Photonic Quantum Chromo-Dynamics

Vortexed photons with finite orbital angular momentum have a distinct mode profile with topological charge at the centre of the mode, while propagating in a certain direction. Each mode with different topological charge of $m$ is orthogonal, in the sense that the overlap integral vanishes among modes with different values of $m$. Here, we theoretically consider a superposition state among 3 different modes with left- and right-vorticies and a Gaussian mode without a vortex. These 3 states are considered to be assigned to different quantum states, thus, we have employed the $\mathfrak{su}(3)$ Lie algebra and the associated SU(3) Lie group to classify the photonic states. We have calculated expectation values of 8 generators of the $\mathfrak{su}(3)$ Lie algebra, which should be observable, since the generators are Hermite. We proposed to call these parameters as Gell-Mann parameters, named after the theoretical physicist, Murray Gell-Mann, who established Quantum Chromo-Dynamics (QCD) for quarks. The Gell-Mann parameters are represented on the 8-dimensional hypersphere with its radius fixed due to the conservation law of the Casimir operator. We have discussed a possibility to explore photonic QCD in experiments and classified SU(3) states to embed the parameters in SO(6) and SO(5).

II. SUMMARY OF THE su(3) LIE ALGEBRA First, we summarise theoretically minimum knowledge on fundamental properties of the su(3) Lie algebra, in order to make our discussions self-contained and clarify our notations.
We wish this will help photonic researchers, who are not familiar with the su(3) algebra, to understand the idea to treat 3 orthogonal quantum states in an equal fashion.It is far from a comprehensive summary, such that interested readers are encouraged to refer to excellent textbooks [1][2][3][4][5]7] .Those who are familiar with the su(3) algebra should skip this section.
The Lie algebra and the Lie group were mathematically developed as early as 1870s, without specific applications in physics [1][2][3][4]6] .The first serious applications in physics found in elementary physics, leading to the discoveries of quarks [5].Obviously, the su (3) Lie algebra and more general representation theories are robust and they will be applicable to various cases.On the other hand, here, we are focussing on applications in photonics, and we will use this example to review fundamental characteristics of su(3) algebra.Thus, we will lose the generality in our construction of the logic, but it will be easier to understand the concept and applications in higher dimensions will be straightforward.

A. Generators of the su(3) algebra
We consider 3 orthogonal quantum states, such as left-and right-vortexed states and the no-vortex state, which are described in the Hilbert space with 3 complex numbers, C 3 .We allow arbitrary mixing of these 3 states, realised by the superposition principle, and the wavefunction could be consider to be normalised 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 5 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 realised by the exponential mapping form the su(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 , which are all Hermite, λ † i = λi (i = 1, • • • , 8), implying their expectation values must be real and observable.We can also uses the bases, êi = λi /2, reflecting underlying the su(2) symmetry among 2 orthogonal states.The bases satisfy the normalisation relationship for the trace, Tr λi • λj = 2δ ij , where δ ij is the Kronecker delta.

B. Commutation relationship
The commutation relationship is obtained by straightforward calculation of basis matrices, and we obtain λi , λj = 2i where the structure constants, C ijk , are listed in Table I.C ijk is an asymmetric tensor, such that odd permutation of indices change its sign.The most of commutation relationship involve only 1 term in the summation on the right hand side of the equation, similar to the su(2) commutation relationship for spin.On the other hand, we must account for 2 terms involved in equations λ6 We also confirm that we have 2 mutually commutable operators, λ1 , λ8 = λ2 , λ8 = λ3 , λ8 = 0 (4) while we see λ1 , λ2 = 0, λ1 , λ3 = 0, λ2 , λ3 = 0.
Therefore, the rank 2 character of the su(3) algebra is confirmed.This is also evident that λ3 and λ8 are already diagonalised in our representation for the basis.
C. Basis operators for the su(2) algebra within the su(3) algebra We are considering 3 orthogonal states for the su(3) algebra, and we can pick up 2 states among 3 available states.There are 3 ways to choose 2 states and each of the pairs of states will form the su(2) Lie algebra.For example, if we choose the first and the second states, corresponding to the left and the right vortexed states, we use bases for describing the su(2) states, since they are equivalent to Pauli matricies, if we neglect the third quantum state for the no-vortex state.These operators, ê(t) i , were originally used for describing isospin for quarks [5].For our applications, they will be useful to describe the rotation between the left and the right vortexed states.The rotation corresponds to mixing the left and the right vortexed states, which will be described in the Poincaré sphere for vortices.
Another pair of states are made of the right vortexed state and the no-vortex state, whose bases are Here, it is important to be aware that we can define a new vector operator of ê(u) 3 , for example, from λ3 and λ8 , since they are basis vector operators in the su(3) algebra, which forms a vector space.We cannot simply add components in the SU(3) Lie group, since SU(3) Lie group is not a vector space.We see that ê(u) These su(2) commutation relationships are summarised as where x = t, u, or v.We have used small letters (x = t, u, or v) for operators describing for a single quanta like a quark or a photon, and we will use capital letters for coherent states of photons (X = T, U, or V ) under Bose-Einstein condensation, where macroscopic number, N , of photons are occupying the same state, latter of this paper.
The su(3) algebra does not contain a non-trivial invariant group.For example, we see such that the internal su(2) groups are connected by commutation relationships and the su(2) algebra is not closed.

D. Ladder operators
We will utilise the Cartan-Dynkin formulation [5] for describing the su(3) states.In the formalism, we consider to fix the quantisation axis rather than isotropic to all directions in the su(2) algebra, and consider ladder operators for rising and lowering the qnautm number along the quantisation axis [5,9].More specifically, we define where t3 stands for the operator of the z component of the rotationally-symmetric su(2) operator t = ( t1 , t2 , t3 ), and t+ and t− are the rising and the lowering operators, respectively, to increment and decrement the quantum number for t3 .We can use t3 and t± instead of ê(t) i (i = 1, 2, 3), and their commutation relationships become t3 , t± = ± t± (24) For applications to isospin, t3 gives 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 realised due to the superselection rule [5], since the superposition state between different charged states is prohibitted.On the other hand, for applications to vortices, we can safely consider the superposition state between the left and the right vortexed states [30,31], such that we can consider arbitrary mixing of left and right vortexed states with arbitrary phase between them.The ladder operators t± correspond to changing the topological charge at the centre 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 the lowering ladder operators, û+ and û− , respectively, for the superposition state between the right vortexed and the no-vortex states, and the z component of the rotationally-symmetric su(2) whose commutation relationships become For the mixing of the left-vortexed and no-vortex states, the corresponding ladder operators v+ and v− , and the z component of the rotationally-symmetric su(2) whose commutation relationships become Here, we have defined 9 operators for ladders and the quantisation components of the 3 sets of su(2) operators ( t, û, v), while only 8 bases are required for the su(3) algebra, due to the traceless requirement.Consequently, we have obtained 1 identity which must be met for all states.This means only 2 quantum numbers are independently chosen, regardless of apparent 3 sets of su(2) states, which is in fact consistent with the rank-2 nature of the su(3) algebra.

E. Hypercharge and topological charge
As we have reviewed above, we can pick up 2 quantum operators from 3 operators, t3 , û3 , and v3 , for describing the su(3) quantum states.If we choose t3 , we can choose û3 or v3 .Alternatively, we can consider the superposition state, made of both û3 andv 3 , whose z Equivalently, we can define the hypercharge operator [5], as meaning that the application of ladder operations by t± will preserve the hypercharge, y.
On the other hand, we find which means û+ increments y and û− decrements y, respectively.We also confirm the same rule for v± as Finally, we obtain the fundamental multiplets (Fig. 1), given by 3 states .
An arbitrary quantum state can be generated by mixing these 3 states upon the superposition principle: multiplying complex numbers to fundamental ket states and summing up.In a standard matrix formulation of quantum mechanics, a general state is given by a row of 3 complex numbers.
For quarks, there is an identity relationship between hypercharge and charge, q, as 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 vortexed states and no-vortex state.We use the orbital angular momentum along the quantisation 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 which becomes 0 for the no-vortex state and 1 for both left-and right-vortexed 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 vorticies with the winding number of 1 or 0, but it will be straightforward to extend our discussions to higher order states.

F. Casimir Operators
There are other conservative properties in the su(3) algebra.We define a Casimir operator, We calculate the commutation relationship, where we have changed the dummy indices at the last line as ik C jik λk λi = ki C jki λi λk = ik C ijk λi λk .Therefore, the Casimir operator, Ĉ1 , obtains the simultaneous eigenstate with the rank-2 states for λi .In fact, we see, from direct calculations, which means that Ĉ1 is actually constant for su(3) states.Here, it is obvious that we have 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 where D ijk is a symmetric tensor, as defined in the anti-commutation relationship, below.
We see that Ĉ2 is also constant in the su(3) algebra, such that the commutation relationship, Ĉ2 , λi = 0, (

G. Anti-Commutation relation
We also obtain the anti-commutation relationship as which is equivalent to where we have abbreviated , as before.The symmetric tensor, D ijk , is shown in Table II.
To prove the identity, we use λi , λj = λi λj + λj λi where the last term becomes ijk Therefore, we obtain which is in fact constant under the su(3) algebra.
Here, our main idea is to assign the 3 states of vortexed modes and no-vortex mode to orthogonal states of the SU(3) states (Fig. 1).The most important part of the vortexed modes for orbital angular momentum is its azimuthal (φ) dependence [32], i.e., the wavefunction of the ray with orbital angular momentum of m is given by which is orthogonal each other for states with different m in a sense, This means that the modes with different orbital angular momentum could be treated as orthogonal states as quantum mechanical states.For our consideration and notation We also assume that all modes have the same polarisation state, such that our SU(3) state is polarised.Later, we will consider the polarisation degree of freedom, which comes from the SU(2) spin of photons [8,9,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25], such that we explore the photonic states with the SU(2) × SU(3) symmetry.
We consider coherent ray of photons emitted from a laser source [17-19, 22, 31, 40, 41], such that a macroscopic number of photons per second, N , is passing through the cross section of the ray.We use capital letters to describe macroscopic observables and expectation values, such as photonic orbital angular momentum [30][31][32][33][34][35][36][37][38] where = h/(2π) is the Dirac constant, defined by the plank constant (h), divided by 2π, while small letters are used for single quantum operator or a normalised parameter, such as a normalised orbital angular momentum operator, for i = 1, 2, and 3.
There is a factor of 2 difference in definition between the orbital angular momentum operator ˆ i and the isospin operator of t3 , but it would be more appropriate to use ˆ i for photonic vortices, since the orbital angular momentum is quantised in the unit of [22][23][24][30][31][32].
We consider the following state, 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 realise this state by an exponential map from su(3) Lie algebra to the SU(3) Lie group which is a phase-shifter with its fast axis rotated for π/4 from the horizontal axis [22], together with another exponential map of which is a rotator.We apply these operators to a unit vector, |ψ 1 , to confirm the SU(2) state made of left-and right-vortexed states.
By calculating a standard quantum mechanical average from |θ l , φ l , for i = 1, 2, and 3, respectively, we obtain Thus, the SU(2) states between left-and right-vortexed states could be shown on the Poincaré sphere for orbital angular momentum [30,31,33,35].Usually, the rotational behaviours similar to spin, yet, it is different from spin.For elementary particles like quarks, states with different charged particles cannot be realised at all, due to the super-selection rule, such that composite particles like a neutron and a proton cannot be their superposition state [5].However, for coherent photons, we can consider a superposition state among different topologically charged states, such that we can mix no-vortex state and vortexed states at an arbitrary ratio in amplitudes with a certain definite phase.Topologically, vortex is well known to be equivalent to a shape of doughnut, which cannot be continuously changed to be a ball.Our challenge could be considered to realise 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 to allow a superposition state of orthogonal states.
Here, we consider the hyperspin coupling, which means that we will explore mixing between vortexed states and no-vortex state.In order to achieve it, an easiest option is to follow the previous approach of the SU(2) state between left and right vortices.We just need to change from λ2 /2 = ê(t) 2 and λ3 /2 = ê(t) 3 to λ5 /2 = ê(v) 2 and ê(v) 3 , respectively, and we define and and we obtain a general SU(3) state, .
Finally, we can calculate the expectation values for all generators of the su(3) Lie algebra, which becomes a vector in an 8-dimensional space, given by .
An arbitrary state of SU(3) is characterised by this vector, similar to the Stokes parameters [11] on the Poincaré sphere [12].The higher dimensional vector of − → λ satisfies the norm conservation upon rotations in 8-dimensional space, which is actually guaranteed from the constant Casimir operator of Ĉ1 = 4/3, as we have seen above.Therefore, an SU(3) state is represented as a point on the hypersphere with the radius of We would like to 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 [26][27][28].In fact, we have the eightfold way [26][27][28] to allow the SU(3) superposition state by changing the amplitudes and the phases of the wavefunction.We can attribute colour charge of red, green, and blue to 3 fundamental states of |ψ 1 , |ψ 2 , and |ψ 3 , similar to QCD [4,10,[26][27][28].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 [44][45][46][47][48].In our photonic QCD, on the other hand, we can discuss an arbitrary superposition state by mixing 3 orthogonal states of left-and right-vorticies and no-vortexed rays.Therefore, we can discuss the SU(3) state before the symmetry broken, or in other words, the symmetry can be recovered without injecting additional energies to the system, similar to the Nambu-Goldstone bosons [44][45][46][47][48].This corresponds to rotate the hyperspin of − → λ in the 8-dimensional Gell-Mann space by using 8 generators of rotation λi (i = 1, • • • , 8) to change the amplitudes and the phases.In experiments, this will be achieved by using rotators and phase-shifters of SU(2) [22-25, 31, 39-42], since we can realise arbitrary rotations of the SU(3) state by using 3 sets of SU(2) rotations, as we have shown above.
Among 8 Gell-Mann parameters, λi , 2 of them are especially important, since the su(3) algebra is rank of 2. One of them is 3 = λ 3 , which determines the average orbital angular momentum along the direction of propagation, z.The other important parameter is which determine the average hypercharge.We confirm the expected maximum and minimum of hypercharge, as max (y 3 ) = 1/3 and min (y 3 ) = −2/3.Hyperchage is simply converted to the topological charge, q t = y 3 + 2/3, and we confirm as max (q t ) = 1 and min (q t ) = 0, as expected for vorticies and no-vortexed state, respectively.

B. Hyperspin with left/right vortex
The Gell-Mann hypersphere contains all practical information on the SU(3) states, in terms of amplitudes and phases.Unfortunately, it is impossible to recognise the 8-dimensional hypersphere for us to see in the world of 3-dimensional space and time.In the previous sub-section, we have seen the coupling between left and right vortices could be represented by the Poincaré sphere for the vortexed photons [30,31], which corresponds to visualise the coupling, controlled by the su(2) generators of ê(t) 1 , ê(t) 2 , and ê(t) 3 .Here, we consider to see another su(2) generators and discuss how hyperspin is represented in a similar way to the Poincaré sphere.
First, we consider the coupling between the left-vortexed state and no-vortex state.This corresponds to take the limit (θ l , φ l ) → (0, 0), and the Gell-Mann parameters become In this case, the parameter λ 3 can take a value between 1 and 0, since the left vortex has topological charge of 1 duet to λ3 |L = |L , while the no-vortex state (|O ), does not have topological charge, as λ3 |O = 0.The superposition state is characterised non-zero average of λ 3 , and if the amount of the right-vortex component is less than that of the left-vortex, λ 3 becomes positive.This corresponds to the net left-circulation of orbital angular momentum.
The other Gell-Mann parameters are given by θ y and φ y .In the limit of the zero right-vortex component, it is convenient to consider the average of the su(2) generating vector, which corresponds to introduce v 3 = (λ 3 + √ 3λ 8 )/4, instead of λ 8 or t q , since only 2 parameters are independent among (t 3 , u 3 , v 3 ) due to the rank-2 character of the su(3) algebra.
Similarly, we also consider to check the coupling between the right-vortex state and the no-vortex state, which corresponds to take the limit of (θ l , φ l ) → (π, 0), and we obtain the Gell-Mann parameters, In this case, the sign of λ 3 changed, compared with the coupling to the left-vortex, since we assigned negative sign for λ3 |R = −|R to the right-vortex, seen from the observer side against the light, coming to the detector [22,31].Therefore, λ 3 can take the value between -1 and 0, which corresponds to the average right-circulation of orbital angular momentum.
For the right circulation, it is useful to calculate the average of the su(2) generating vector, û as which becomes the same formula for v, when only 1 chirality (i.e., left or right vortex) is involved.In fact, the parameters θ y and φ y account for the relative phase and the amplitudes between the state with |m| = 1 and the state with m = 0 without including the difference in chiralities.Here, u 3 is introduced by u 3 = (−λ 3 + √ 3λ 8 )/4, and it satisfies the conservation Consequently, we have obtained 3 vectors, (t, u, v), where t = ( 1 , 2 , 3 )/2 is coming from the average orbital angular momentum.Each vector of t, u, or v is 3-dimensional, such that they are represented by the Poincaré spheres.However, care must be taken in the radiuses of the Poincaré spheres, since they depend on the relative amplitudes, determined by θ l and θ y .This comes from the mutual dependence among 3 sets of the su(2) algebra, since the su(3) algebra does not contain the non-trivial invariant group, as we confirmed above.As a result, we obtained 3 mutually dependent spheres, which have 3 × 3 = 9 parameters, with 1 identity of v 3 = u 3 + t 3 .For visualisation purposes, the 3 Poincaré spheres with variable radiuses might be practically more useful for humans, living in 3 spatial dimensions, rather than 8-dimensional Gell-Mann hypersphere of the constant radius of 2/ √ 3, whose surface is equivalent to 7-dimensional spherical surface of S 7 , given by real numbers, with fixed radius in 8-dimensional space.
C. Hyperspin embedded in SO (6) Gell-Mann parameters in SO( 8) are useful to understand the coupling between |L , |R , and |O .However, we can easily recognise that the generators of su(3) cannot span the whole hypersurface of SO (8).For example, parameters λ i (i = 1, • • • , 7), except for λ 8 , cannot take values above 1 nor below -1, while the radius of 2/ √ 3 is larger than 1.This clearly shows a point like (2/ √ 3, 0, • • • , 0) cannot be covered at all, such that SO( 8) is much larger than parameter space required to represent the photonic states, composed of 3 orthogonal states.
Then, let's consider the number of freedom, required for mixing |L , |R , and |O .In general, we should consider variable density of photons, since the radius of the Poincaré sphere depends on the output power of the ray [8, 9, 11-25, 31, 40, 41].Then, photons in coherent state are represented by 1 complex number per orthogonal degree of freedom for the component of the wavefunction.We are considering for fixed polarisation state, while we have 3 orthogonal states for vorticies, and therefore, we have 6 degrees of freedom (Table III).
These 6 degrees of freedom are attributed to corresponding physical parameters (Table III). 1 degree is assigned to the power density of the ray, and another is used for the global U(1) phase, which will not play a role for expectation values of the Gell-Mann hypersphere.
2 degrees of freedom are required for describing the superposition state for orbital angular momentum, which can be shown in the Poincaré sphere with variable radius (Fig. 2 (a)).
Therefore, the rest of the remaining 2 parameters should be assigned to hyperspin to account for the mixing of |L and/or |R with |O .This picture is consistent with the wavefunction of |θ l , φ l ; θ y , φ y , where θ y and φ y account for hyperspin.On the other hand, we used 8 Gell-Mann parameters for describing the superposition state from the expectation values.
All 8 parameters are required to understand the full rotational ways on the Gell-Mann hypersphere, however, less parameters are required to scan the full wavefunction over the expected Hilbert space of S 5 .Here, we try to reduce the number of Gell-Mann parameters to embed hyperspin in SO (6).The goal is to represent hyperspin as shown on the Poincaré sphere (Fig. 2 (b)), which should be enough for showing θ y and φ y , topologically.
A hint is found in Gell-Mann parameters, λ 4 , • • • , λ 7 , which keep the magnitude, upon changing other parameters of θ l , φ l , and φ y .Therefore, we have a chance to eliminate λ 6 and λ 7 by the renormalising the operators for the su(3) algebra.
By inspecting λ 4 , • • • , λ 7 , we realise the phases of φ y and φ l are coupled in a mixed form.If we could convert φ y ± φ l /2 → φ y , the rest of parameters are easily converted upon rotations.This could be achieved, if we remember the rotation matrices of form a group to satisfy the associative requirement Then, we obtain whose reverse relationship becomes By using these formulas, we define Then, we could successfully convert φ y ± φ l /2 → φ y , as intended.Finally, we can rotate between λ 4 and λ 6 to eliminate λ 6 by defining Similarly, we define In order to obtain these expectation values for Gell-Mann parameters, we should renormalise the original su(3) basis operators to define If we use these operators, the Gell-Mann parameters become such that we could successfully remove λ 6 and λ 7 .These parameters are equivalent to use photonic orbital angular momentum and hyperspin which can be shown on 2 Poincaré spheres with the radiuses of cos 2 (θ y /2) and 1/2, respectively, instead of original 3 spheres.This is consistent with 4 degrees of freedom for orbital angular momentum and hyperspin, as confirmed before (Table III), and it is also expected from the rank-2 nature of su(3) algebra, which requires only 2 sets of su(3) among 3 sets of ( t, û, v).In practice, we do not know the angles of θ l and φ l , a priori, such that the angles are obtained from expectation values or experimental results.
Finally, we could successfully embed Gell-Mann parameters in SO(6) to renormalise which satisfies the conservation law of the norm, which was required from the constant Casimir operator of Ĉ1 .

D. Alternative Coherent States
We have used D(v) 2 (θ y ) and D(v) 3 (φ y ) to define an arbitrary state, but we can define alternative coherent state, using original su Using this coherent state, we obtain the Gell-Mann parameters as expectation values, We will embed Gell-Mann parameters into SO (6) for this coherent state in the same way with the previous subsection.To achieve such a conversion, we need to transfer The rest of the calculations are exactly the same with the previous subsection, and we can use the same renormalised operators of λ4 λ5 , while we remove λ6 = 0 and λ7 = 0.
Then, the renormalised Gell-Mann parameters become which keep − → unchanged, while hyperspin becomes This just corresponds to change the azimuthal angle, φ y → √ 3φ y /2, in the Poincaré sphere of Fig. 2 (b).Consequently, we could embed Gell-Mann parameters in SO (6) as which also keep the norm upon arbitrary rotations in 6-dimensional space in SO (6).In the practical experiments, however, it will be more complex, if we set up a rotator for D8 (φ y ), since 3 waves are involved rather than 2 waves.In conventional optical experiments, various splitters and combiners are prepared for 2 waves such that it is much easier to rely on SU(2) rotations, including D(v) 3 (φ y ) and D(u) 3 φ y , such that we do not have to stick on using original bases of λi for SU(3) states.

IV. EMBEDDING IN SO(5)
For the complete description of the eightfold way to rotate the SU(3) states, Gell-Mann parameters in SO (8) are more useful to understand the differences in phases and amplitudes among |L , |R and |O .On the other hand, SO(8) is too larger to show the nature of the wavefunction, made of 3 complex numbers (C 3 ) with its norm conserved to cover S 5 in Hilbert space.
We could successfully reduce the dimension of Gell-Mann parameters from SO(8) to SO (6) or SO(3) × SO(3) to represent SU(3) states, in terms of orbital angular momentum and hyperspin, as expectation values.On the other hand, we have just 4 parameters (θ l , φ l , θ y , and φ y ), such that we have a chance to reduce 1 more dimension to represent on S 4 in SO (5).
In the similarity with the SU(2) states, one of the degree of freedom in S 5 would be coming from the global phase, such that we may have a chance to represent the expectation values on S 4 in SO (5).However, we could not establish a surjective mapping from SO(6) to SO(5) purely upon rotations using our bases of su(3), because the expectation values of λ i (i = 1, • • • , 7) cannot be larger than 1, while we needed to renormalise λ 8 to combine with λ 4 and λ 5 .Then, we have focussed on the conservation relationships of and consider a following non-surjective mapping from SO(6) to SO(5), as we renormalise which preserve We also confirm that the renormalised Gell-Mann parameters conserve the norm which is consistent with the constant Casimir operator.Consequently, expectation values are embedded on a compact Gell-Mann hypersphere of S 4 in SO(5).It is well established a photonic crystal is an excellent test bed to explore a cavity Quantum Electro-Dynamics (QED) in an artificial environment [51].Here we consider an analogue to a cavity QED as a cavity QCD.We can construct a one-dimensional cavity, for example, as a Fabry-Perot interferometer, where |L , |R , and |O states are realised.The ray is propagating in the cavity along z, and reflected back to propagate along the opposite direction of −z.The chiralities of spin and orbital angular momentum will be reversed upon reflections [15][16][17][18][19][20][21][22][23][24][25], such that the state along −z would be a conjugate state to the state along z.Consequently, we will be able to construct multiplets similar to mesons, made of quarks and anti-quarks [4,5,9,10].For quarks, an individual quark is very difficult to be where we have assumed θ is infinitesimally small and considered only the first order in the expansion, and Fa is an adjoint operator, whose matrix element becomes ( Fa ) bc = f abc , which is a matrix of (n 2 − 1) × (n 2 − 1).Therefore, the rotation of the wavefunction in SU(n) becomes the rotation of the corresponding expectation value in SO(n 2 − 1), as we expected.
We have also checked its validity in the second order of θ as and therefore, the above formula is also valid in the second order.Actually, this is the reflection of the differentiability of the Lie group, which was originally called as an infinitesimal group.Once a formula is derived in the infinitesimal small value, it is straightforward to extend it to the finite value.In our case, we can repeat the infinitesimal amount of rotation with the angle of θ/N , while we can repeat N times, and we take the limit N → ∞ as Xb F = lim Therefore, we have proved that the quantum mechanical rotation of the wavefunction in SU(N), which is given by C n on S ( n − 1) upon the normalisation, will rotate the expectation value of the generator, which is a vector of R n 2 −1 , in SO(n 2 − 1), using the adjoint operator of su(3) Lie algebra.

VI. CONCLUSION
We have proposed to use photonic orbital angular momentum for exploring the SU (3) states as a photonic analogue of QCD.We have shown that the 8-dimensional Gell-Mann hypersphere in SO(8) characterises the SU(3) state, made of left-and right-vortexed photons and no-vortexed photons.There are several ways to visualise the Gell-Mann hypersphere, and we have calculated expectation values for orbital angular momentum and defined hyperspin to represent the coupling between vortexed and no-vortexed states, which could be shown on 2 Poincaré sphere or 1 hypersphere in SO (6) or SO (5).We believe the proposed superposition state of photons are useful to explore photonic many-body states to have some insights on the nature of the symmetries in photonic states.

FIG. 1 .
FIG. 1. SU(3) states with photonic orbital angular momentum.(a) Fundamental multiplet of the su(3) algebra.Fundamental basis states of |ψ 1 , |ψ 2 , and |ψ 3 are shown on the (t 3 , t 8 ) plane, characterised by their quantum numbers.t 3 is known as isospin for quarks.The states are shown by points, given by the eigenvalues, which are separated by the same distance and form an equilateral triangle as their topology, implying the states are treated in an equal footing in the su(3) algebra.We can use u 3 , v 3 , or hyperchrage of y = 2(u 3 + v 3 )/3 instead of t 8 , but only 2 vectors are required to span the t 3 -t 8 plane due to the rank 2 character of the su(3) algebra.(b) Bending of the quantisation axis of SU(2) to form SU(3) states.The quantum number ( 3 ) of orbital angular momentum along the quantisation axis is usually characterised by SU(2) states, as shown by the upper diagram.By allowing the SU(2) rotation between left-and right-vortexed states, we will effectively bend the 3 to realise the superposition state.By combining superposition states with no-vortex state, we will mix the 3 orthogonal states to realise SU(3) states.
symmetry and 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 of ∆m = ±1, and instead, ∆m = ±2 is required.This could be achieved by using a spiral phase plate[43] with topological charge of m = 2.Alternatively, it is possible to make a superposition state between |L = |1 and |R = | − 1 , and SU(2) states could be realised by controlling the amplitudes and the phases[30,31].For our considerations in the SU(3) states, this corresponds to bend the quantization axis, ˆ 3 , for allowing 3 states to couple each other (Fig.1(b)).Now, we proceed to consider coupling between the no-vortex state and left-or right-vortex states.This corresponds to change the hypercharge and the topological charge.We would like to propose call these SU(2) couplings to hyperspin, since they exhibit spin-like SU(2)

FIG. 2 .
FIG. 2. Renormalisation of Gell-Mann parameters.8-dimensional Gell-Mann hypersphere could be reduced to 2 Poincaré spheres for (a) photonic orbital angular momentum and (b) hyperspin.The radius of the Poincaré sphere for orbital angular momentum is cos 2 (θ y /2), while it is 1/2 for hyperspin.The maximum and minimum of y 3 correspond to hypercharge of 1/3 and -2/3, which are equivalent to topological charge of 1 (pure vortex of |L or |R ) and 0 (no vortex, |O ), respectively.

2 θy 2 sin (θ l ) sin (φ l ) cos 2 θy 2 cos (θ l ) cos 2 θy 2 sin 3 2
cos (θ y ) in closed loops, the expectation values of SU(2) states are represented on the sphere, represented by the SO(3) group.Consequently, the original topology of the wavefunction on S 3 is reduced to the Poincaré sphere of S 2 in expectation values.

2 θy 2 sin 2 cos (θ l ) cos 2 θy 2 4 3 − cos 2 θy 2 cos (φ y ) sin (θ y ) 4 3 − cos 2 θy 2 sin
(θ l ) sin (φ l ) cos 2 θy (φ y ) sin (θ y ) observed in experiments due to the strong confinements in composite materials of mesons and baryons.On the other hand, we expect an opposite behaviour, since photons trapped inside the cavity are difficult to observe as is, while photons escaping from the cavity are observed and analysed by detectors.This corresponds to see an individual quark, which is a ray of photons propagating at either z or −z.It is quite hard to observe the composite meson analogue, which is realised inside the cavity and it would be difficult to allocate detectors to see photons propagating along the opposite directions at the same time, which would require a transparent detector.But, it would not be essential to observe within the cavity, since we can examine the state inside the cavity from the photons escaping from the both ends.The cavity QCD experiments will open to explore SU(3) and SU(2)×SU(3) is transferred in the final state as Xb F = F| Xb |F .= I|e i Xaθ Xb e −i Xaθ |I .≈ I|(1 + i Xa θ) Xb (1 − i Xa θ)|I + O(θ 2 ) ≈ I|( Xb + iθ[ Xa , Xb ])|I + O(θ 2 ) ≈ (δ bc − c f abc θ) Xc I + O(θ 2 ) ≈ c e − Faθ bc Xc I + O(θ 2 ),

TABLE III .
Degrees of freedom for photons with 3 orthogonal states.
Poincaré rotator, which allows an arbitrary rotation of polarisation state by realising SU(2) rotations in a combination of half-and quarter-wave plates and phase-shifters[31, 40-  42].If we use the Poincaré rotator for the ray with SU(3) states of vortices under certain polarisation, we can realise the SU(2)×SU(3), since spin and orbital angular momentum are different quantum observables, such that a general state is made of a direct product state for spin and orbital angular momentum.We can also envisage to realise a state made by a sum of these states with different spin and orbital angular momentum.For example, if we realise the SU(2) state of left-and right-vortices and assign horizontally and vertically polarised states, respectively, we can realise both singlet and triplet states by controlling the phase among 2 different many-body states.
V. DISCUSSIONS A. SU(2)×SU(3) and higher dimensional systems So far, we have assumed the ray is polarised such that the polarisation state is fixed.We can control the polarisation state by a phase-shifter and a rotator.We have recently proposed a B. Cavity QCD and photonic mesons