applications in isospin, t ̂ 3 provides the fixed isospin value (t ₃) for each elementary particle, such as a proton (t ₃ = 1/2) and a neutron (t ₃ = −1/2), a deuterium (D, t ₃ = 0), and a tritium (T, t ₃ = 1/2). In elementary particle physics, a superposition state between a proton and a neutron, for example, is not realized due to the superselection rule [5] since the superposition state between different charged states is prohibited. On the other hand, for applications in vortices, we can safely consider the superposition state between the left- and right-twisted states [35, 63], such that we can consider arbitrary mixing of left- and right-twisted states with an arbitrary phase between them. The ladder operators t ̂ ± correspond to changing the topological charge at the center of the vortices for changing its orbital angular momentum from the left to the right circulation or vice versa.

Similarly, we consider the rising and lowering ladder operators, u ̂ + and u ̂ − , respectively, for the superposition state between the right-twisted and no-vortex states, and the z component of the rotationally symmetric s u ( 2 ) operator u ̂ = ( u ̂ 1 , u ̂ 2 , u ̂ 3 ) : u ̂ ± = 1 2 λ ̂ 6 ± i λ ̂ 7 , (34) u ̂ 3 = 1 4 − λ ̂ 3 + 3 λ ̂ 8 , (35) whose commutation relationships become u ̂ 3 , u ̂ ± = ± u ̂ ± , (36) u ̂ + , u ̂ − = 2 u ̂ 3 . (37)

For the mixing of the left-twisted and no-vortex states, the corresponding ladder operators, v ̂ + and v ̂ − , and the z component of the rotationally symmetric s u ( 2 ) operator v ̂ = ( v ̂ 1 , v ̂ 2 , v ̂ 3 ) become v ̂ ± = 1 2 λ ̂ 4 ± i λ ̂ 5 , (38) v ̂ 3 = 1 4 λ ̂ 3 + 3 λ ̂ 8 , (39) whose commutation relationships become v ̂ 3 , v ̂ ± = ± v ̂ ± , (40) v ̂ + , v ̂ − = 2 v ̂ 3 . (41)

Here, we defined nine operators for ladders and the quantization components of the three sets of s u ( 2 ) operators ( t ̂ , u ̂ , v ̂ ) , while only eight bases are required for the s u ( 3 ) algebra due to the traceless requirement. Consequently, we obtained one identity, v ̂ 3 = u ̂ 3 + t ̂ 3 , (42) which must be met for all states. This means that only two quantum numbers are independently chosen, regardless of apparent three sets of SU(2) states, which is, in fact, consistent with the rank-2 nature of the s u ( 3 ) Lie algebra.

As discussed previously, we select two quantum operators from three operators, t ̂ 3 , u ̂ 3 , and v ̂ 3 , for describing the SU(3) quantum states. If we choose t ̂ 3 , we can choose u ̂ 3 or v ̂ 3 . Alternatively, we can consider the superposition state, made of both u ̂ 3 and v ̂ 3 , whose z component becomes λ ̂ 8 = 2 3 u ̂ 3 + v ̂ 3 . (43)

Equivalently, we define the hypercharge operator [5] as y ̂ = 1 3 λ ̂ 8 , (44) = 2 3 u ̂ 3 + v ̂ 3 = 4 3 u ̂ 3 + 2 3 t ̂ 3 = 4 3 v ̂ 3 − 2 3 t ̂ 3 , (45) which was indispensable to understand quarks, their 2-body (3-body) compounds of meson, and the 3-body compounds of baryons. They are commutative with the other three operators, y ̂ , t ̂ 3 = y ̂ , u ̂ 3 = y ̂ , v ̂ 3 = 0 , (46) meaning that hypercharge could be the simultaneous quantum number with the other parameter. Therefore, we expect that the eigenstate is labeled by the quantum numbers, t ₃, u ₃, and v ₃ with y = (u ₃ + v ₃)/3, to satisfy y ̂ t 3 , u 3 , v 3 = y t 3 , u 3 , v 3 = y t 3 , y . (47)

We also confirm the identity y ̂ , t ̂ ± = 2 3 u ̂ 3 , t ̂ ± + 2 3 v ̂ 3 , t ̂ ± = 0 , (48) which ensures that y ̂ t ̂ ± t 3 , y = y t ̂ ± t 3 , y , (49) meaning that the application of ladder operations by t ̂ ± preserves the hypercharge, y.

On the other hand, we find that y ̂ , u ̂ ± = 2 3 u ̂ 3 + v ̂ 3 , u ̂ ± = ± u ̂ ± , (50) which leads to y ̂ u ̂ ± t 3 , y = y ± 1 u ̂ ± t 3 , y , (51) which means u ̂ + increments y and u ̂ − decrements y, respectively. We also confirm the same rule for v ̂ ± as y ̂ , v ̂ ± = 2 3 u ̂ 3 + v ̂ 3 , v ̂ ± = ± v ̂ ± , (52) which corresponds to y ̂ v ̂ ± t 3 , y = y ± 1 v ̂ ± t 3 , y . (53)

Finally, we obtain the fundamental multiplets (Figures 2A, B), given by three states, | ψ 1 〉 = t 3 = 1 2 , t 8 = 1 3 3 2 ⟩ = 1 0 0 , (54) | ψ 2 〉 = t 3 = − 1 2 , t 8 = 1 3 3 2 ⟩ = 0 1 0 , (55) | ψ 3 〉 = t 3 = 0 , t 8 = − 2 3 3 2 ⟩ = 0 0 1 . (56)

An arbitrary quantum state can be generated by mixing these three states using the superposition principle: multiplying complex numbers to fundamental ket states and adding up. In a standard matrix formulation of quantum mechanics, a general state is given by a row of three complex numbers.

For quarks, there exists an identity relationship between hypercharge and charge, q, as q = t 3 + 1 2 y . (57)

Therefore, one can use charge instead of hypercharge for an alternative quantum number.

For our applications to photonic orbital angular momentum, we consider superposition states between left- and right-twisted states and no-vortex state. We use the orbital angular momentum along the quantization axis, z, which is the direction of the propagation, as the first quantum number, instead of the isospin of t ₃. For the second quantum number, instead of hypercharge, we choose the topological charge, defined by q t = y + 2 3 , (58) which becomes 0 for the no-vortex state and 1 for both left- and right-twisted states. The topological charge corresponds to the winding number of the mode at the core, propagating along a certain z direction. It is also linked to the magnitude of photonic orbital angular momentum. In this paper, we only consider vortices with a winding number of 1 or 0, but it will be straightforward to extend our discussions to higher-order states.

There are other conservative properties in the s u ( 3 ) Lie algebra. We define a Casimir operator as C ̂ 1 = 1 4 ∑ i = 1 n λ ̂ i 2 = ∑ i = 1 n e ̂ i 2 . (59)

We calculate the commutation relationship as follows: C ̂ 1 , λ ̂ j = 1 4 ∑ i = 1 n λ ̂ i 2 , λ ̂ j , (60) = 1 4 ∑ i = 1 n λ ̂ i 2 λ ̂ j − λ ̂ j λ ̂ i 2 , (61) = 1 4 ∑ i = 1 n λ ̂ i λ ̂ i λ ̂ j − λ ̂ j − λ ̂ i λ ̂ i , (62) = 1 4 ∑ i = 1 n λ ̂ i λ ̂ i , λ ̂ j − λ ̂ j , λ ̂ i λ ̂ i , (63) = 2 i 4 ∑ i k C ijk λ ̂ i λ ̂ k − C jik λ ̂ k λ ̂ i , (64) = 0 , (65) where we changed the dummy indices in the last line as ∑ i k C jik λ ̂ k λ ̂ i = ∑ k i C jki λ ̂ i λ ̂ k = ∑ i k C ijk λ ̂ i λ ̂ k . Therefore, the Casimir operator, C ̂ 1 , obtains the simultaneous eigenstate with the rank-2 states for λ ̂ i . In fact, we observe, from direct calculations, that C ̂ 1 = 1 + 1 3 0 0 0 1 + 1 3 0 0 0 1 + 1 3 = 4 3 , (66) which means that C ̂ 1 is constant for SU(3) states. Here, it is obvious that we abbreviated the unit matrix of 3 × 3, 1 ₃, multiplied with 4 3 = 4 3 1 3 , for simplicity.

There is another Casimir operator, defined by C ̂ 2 = ∑ ijk D ijk t ̂ i t ̂ j t ̂ k = 1 8 ∑ ijk D ijk λ ̂ i λ ̂ j λ ̂ k , (67) where D _ijk is a symmetric tensor as defined in the anti-commutation relationship given as follows. We observe that C ̂ 2 is also constant in the s u ( 3 ) Lie algebra, such that the commutation relationship C ̂ 2 , λ ̂ i = 0 , (68) vanishes.

We also obtain the anti-commutation relationship as λ ̂ i , λ ̂ j = 4 3 δ i j + 2 ∑ k = 1 8 D ijk λ ̂ k , (69) which is equivalent to e ̂ i , e ̂ j = 1 3 δ i j + ∑ k = 1 8 D ijk e ̂ k , (70) where 4 3 δ i j should be considered as 4 3 δ i j 1 3 , as before. The symmetric tensor, D _ijk , is shown in Table 2.

By multiplying λ ̂ k with the anti-commutation relationship, we obtain λ ̂ i , λ ̂ j λ ̂ k = 4 3 δ i j λ ̂ k + 2 D i j k ′ λ ̂ k ′ λ ̂ k . (71)

We take the trace of the equation, while using Tr ( λ ̂ i ) = 0 and Tr ( λ ̂ i λ ̂ j ) = 2 δ i j , and we obtain D ijk = 1 4 Tr λ ̂ i , λ ̂ j λ ̂ k . (72)

Similarly, we also obtain C ijk = 1 4 i Tr λ ̂ i , λ ̂ j λ ̂ k (73) from the commutation relationship.

Finally, we show that C ̂ 2 = C ̂ 1 2 C ̂ 1 − 11 6 = 10 9 . (74)

To prove the identity, we use λ ̂ i , λ ̂ j = λ ̂ i λ ̂ j + λ ̂ j λ ̂ i (75) and λ ̂ i , λ ̂ j = λ ̂ i λ ̂ j − λ ̂ j λ ̂ i . (76)

By adding these equations, we obtain λ ̂ i , λ ̂ j + λ ̂ i , λ ̂ j = 2 λ ̂ i λ ̂ j , (77) which becomes 4 3 δ i j + 2 D ijk λ ̂ k + 2 i C ijk λ ̂ k = 2 λ ̂ i λ ̂ j (78) from commutation and anti-commutation relationships. Then, we multiply a factor of λ ̂ i λ ̂ j and sum up to obtain ∑ i j λ ̂ i 2 λ ̂ j 2 = 2 3 ∑ i λ ̂ i 2 + ∑ ijk D ijk λ ̂ i λ ̂ j λ ̂ k − i ∑ ijk C ijk λ ̂ i λ ̂ j λ ̂ k , (79) where the last term becomes ∑ ijk C ijk λ ̂ i λ ̂ j λ ̂ k = 1 2 ∑ ijk C ijk λ ̂ i λ ̂ j λ ̂ k + C ikj λ ̂ i λ ̂ k λ ̂ j = 1 2 ∑ ijk C ijk λ ̂ i λ ̂ j λ ̂ k − C ijk λ ̂ i λ ̂ k λ ̂ j = 1 2 ∑ ijk C ijk λ ̂ i λ ̂ j λ ̂ k − λ ̂ k λ ̂ j = 1 2 ∑ ijk C ijk λ ̂ i λ ̂ j , λ ̂ k = i ∑ ijk C ijk C jkl λ ̂ i λ ̂ l = i ∑ ijk C jki C jkl λ ̂ i λ ̂ l = 3 i ∑ i λ ̂ i 2 , (80) which leads to ∑ i j λ ̂ i 2 λ ̂ j 2 = ∑ ijk D ijk λ ̂ i λ ̂ j λ ̂ k + 2 3 + 3 ∑ i λ ̂ i 2 . (81)

Therefore, we obtain C ̂ 2 = C ̂ 1 2 C ̂ 1 − 11 6 = 10 9 , (82) which is in fact constant under the s u ( 3 ) Lie algebra.

Now, we discuss how to classify the superposition state between left- and right-twisted states and no-vortex state using the s u ( 3 ) Lie algebra and the SU(3) Lie group. We assume Laguerre–Gauss modes with a topological charge of q _t = 1 for both left- and right-twisted states [23, 34, 35, 39, 40, 47–51, 63, 65, 71–73] for simplicity.

Here, our main idea is to assign the three states of twisted modes and no-vortex mode to orthogonal states of the SU(3) states (Figures 1, 2). The most important part of the twisted modes for orbital angular momentum is its azimuthal (ϕ) dependence [34]; i.e., the wavefunction of the ray with orbital angular momentum of m is given by ⟨ ϕ | m ⟩ = e i m ϕ , (83) which is orthogonal to each other for states with different charges of m in a sense, ⟨ m ′ | m ⟩ = ∫ 0 2 π d ϕ 2 π e i ( m − m ′ ϕ = δ m , m ′ . (84)

This means that the modes with different orbital angular momentum could be treated as orthogonal quantum mechanical states. For our consideration and notation [24–27, 63, 70, 74–76], we assign left- and right-twisted states as |L⟩ = |1⟩ = |ψ ₁⟩ and |R⟩ = | − 1⟩ = |ψ ₂⟩, respectively, and the no-vortex Gaussian state as |O⟩ = |0⟩ = |ψ ₃⟩. These states are also considered to have different topological charge, such as red, blue, and green (Figure 1).

We also assume that all modes have the same polarization state, such that our SU(3) state is polarized. Then, we consider the polarization degree of freedom, which comes from the SU(2) spin of photons [8, 9, 11–21, 24–27], such that we explore the photonic states with the SU(2) × SU(3) symmetry.

We consider a coherent ray of photons emitted from a laser source [17–19, 24, 63, 70, 75], such that a macroscopic number of photons per second, N, pass through the cross section of the ray. We use upper-case letters to describe macroscopic observables and expectation values, such as photonic orbital angular momentum [23, 34, 35, 39, 40, 47–51, 63, 65, 71–73], L ̂ i = ℏ N ℓ ̂ i = ℏ N λ ̂ i , (85) where ℏ = h/(2π) is the Dirac constant, defined by the Plank constant (h), divided by 2π, while lower-case letters are used for a single-quantum operator or a normalized parameter, such as a normalized orbital angular momentum operator, ℓ ̂ i = λ ̂ i , (86) for i = 1, 2, and 3.

There is a difference of factor of 2 in the definition between the orbital angular momentum operator ℓ ̂ i and the isospin operator of t ̂ 3 , but it would be more appropriate to use ℓ ̂ i for photonic vortices since the orbital angular momentum is quantized in the unit of ℏ [24–26, 34, 35, 63].

First, let us review the SU(2) coupling between left- and right-twisted states [35, 63]. For this, we consider the following state: | θ l , ϕ l 〉 = e − i ϕ l 2 ⁡ cos θ l 2 e + i ϕ l 2 ⁡ sin θ l 2 0 , (87) where the amplitudes of left- and right-vortex states are controlled by the polar angle of θ _l and the phase is defined by ϕ _l . We can realize this state using an exponential map from the s u ( 3 ) Lie algebra to the SU(3) Lie group: D ̂ 2 θ l = exp − i λ ̂ 2 θ l 2 , (88) = cos θ l 2 − sin θ l 2 0 sin θ l 2 cos θ l 2 0 0 0 0 , (89) which is a phase-shifter with its fast axis rotated for π/4 from the horizontal axis [24], together with another exponential map of D ̂ 3 ϕ l = exp − i λ ̂ 3 ϕ l 2 , (90) = exp − i ϕ l 2 0 0 0 exp i ϕ l 2 0 0 0 0 , (91) which is a rotator. We apply these operators to a unit vector, |ψ ₁⟩, to confirm the SU(2) state, | θ l , ϕ l 〉 = D ̂ 3 ϕ l D ̂ 2 θ l | ψ 1 〉 , (92) made of left- and right-twisted states.

By calculating a standard quantum mechanical average from |θ _l , ϕ _l ⟩, such that ℓ i = λ i = ⟨ ℓ ̂ i ⟩ = ⟨ λ ̂ i ⟩ = ⟨ θ l , ϕ l | λ ̂ i | θ l , ϕ l ⟩ (93) for i = 1, 2, and 3, respectively, we obtain λ 1 = sin θ l cos ϕ l , (94) λ 2 = sin θ l sin ϕ l , (95) λ 3 = cos θ l . (96)

Thus, the SU(2) states between left- and right-twisted states could be shown on the Poincaré sphere for orbital angular momentum [35, 39, 40, 63, 72]. Usually, the rotational symmetry and the corresponding orbital angular momentum are considered by the SU(2) symmetry since the states between |L⟩ = |1⟩ and |R⟩ = | − 1⟩ cannot be transferred by the change in Δm = ±1, and instead, Δm = ±2 is required. This could be achieved by using a spiral phase plate [37] with a topological charge of m = 2. Alternatively, it is possible to create a superposition state between |L⟩ = |1⟩ and |R⟩ = | − 1⟩, and SU(2) states could be realized by controlling the amplitudes and phases [35, 63]. For our considerations in the SU(3) states, this corresponds to bending the quantization axis, ℓ ̂ 3 , for allowing three states to couple among each other (Figures 2A, B).

Next, we consider coupling between the no-vortex state and left- or right-vortex states. This corresponds to changing the hypercharge and topological charge. We call these SU(2) couplings hyperspin since they exhibit spin-like SU(2) behaviors, yet they are different from spin. For elementary particles, such as quarks, states with different charged particles cannot be realized at all due to the superselection rule, such that composite particles, such as a neutron and a proton, cannot be in their superposition state [5]. However, for coherent photons, we consider a superposition state among different topologically charged states, such that we can mix the no-vortex state and twisted state at an arbitrary ratio in amplitudes with a certain definite phase. Topologically, the vortex is well known to be equivalent to a shape of a doughnut, which cannot be continuously changed to a shape of a ball. Our challenge could be considered to realize a superposition state between a doughnut and a ball, which is impossible classically, while we would have a chance since photons are elementary particles with a wave character allowing a superposition state of orthogonal states.

Here, we consider the hyperspin coupling, which means that we explore mixing between twisted states and no-vortex state. In order to achieve this, the easiest option is to follow the previous approach of the SU(2) state between left and right vortices. We simply need to change from λ ̂ 2 / 2 = e ̂ 2 ( t ) and λ ̂ 3 / 2 = e ̂ 3 ( t ) to λ ̂ 5 / 2 = e ̂ 2 ( v ) and e ̂ 3 ( v ) , respectively, and we define D ̂ 2 v θ y = exp − i e ̂ 2 v θ y , (97) = cos θ y 2 0 − sin θ y 2 0 0 0 sin θ y 2 0 cos θ y 2 , (98) and D ̂ 3 v ϕ y = exp − i e ̂ 3 v ϕ y , (99) = exp − i ϕ y 2 0 0 0 0 0 0 0 exp i ϕ y 2 . (100) We obtain a general SU(3) state, | θ l , ϕ l ; θ y , ϕ y 〉 = D ̂ 3 ϕ l D ̂ 2 θ l D ̂ 3 v ϕ y D ̂ 2 v θ y | ψ 1 〉 = e − i ϕ y 2 e − i ϕ l 2 ⁡ cos θ l 2 cos θ y 2 e − i ϕ y 2 e + i ϕ l 2 ⁡ sin θ l 2 cos θ y 2 e + i ϕ y 2 ⁡ sin θ y 2 . (101)

Finally, we can calculate the expectation values for all generators of the s u ( 3 ) Lie algebra, which becomes a vector in an eight-dimensional space, given by λ ⃗ = λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 = sin θ l cos ϕ l cos 2 θ y 2 sin θ l sin ϕ l cos 2 θ y 2 cos θ l cos 2 θ y 2 cos ϕ y + ϕ l 2 sin θ y cos θ l 2 sin ϕ y + ϕ l 2 sin θ y cos θ l 2 cos ϕ y − ϕ l 2 sin θ y sin θ l 2 sin ϕ y − ϕ l 2 sin θ y sin θ l 2 − 3 6 + 3 2 cos θ y . (102)

An arbitrary state of SU(3) is characterized by this vector, which is similar to the Stokes parameters [11] on the Poincaré sphere [12]. The higher-dimensional vector of λ ⃗ satisfies the norm conservation ∑ i = 1 8 λ i 2 = 4 3 , (103) upon rotations in the eight-dimensional space, which is guaranteed from the constant Casimir operator of C ̂ 1 = 4 / 3 , as shown previously. Therefore, an SU(3) state is represented as a point on the hypersphere with the radius of ∑ i = 1 8 λ i 2 = 2 3 . (104)

We propose this hypersphere as the Gell-Mann hypersphere, named after Gell-Mann who found the SU(3) symmetry of baryons and mesons, leading to the discovery of quarks [28–30]. In fact, we have the eightfold way [28–30] to allow the SU(3) superposition state by changing the amplitudes and the phases of the wavefunction. We can attribute color charge of red, green, and blue to three fundamental states of |ψ ₁⟩, |ψ ₂⟩, and |ψ ₃⟩, respectively, similar to QCD [4, 10, 28–30]. In QCD for quarks, only certain sets of multiplets, such as baryons and mesons, are observed as stable bound states of quarks, due to the spontaneous symmetry breaking of the universe [77–81]. In our photonic QCD, on the other hand, we can discuss an arbitrary superposition state by mixing three orthogonal states of left- and right-vortices and no-twisted rays. Therefore, we can discuss the SU(3) state before the symmetry is broken; in other words, the symmetry can be recovered without injecting additional energies to the system, similar to the Nambu–Goldstone bosons [77–81]. This corresponds to the rotation of the hyperspin of λ ⃗ in the eight-dimensional Gell-Mann space by using eight generators of rotation λ ̂ i (i = 1, ⋯, 8) to change the amplitudes and phases. In experiments, this is achieved by using rotators and phase-shifters of SU(2) [24–27, 63, 70, 74–76] since we can realize arbitrary rotations of the SU(3) state by using three sets of SU(2) rotations, as shown previously.