We explain the simple method to change the polarisation state together with the intensity in the Stokes space. We assume specific experimental parameters to make the argument easy to understand, but it is straightforward to change parameters and to construct more generic theories. First, we prepare the input wave with the power of P _in = 1.5 mW with the diagonally (D) polarised state, such that the input state of |Input⟩ is prepared as P in , π 2 , 0 = P in 2 1 1 . (11) Then, we split the input wave into two waves by using a polarisation-independent directional fibre-to-fibre coupler (FFC). We used the FFC of the splitting ratio of 90:10, which means that 90% of the signal is transmitted to the through port 3 and 10% is coupled to the tap port 4, when we inserted from the input port 1, while the isolated port 2 is not used. We define the coupling constant of α = 0.1 to account for the FFC, and the splitting is simply defined by a matrix operation, | Port   3 〉 | Port  4 〉 = 1 − α − α α 1 − α | Port   1 〉 | Port  2 〉 , (12) where two components of polarisation states |P _in, π/2, 0⟩ is used for |Port 1⟩, and |Port 2⟩ is 0 = (0,0) ^t , where ^t stands for the transpose of the vector. The output from the tap port 4 is used to manipulate its polarisation state by a phase-shifter, D ̂ n ̂ 1 , δ ϕ p = 1 ⁡ cos δ ϕ p 2 − i σ ̂ 3 ⁡ sin δ ϕ p 2 , (13) which is the rotation along S ₁ in the S ₂–S ₃ plane [81]. We defined the rotation angle of δϕ _p since this corresponds to the rotation along the poloidal direction, as we shall see as follows. We have previously shown that we can construct a passive phase-shifter by a combination of two quarter-wave plates (QWPs) and two half-wave plates (HWPs) [81]. The amount of rotation in SO(3) is determined by the physical angle of the rotation (δΨ_p) of one of the HWPs as δϕ _p = 4δΨ_p [81]. Then, the output state of the tap port 4 becomes | P o r t 4 ′ 〉 = D ̂ n ̂ 1 , δ ϕ p | Port   4 〉 , (14) while the output from the through port 3 is preserved to keep its polarisation state as |Port 3′⟩ = |Port 3⟩. Then, we recombine the through port 3 and tap port 4 by the inverse arrangement, | P o r t 1 ′ 〉 | P o r t 2 ′ 〉 = 1 − α α − α 1 − α | P o r t 3 ′ 〉 | P o r t 4 ′ 〉 , (15) to expect that the port 1′ is the main output, while the port 2′ is not used. Finally, the spin expectation values of |Port 1′⟩ are calculated, as shown in Figure 1.

We confirmed that the polarisation states are mostly located near the diagonally polarised state, |D⟩ since we assumed 90% of the input for the through port 3 is preserved. Without the phase-shift of the tap port 4, we confirmed that the polarisation state and the intensity are not affected by the input state. On the other hand, if we closely look at the circular trajectory, we confirm that the trajectory is inside the original Poincaré sphere, which means that the radius, corresponding to the intensity, is successfully decreased. The reduction in energy is confirmed in S ₀ (Figure 1B), which becomes minimum at δϕ _p = 2π for the rotation of the tap port 4, where the polarisation state is purely diagonally polarised. This results from the double covering of SU(2) to SO(3), discussed previously [75]. The circular rotation means that the input state of |D⟩ returns to the original state in the SO(3) space; however, in the real physical space, it is enough for the complex electric field for the linearly polarised diagonal state to rotate only for the rotation angle of π to become the diagonal state upon the rotation. This means that the electric field will be flipped to change the sign, which is not visible on the original Poincaré sphere. On the other hand, this sign change is observable by using the portion of the original input wave, which is the role of the transmitted wave through port 3. The destructive interference becomes maximum at δϕ _p = 2π, yielding the minimum of S ₀. The entire trajectory requires the 4π-rotation to close, as expected for the SU(2) group, and it became a polarisation circle ( S 1 ) , which resides in the S ₂–S ₃ plane. The operation process corresponds to a Mach–Zehnder interferometer with a polarisation control, associated with a phase-shift, and we term it polarisation interferometer (Figure 2). The poloidal polarisation circle is realised after the second FFC as shown in Figure 2.

We acquire a scheme to control the radius of the polarisation states, simply by mechanically rotating a HWP, which corresponds to the change in the intensity. The total energy must be conserved upon the linear operations, and the reduced intensity exits from the isolated port 2′, which is terminated. Consequently, the output intensity from the output port 1′ could be reduced upon addition to our polarisation interferometer. Thus, if we focus on the output waves, the system is not only based on unitary operations but also allows a loss mechanism to reduce the radius, the direction of which is perpendicular to the surface of the sphere along polar and azimuthal directions, controlled by θ and ϕ for the polar coordinate or by γ and δ in the HV bases. In this sense, our system is non-Hermitian, and the original rotational symmetry of the polarisation state is also broken to expect a loss in a particular direction. In this way, we obtain a method to potentially scan an entire Euclidean coordinate in the Stokes space inside the original Poincaré sphere, spanned by the radius of the input wave. Due to the 3D nature of the Stokes space, we can consider various topologically non-trivial structures, which we will explore as follows.

As the first non-trivial topological structure of polarisation states, we explain how to construct a polarisation torus as a set of polarisation states in the Stokes space. In topology, a torus is made of T 2 ≅ S 1 × S 1 , which requires topological groups to describe two circular rotations, orthogonal to each other. In the previous subsection, we constructed rotators to describe the rotation along the poloidal direction, and therefore, we just need to add rotators to describe the rotations along the toroidal direction (Figure 3). This is achieved simply by applying a conventional rotator along the S ₃ axis, D ̂ n ̂ 3 , δ ϕ t = 1 ⁡ cos δ ϕ 2 − i σ ̂ 2 ⁡ sin δ ϕ t 2 , (16) where δϕ _t is the amount of rotation along the toroidal direction to the poloidal polarisation circle. We have previously shown that two successive operations of HWPs are equivalent to a proper rotator operation to form the SO(2) group [81] rather than a pseudo-rotator realised by one rotated HWP [60–62], and we just need to mechanically rotate one of the HWP to realise the target amount of rotation along the S ₃ axis (the last part of rotating optical plates in Figure 2). The output polarisation state after the operation becomes | O u t p u t 〉 = D ̂ n ̂ 3 , δ ϕ t | P o r t 1 ′ 〉 , (17) and we can finally calculate the Stokes parameters from this output state (Figure 3).

We confirmed that the calculated vectorial components of Stokes parameters form a polarisation torus (Figures 4A–C), and the intensities of the output satisfy S 0 = S 1 2 + S 2 2 + S 3 2 since we consider a coherent state. As shown in the colour map of Figure 4A, S ₀ depends on the location in the torus, and thus, we could realise a non-trivial topological structure with g = 1. For the definition of a torus, the states inside the torus are empty, as observed in Figure 4C. It is also evident that the torus is compact and so closed as a set, such that even if we extend the amount of rotations for δϕ _p and δϕ _t beyond 4π and 2π, respectively, we cannot generate new polarisation states outside the torus. It is also evident from the construction of the torus that D ̂ ( n ̂ 1 , δ ϕ p ) and D ̂ ( n ̂ 3 , δ ϕ t ) from U ( 1 ) ≅ S 1 groups, such that the two successive operations could be performed by one operation, and the inverse of a rotation could be defined. This was also true for a spherically symmetric Poincaré sphere, where the rotator and the phase-shifter are physical realisations of Lie group operations of SU(2) [75, 81, 82]. In the present case, the torus is realised upon the interference to reduce its intensity, which is a non-reversal process, such that these rotational operations must be completed before the interference occurred. As far as rotations are made before the interference, we can consider alternative operations. For example, we could first rotate the polarisation state of the input along the S ₃ axis to control along the toroidal direction, and then, we could split it into two waves to allow the poloidal rotation.

We have also plotted the calculated Stokes parameters on the Poincaré sphere after normalisation at each output state (the inset of Figure 4A). In this case, we cannot discuss differences in relative intensities, and the trajectories of the torus collapsed to form a belt on the Poincaré sphere with no width. The mapping corresponds to a projection from R 3 to S 2 , and the information on the radius, corresponding to N or P, will disappear. One can always consider this mapping from a torus to a belt, but this ends up considering non-trivial topological structures in R 3 . We will return to this point when we discuss the Chern number [37], the Pancharatnam–Berry phase [38, 39], and the Gauss–Bonnet theorem [44] towards the end of this paper. Here, we emphasize the non-trivial topological feature that appeared in the Stokes space, which uses R 3 in SO(3) rather than C 2 for U(2) wavefunctions, and for bosons, it is meaningful to consider the difference in N due to their Bose–Einstein statistics.

As shown in Figures 5A–F, we have also calculated the polarisation torus by using the rotated QWP as the final rotation in Figure 2 instead of the rotator. In this case, the trajectories by a rotated QWP for the input of the D-state assume an “8”-like structure with two holes [81], such that the rotation of the polarisation circle of Figure 1A upon the rotated QWP forms two toruses (Figure 5A). Unfortunately, this structure is not a torus of g = 2, but it is simply two toruses of g = 1 overlapping each other, since the simple moves of the circle intersect near the D-state. As shown in Figure 5C, the trajectories have cross-sections, such that the topological feature is closed. In order to claim that the structure is a torus of g = 2, the inside of the structure must be a complete hollow without intersections. By introducing an ellipticity for the input polarisation state, we can introduce the asymmetry for the “8”-like structure (Figure 5B) with a larger hole and a smaller hole [70]. However, this asymmetry also has intersections to separate the hollows. Nevertheless, closed links, made by trajectories of polarisation states, are located on the surface of the torus (g = 2) without intersections, as shown in Figures 5D, E. Figure 5D was drawn for four values of the phase-shifters, while the QWP was fully rotated for each link. In experiments, we have confirmed the trajectories of Figure 5D for the input of the diagonally polarised state. Figure 5E was obtained by gradual rotations of QWP while phase-shifting at the interferometer. In this example, 25 rotations were made by phase-shifters to rotate the torus of g = 2, locally long the poloidal direction, while the entire link is closed at the diagonally polarised state. Similarly, we can realise a trefoil knot [55, 102, 103] by closing the link on the surface of the torus (g = 1), while phase-shifting in the interferometer (Figure 5F).

The experimental Stokes parameters are shown in Figure 6. We have set δϕ _p at 0, π, 2π, and 3π, which corresponds to the rotation of HWP δΨ_p at 0, π/4, π/2, and 3π/4, respectively. At each δϕ _p, we have mechanically rotated the HWP of the rotator to change δϕ _t and obtained the trajectories by recording the Stokes parameters using PM, while rotating physically. As expected, from the proper rotator operation of two successive operations of HWPs to form SO(2) [81], each trajectory is a circle, located parallel to the S ₁–S ₂ plane (Figure 6B). On the other hand, the change in δϕ _p corresponds to the rotation along the poloidal direction, and the intensity has been changed upon the interference, which is confirmed by the small empty region in the cross-section of the torus (Figure 6C). The maximum output power was 1.5 mW, such that the minimum insertion loss was about 0.8 dB, and the overall feature of the observed polarisation torus is in reasonable agreement with the calculated results (Figure 3).

Now, we can provide more details on the realisation of the polarisation torus based on the U(2) theory along with our experimental preparations of HWPs and QWPs. We explain the free space operations in the polarisation interferometer of Figure 2. We specify the alignment of these waveplates by the angle of the fast axis (FA), measured from the horizon seen from the detector side (opposite to the LD source side) of the plates [75, 81]. In our convention, we assume that the angle is 0, if the FA is aligned horizontally, and equivalently, the slow axis (SA) is aligned vertically, and the direction of rotation is positive, if it is rotated anti-clock-wise, as seen from the top of the detector side. The first free space operations were conducted by a sequential application of QWP (whose FA is aligned to −pi/4), HWP (aligned horizontally), HWP (rotated δΨ_p), and finally QWP (whose FA is aligned to pi/4), and these operations are given by | P o r t 4 ′ 〉 = D ̂ n ̂ 2 , π / 2 D ̂ n ̂ 3 , 2 δ Ψ p D ̂ n ̂ 1 , π D ̂ n ̂ 3 , − 2 δ Ψ p D ̂ n ̂ 1 , π D ̂ n ̂ 2 , − π / 2 | P o r t 4 〉 , (18)

whose operations are equivalent to D ̂ ( n ̂ 1 , δ ϕ p ) , and the physical rotation of π for δΨ_p is enough to realise the equivalent rotation of 4π for δϕ _p along the poloidal direction.

Similarly, the second free space operation can be decomposed into two sequential operations of the HWP (aligned horizontally) and HWP (rotated δΨ_t), | O u t p u t 〉 = D ̂ n ̂ 3 , 2 δ Ψ t D ̂ n ̂ 1 , π D ̂ n ̂ 3 , − 2 δ Ψ t D ̂ n ̂ 1 , π | P o r t 1 ′ 〉 , (19)

to realise the proper SO(2) rotation [81] by D ̂ ( n ̂ 3 , δ ϕ t ) , and the physical rotation of π/2 for δΨ_t is enough to realise the equivalent rotation of 2π for δϕ _t along the toroidal direction.

A torus is obviously a topological structure, such that it is expected that the topological structure is strong against distortions. As the first step to confirm the topological robustness, we have inserted the additional QWP with its FA rotated π/4 towards the end of the device, such that the output becomes | P o r t 4 ′ 〉 = D ̂ n ̂ 2 , π / 2 D ̂ n ̂ 1 , δ ϕ p | P o r t 4 〉 , (20) which means that the polarisation states should be rotated along the S ₂ axis with the amount of π/2, and consequently, the S ₁–S ₂ plane is rotated to be the S ₂–S ₃ plane. The experimental results on the rotated torus are shown in Figure 7. We confirmed that the structure of the torus remained unchanged in the Stokes space upon the rotation, while the toroidal direction is now located parallel to the S ₂–S ₃ plane.

Similarly, we have also confirmed that the rotation along the S ₁ axis preserves the topological structure of the torus. This was realised by adding the QWP with its FA aligned horizontally, and the expected output state becomes | P o r t 4 ′ 〉 = D ̂ n ̂ 1 , π / 2 D ̂ n ̂ 1 , δ ϕ p | P o r t 4 〉 , (21) and the experimental results are shown in Figure 8. In this case, the principal axis of the toroidal rotation is along the S ₂ axis, which is orthogonal to the axes of previous toruses (Figures 6, 7). Therefore, the torus was not distorted upon the applications of rotated QWPs.

Theoretically, the phase-shifter and the rotator merely rotate Stokes parameters upon the application of D ̂ ( n ̂ , δ ϕ ) , such that it results in rotations of the vectorial point (S ₁, S ₂, S ₃) along some direction n ̂ with the amount of δϕ. This linear and unitary operation cannot change the topology of a set of points in the Stokes space, such that a torus or a sphere would be transferred to the same topological structure, respectively, without changing its genus.

We have also tried to realise the double-connected toruses with g = 1 by a rotated QWP. Here, we expect the output state of | O u t p u t 〉 = D ̂ n ̂ 3 , 2 δ Ψ t D ̂ ( n ̂ 1 , π / 2 ) D ̂ n ̂ 3 , − 2 δ Ψ t | P o r t 1 ′ 〉 , (22) which makes two connected toruses. The corresponding experimental results are shown in Figure 9. This corresponds to the theoretical calculations of Figure 5D. As shown by the colour difference of Figure 5D, the intensities of each trajectory are different, meaning that the trajectories are not intersecting at the D-state, and therefore, the trajectories are located on the surface of the torus of g = 2. If we increase the number of steps for the poloidal control apart from 4 (Figure 9), the trajectories will be eventually filled, as we have discussed for Figures 5A–C. In order to overcome the closure of the surface, the linked trajectory to cover the surface of the torus of g = 2 (Figure 5E) is required. In our current experimental setup, it was difficult to confirm the trajectory of Figure 5E since we needed to manually rotate the waveplates physically, such that it was tough to allow correlated simultaneous rotations for both poloidal and toroidal directions. In order to allow these experiments, we need dynamic control of polarisation states by optical modulators. We are developing a dynamic polarisation controller by using an LiNbO₃ optical modulator [82]. It will be possible in the future to allow the complicated polarisation control by combining the proposed polarisation interferometer together with the dynamic Poincaré rotator.

We consider how to realise a Möbius strip in the Stokes space. A Möbius strip [41, 44, 48] is made of a strip with the one end, flipped for connecting to the other end. We consider the same setup as that shown in Figure 2, and we just need to change the optical components in the free space regions. For a Möbius strip, we need to prepare a line segment, rather than a circle (Figure 1), prepared for the torus. Such a line segment can be made by the simple phase-shift to the wave out of the tap port 4 by the phase-shift of D ̂ ( n ̂ 2 , δ ϕ p ) along the S ₂ axis. This operation will not change the input polarisation state of the D-state, while the phase is shifted, which changes the intensity of S ₀. The line segment should be rotated for the toroidal direction along the S ₁ axis, such that the operations become | P o r t 4 ′ 〉 = D ̂ n ̂ 1 , δ ϕ t D ̂ n ̂ 2 , δ ϕ p | P o r t 4 〉 . (23) After these operations, the bypassed wave should be recombined by the subsequent FFC with the output from the through port 3. The combined wave should be rotated by the final rotator with the amount of δϕ _t, which is the same operation as that for the torus. The Stokes parameters were calculated from these U(2) wavefunctions, and the trajectories become the polarisation Möbius strip, as shown in Figure 10A. We can recognise a standard feature of a Möbius strip, designed in the Stokes space.

In our original consideration for the torus, we have controlled δϕ _p from 0 to 4π, while δϕ _t was changed from 0 to 2π. These parameters also work for the Möbius strip, and this will cover the Möbius strip twice, due to the collapsing of the pore of the torus for the strip. Consequently, the half-rotation for δϕ _p is enough to realise the Möbius strip. It is even better to consider the quarter-rotation of δϕ _p from 0 to π, and instead, the double rotation of δϕ _t from 0 to 4π could be considered. In this case, it is easier to track the trajectory, controlled by δϕ _t. For example, if we start from the point, realised by δϕ _p = 0 and δϕ _t = 0, the original input state is recovered, which is located at the edge of the Möbius strip, and the intensity is maximised. Then, it is easy to see the trajectory (Figure 10B) of showing how δϕ _t changes the point, moving from the outer edge to the inner edge, and coming back to the original point upon the application of the 4π-rotation. On the other hand, if we start from δϕ _p = π and δϕ _t = 0, it is located at the centre of the Möbius strip, and the trajectory becomes the circle upon the change of δϕ _t from 0 to 2π, and it rotates twice upon the 4π-rotation (Figure 10C).

It is well-known that a Möbius strip cannot be assigned its orientation, which is evident from the fact that we cannot distinguish the front surface from the back surface. On the other hand, we can define its chirality, which depends on how the line segment could be connected upon rotations in our case. The Möbius strip, as shown in Figure 10, is defined to be right-handed because the line segment was rotated to the right side, seen from the direction, opposite to the toroidal rotation. Correspondingly, the left-handed Möbius strip could be considered, simply by the opposite rotation to yield | P o r t 4 ′ 〉 = D ̂ n ̂ 1 , δ ϕ t D ̂ n ̂ 2 , − δ ϕ p | P o r t 4 〉 , (24) which was used to calculate its Stokes parameters, as shown in Figure 11. If we focus on the trajectory of the edge, starting from the outer edge at δϕ _p = 0 and δϕ _t = 0, polarisation states rotate along the bottom (negative S ₃), upon the toroidal direction, to arrive at the inner edge, and continue to go up to the top (positive S ₃), towards going back to the original point. This is the opposite chirality to that of the right-handed Möbius strip, shown in Figure 10. Therefore, the chirality could be controlled, when we realise the Möbius strip by defining the sign of rotation to flip the line segment.

Next, we consider realising a Hopf link using polarisation states. A Hopf link is made of two rings, which are completely disconnected, while one ring intersects with the other (Figure 12). We think it is impossible to realise it solely from 1 wavelength since we cannot allow two different points with different N (or equivalently, P), while keeping the same angles for γ and δ. Therefore, trajectories, controlled by the polarisation states, would be continuous in the Stokes space for 1 wavelength. However, if we allow wavelength-division multiplexing (WDM) in the SMF, we can separately manipulate polarisation states for multiple wavelengths, leading the way to realise optical Hopf links. We just need to adjust relative powers and polarisation states for both wavelengths by considering to establish the topology between two waves.

As an example, we consider a polarisation circle realised by the polarisation interferometer, shown in Figure 2. Here, we assume 1 wavelength of approximately 1530 nm with the input power of 1.5 mW to be controlled by the polarisation interferometer. The output of the tap port 4 is now controlled as | P o r t 4 ′ 〉 = D ̂ n ̂ 3 , δ ϕ p | P o r t 4 〉 , (25) while we do not need to rotate along the toroidal direction, such that we do not insert any optical components in the second free space region. This will create a polarisation circle in the S ₁–S ₂ plane, perpendicular to the direction of S ₃ (the small circle of Figure 12A). The polarisation circle in the Stokes space is a line segment in the normalised Poincaré sphere (the blue line in the inset of Figure 12A). Next, we just need to prepare another polarisation circle by using a different wavelength of approximately 1550 nm at the power of 1.0 mW, which could be controlled by proposed Poincaré rotators (phase-shifters to control δ) [81, 82], separately, to allow circular changes of polarisation states, located in the S ₂–S ₃ plane, perpendicular to the direction of S ₁ (a large circle of Figure 12A). After constructing these waves in the SMFs, we can combine these by a polarisation-dependent FFC, with an appropriate power splitting ratio. These two polarisation circles do not touch each other in Stokes space, forming a Hopf link (Figure 12). On the other hand, if we plot these states by normalising Stokes parameters to have a unit radius, these two circles are connected (the inset of Figure 12). Therefore, it is important to distinguish the power difference of these waves.

Similarly, we have also calculated a Hopf link by assuming the different ratio of 50:50 (α = 0.5), as shown in Figure 12B, while the other parameters were the same as those for Figure 12A. In this case, the larger polarisation circle is realised upon the interference, due to the larger power, propagating into the tap port 4. As a consequence of the interference, the minimum output power could be 0, which is why the heart-like dip is realised near the origin of the Stokes space.

This heart shape affects the torus structure if the polarisation states are further controlled upon the toroidal rotation, as shown in Figure 13. Here, we have assumed the splitting of 50:50 (α = 0.5) for an input of the single wavelength at 1530 nm with the power of 1.5 mW. We expect that the heart-shaped polarisation circles are making trajectories upon rotations to the toroidal direction, with the amount of δϕ _t changed from 0 to 3π/2. Due to its feature, we call it a polarisation apple to have seed-like regions due to the heart-shaped dip near the origin. This is a remarkable difference between the mathematical overlapping of two circles upon rotations. In our case, we realise the circular heart-shaped circles upon the interference, such that the intensity near the origin becomes 0 due to the complete destructive interference between two separated waves with the sign change upon the SU(2) rotation.

We have several topologically non-trivial structures as polarisation states in the Stokes space, compared with the Poincaré sphere. Polarisation results from spin expectation values [56–75, 77, 81, 82], such that the non-trivial polarisation states are determined by the broken rotational symmetry for the polarisation states. These topologically non-trivial features are robust against the rotationally symmetric disturbances. For example, the polarisation-independent loss in the SMF cannot change the topology of the polarisation states. Moreover, spherically symmetric operations of phase-shifters and rotators can rotate the topological structures such as toruses (Figures 6–8), Möbius strip (Figures 10, 11), and Hopf links (Figure 12), but still, the relative topology within these structures will be kept upon rotations. It is a polarisation-dependent loss to cut these topological features. Nevertheless, it is not so easy to change the topology since a simple insertion of a polariser, for example, will completely destroy the polarisation structure, ending up to be one point in the Stokes space. We need to make an optical scissor to allow an arbitral cutting of topological polarisation states. In order to reduce the intensity of the targeted polarisation states, we need to observe the polarisation states by using a polarimeter, which corresponds to observing the wavefunctions to expect the collapse of the wavefunction. For coherent photons, we can observe a bypassed contribution via a tap port, while keeping the contribution in the through port [82], but still we need to prepare a complicated photonic circuit to allow the splitting, delay, and manipulation of the loss. These difficulties result from robust correlations among the bits in the pulse stream to form non-trivial topological polarisation states. Here, we discuss how the broken symmetry in the bulk is corresponding to the edge state [4–6, 14, 22, 41–44]. More specifically, we return to the case of the polarisation torus and consider how the polarisation states could be connected to the original Poincaré sphere.

The torus is obviously distinct from the sphere because of the non-zero genus. For the pulse streams of light, coming out of the proposed device, which is the polarisation interferometer with the polarisation rotator (Figure 2), the Stokes parameters of the pulse represent one of the points on the polarisation torus, such that the pulse streams are considered to form the bulk state of the polarisation torus, characterised by g = 1. On the other hand, in the standard SMF with the rotational symmetry, the polarisation states are well-known to be characterised by the Poincaré sphere with g = 0 as a different bulk state. If we would like to connect the torus to the Poincaré sphere, we need to prepare the edge state, where the pore of the torus is closed, such that the polarisation circle (Figure 1A) is closed to be 1 point, which must be robust against disturbances to rotate the polarisation state.

We can generate such an edge state, simply by changing the rotations in the Stokes space. One practical implementation is to use the same setup of Figure 2, while we introduce the amplitude controller, A ̂ δ ϕ p = 1 ⁡ cos δ ϕ p 2 , (26) in the first free space region. The amplitude controller is not a unitary operator, such that this operator does not belong to a family of SU(2). It is an operator of U(2), except for the parameters, δϕ _p = π, 3π, ⋯, where the operator becomes 0, such that the norm is controlled upon the operation. The amplitude controller is made of polarisation splitters to change the amplitude of each polarisation state, independently, while each polarisation state would be inserted into a polarisation rotator [75, 81, 82] to change its polarisation in the S ₁-S ₂ plane, independently, picking up only the original polarisation state after the rotation, and finally recombining orthogonal polarisation states in a combiner. Alternatively, the amplitude controller is simply made of a polarisation-independent Mach–Zehnder interferometer to control the amplitudes for both polarisation components, simultaneously, while keeping the polarisation. The amplitude controller could also be defined as B ̂ δ ϕ p = σ ̂ 1 ⁡ sin δ ϕ p 2 , (27) which accompanies a swapping of the polarisation states in addition to the polarisation rotation.

After the amplitude control of the contribution in the tap port 4, the output polarisation state is further controlled by a QWP, yielding | P o r t 4 ′ 〉 = D ̂ ( n ̂ 1 , π / 2 ) A ̂ δ ϕ t | P o r t 4 〉 , (28) which is combined with the contribution with the through port 3. Finally, the recombined state is rotated along the S ₂ axis instead of the S ₃ axis by the polarisation rotator | O u t p u t 〉 = D ̂ n ̂ 2 , δ ϕ t | P o r t 1 ′ 〉 . (29) These operations will create a topological Dirac cone near the D-state. However, it is intriguing to illustrate it near the north pole (in our convention, the left circularly polarised state [75]), which could be achieved simply by a π/2-rotation along S ₁. Alternatively, we can apply | P o r t 4 ′ 〉 = A ̂ δ ϕ t | P o r t 4 〉 , (30) to the tap port 4 without a QWP, while the through port 3 is phase-shifted by a QWP to be | P o r t 3 ′ 〉 = D ̂ ( n ̂ 1 , π / 2 ) | P o r t 3 〉 , (31) and then, the recombined states should be rotated along the S ₃ axis, just like creating a torus before.

The calculated Stokes parameters in this way are shown in Figure 14, where a topological Dirac cone is recognised in the Stokes space. Bosonic Dirac bosons were previously discussed [106, 107] in realising a single-particle spectrum of a boson. Here, we are not discussing a single-particle energy spectrum in the momentum space. Instead, we are considering the many-body energy (S ₀) of a bit in a pulse stream, generated from a device, and the change in energy is described against the spin expectation values rather than momentum for an energy band. Due to the coherent nature of bosons with no charge, photons in the same bit do not interact with each other, but the energy difference in S ₀ could be considered the difference in number of photons in each bit. As shown in Figure 14A, the polarisation circle, generated by a rotator, is closed at the Dirac point, where the light cone is closed. At the Dirac point, the polarisation state is not changed upon rotations along the S ₃ axis, such that this edge state to close the circle is topologically robust against rotations in the S ₁–S ₂ plane. We have also plotted the Dirac cone in the space for (S ₀, S ₁, S ₂), as shown in Figure 14B. We confirmed the linear energy change, seen from S ₀, against the radius of the polarisation circle in the plane, parallel to the S ₁-S ₂ plane. For the constant energy in S ₀, or equivalently, for the same number of photons in a bit, the bit is characterised in the polarisation circle of the Dirac cone, which changes the helical spin expectation values (S ₁, S ₂) upon rotations by a rotator along the S ₃ axis.

If we would like to change g = 1 of a torus to g = 0 of a sphere, it is inevitable to make such a Dirac point, where the polarisation state is robust against the rotations. We think the Dirac point corresponds to the edge state, while the polarisation torus and Poincaré sphere are bulk states. This is the bulk-edge correspondence [4–6, 14, 22, 41–44] for our topological polarisation states.

As an example, we have connected a polarisation torus to a Poincaré sphere via topological Dirac cones, as shown in Figure 15. Here, we have assumed the maximum power of 1.5, 1.05, and 0.75 mW for the polarisation torus, Dirac bosons, and the inner Poincaré sphere, respectively. The number of connected Dirac cones was 4 in this example, but it is not limited to this particular number. The number of edge states simply depended on our experimental setup and feasibility on how to close the pore generated in the torus. In the example of Figure 15A, the torus is continuously connected to four Dirac cones, with four Dirac points to close the pore, and the inner light cone is continuously connected to a Poincaré sphere with a smaller radius (S ₀). The torus (g = 1) and the sphere (g = 0) describe different bulk states, respectively, while the Dirac points are edge states. Even if we rotate the whole structure in the 3D (S ₁, S ₂, S ₃) space, the topology of these states will not be changed at all, and the polarisation-independent loss merely changes the scale (radius by S ₀), such that the topology will not be changed, either. Consequently, these topological features in the Stokes space will be robust during the propagation in the SMF, regardless of the polarisation rotations and the loss.

Another method to close the torus is to introduce a phase-shift. For example, we can introduce a phase change upon the toroidal rotation as | P o r t 4 ′ 〉 = D ̂ n ̂ 2 , δ ϕ t D ̂ n ̂ 1 , δ ϕ p | P o r t 4 〉 , (32) which induces the destructive rotation upon the toroidal rotation, leading to a generation of a Dirac point at the diagonally polarised state (Figure 15B). The structure of Figure 15B is not a torus anymore since the pore is closed, but we can introduce the toroidal phase change upon the dynamic operation [82]. Then, the polarisation torus can be dynamically switched to the conventional Poincaré sphere, continuously, along with the time evolution. By considering time as another coordinate, inspired by the time crystal [34, 35], we can dynamically switch polarisation structures with different genera. It is important to recognise that the edge must exist between different topological structures, and in order to close the torus continuously for the sphere, we need at least one Dirac point.

We consider the topological invariance discussed in this paper. In the physics of topological materials, the Chern number [37] is the topological invariance to characterise the non-trivial quantum states. The Chern theorem [37] was established through a generalisation to C numbers for wavefunctions [1, 2, 4–6, 8, 11, 12, 14, 15, 36, 38–44], while the classical Gauss–Bonnet theorem is valid in R numbers. We focus on the polarisation torus to identify the topological invariance.

First, we evaluated the Chern number for the polarisation torus. To calculate the Chern number, we need to integrate the Pancharatnam–Berry phase of γ along a closed loop. We consider the poloidal rotation, as shown in Figure 2A, whose output state after the polarisation interferometer is given by | ϕ p 〉 = | P o r t 1 ′ 〉 , (33) for the total rotation of ϕ _p, and we consider the small deviation of δϕ _p from ϕ _p, and the Berry connection [4–6, 38–41, 44] is defined by d A = 〈 ϕ p | ϕ p + δ ϕ p 〉 − 〈 ϕ p | ϕ p 〉 〈 ϕ p | ϕ p 〉 ≡ 〈 δ δ ϕ p 〉 ⋅ δ ϕ p , (34) which yields the Pancharatnam–Berry phase as i γ = ∮ d A (35) to give the Chern number C = γ 2 π = ∮ d A 2 π i , (36) which is the winding number for the wavefunction along the closed trajectory in the Hilbert space. We confirmed that the Chern number for the topological torus is 0 upon numerical calculations. This is confirmed on the normalised Poincaré sphere of Figure 4 (and the blue line in the inset of Figure 12A) because the circular rotation in the Stokes space along the radial direction (Figure 2A) simply corresponds to the line integration in the normalised Poincaré sphere, whose solid angle vanishes. Due to the uncertainty of 4π in solid angle and the nature of the 2-level systems, the Chern number of polarisation torus is given by integers ( Z ) ; thus, we obtain C ∈ Z . The line integration over the poloidal direction can be converted into the surface integration over the torus, such that the integer Chern number characterises the nature of the polarisation torus. This is exactly the same as that for the rotationally symmetric Poincaré sphere, such that we have no difference in the wavefunction, which is not surprising in the definition of the Berry connection, which is defined as the overlap of the normalised wavefunctions upon a trajectory over the phase space. If we take the integration contour over the toroidal direction, rather than the poloidal direction, the Pancharatnam–Berry phase becomes finite in agreement with the solid angle, defined by the toroidal trajectory. However, in this case, the toroidal loop cannot cover the whole surface of the torus, such that we cannot apply the Stokes theorem to characterise the topology of the torus. Consequently, it is reasonable to use the contour over the poloidal direction, and the Chern number of the torus is the same as that of the full sphere of the Poincaré sphere. Therefore, the normalised SU(2) wavefunction is not useful to characterise the polarisation torus, since the non-trivial nature of the polarisation torus in topology could be considered only when we take the variable radius of the U(2) wavefunction into account for the coherent many-body states with Bose–Einstein statistics.

In fact, the topological nature of the polarisation torus is appeared in the Stokes space, where the spin expectation values could take potentially any values in R 3 . For characterising the topology in the real space, we can use the Gauss–Bonnet theorem to obtain the Euler number, χ = 2 1 − g = ∫ d S 2 π K , (37) where K is the product of the minimum and maximum curvatures on the surface and dS is the infinitesimal surface area. For the torus, the curvatures upon the toroidal direction change their sign, such that the integration becomes 0, yielding χ = 0 and g = 1. The values are completely different for a sphere to have χ = 2 and g = 0. Thus, the Euler number and the genus should be appropriate as topological invariants in the polarisation torus. Other topological features are also considered in R 3 , such that these numbers will be useful to consider polarisation states in the Stokes space.