Abstract
Vector beams, nonseparable in spatial mode and polarisation, have emerged as enabling tools in many diverse applications, from communication to imaging. This applicability has been achieved by sophisticated laser designs controlling the spin and orbital angular momentum, but so far is restricted to only twodimensional states. Here we demonstrate the first vectorially structured light created and fully controlled in eight dimensions, a new stateoftheart. We externally modulate our beam to control, for the first time, the complete set of classical Greenberger–Horne–Zeilinger (GHZ) states in paraxial structured light beams, in analogy with highdimensional multipartite quantum entangled states, and introduce a new tomography method to verify their fidelity. Our complete theoretical framework reveals a rich parameter space for further extending the dimensionality and degrees of freedom, opening new pathways for vectorially structured light in the classical and quantum regimes.
Introduction
In recent years, structured light, the ability to arbitrarily tailor light in its various degrees of freedom (DoFs), has risen in prominence^{1,2,3}, particularly the vectorially structured light^{4,5,6}, which is nonseparable in spatial mode and polarisation. A popular example is vector vortex beams, a vectorial combination of spin and orbital angular momentum (OAM) states, as a form of a twodimensional classically entangled state^{7,8,9,10}. Sharing the same hallmark of nonseparability of quantum entanglement, the classically entangled vector beam is more than simple mathematical machinery and can extend a myriad of applications with quantumclassical connection. Such states of vectorially structured light have been created external to the source through interferometric approaches by spin–orbit optics^{11,12,13,14}, as well as by customised lasers^{15} including custom fibre lasers^{16}, intracavity geometric phase elements in solidstate lasers^{17,18,19,20} and custom onchip solutions^{21,22,23}. The resulting beams have proved instrumental in imaging^{24}, optical trapping and tweezing^{25,26}, metrology^{27,28,29}, communication^{30,31} and simulating quantum processes^{8,9,10,32,33,34,35,36}. In the quantum regime, they are referred to as hybrid entangled states and have likewise been used extensively in quantum information processing and cryptography^{37,38,39}.
Despite these impressive advances, the prevailing paradigm is limited in twoDoF (bipartite) and twodimensional (2D) classically entangled states of light, the classical analogy to twophoton qubit entanglement, which has proved useful in describing such beams as states on a sphere^{40,41}. The ability to access more DoFs and arbitrarily engineered highdimensional state spaces with vectorial light would be highly beneficial, opening the way to many exciting applications in simulating multipartite quantum processes in simpler laboratory settings^{42,43}, in advancing our understanding of spin–orbit coupling through new coupling paradigms between spin and the trajectory of light^{44,45}, and in accessing more DoFs and dimensions in the singlephoton and coherent states for highcapacity communication^{46,47,48,49}. To do so requires the creation and control of new DoFs in vectorially structured light.
Existing vectorial control is very powerful^{6} but does not easily extend the DoFs. One could carry out the spatial manipulation of light to partition the spatial DoFs, e.g., the scalar modes into their two indices (n and m for the HermiteGaussian modes, p and ℓ for the LaguerreGaussian modes), but the DoFs remain limited to three^{39,50}, and independent control is practically impossible with the present tools, e.g., how can one change the phase of only the radial modes and not the azimuthal modes in the LaguerreGaussian basis? It is possible to extend the DoFs by the timefrequency or wavelength control of light^{51,52,53}, but this is nontrivial and involves nonlinear materials. One could split the beam into multiple paths^{54,55,56,57,58,59,60,61} but then the DoFs would no longer be intrinsic to one paraxial beam and control would become increasingly complicated and problematic. A recent work extended the DoFs up to three but still limited to 2D states^{62}, that cannot be fully controlled in highdimensional space. The open challenge is therefore to find unlimited DoFs that are easy to control, intrinsic to a paraxial beam, and have the potential to access high dimensions with classical light, a topic very much in its infancy.
Here, we introduce the notion that the intrinsic DoFs from a raywave duality laser can be marked and controlled for highdimensional multipartite (multiDoF) classically entangled states of vectorial light. We operate a laser in a frequencydegenerate state that is known to produce multiple raylike trajectories but in a single wavelike paraxial beam as a spatial wave packet of SU(2) coherent state^{63,64,65,66} and vectorise it by using offaxis pumping and an anisotropic crystal^{62,67,68}. However, the ray states in this beam cannot be tuned independently and therefore increase the difficulty of the arbitrary control of these highdimensional states. We propose the combination of raywave duality in a laser beam and external digital modulation to overcome this paradigm. This method allows us to produce new forms of vectorial structured light not observed before, in stark contrast to conventional vector beams expressed by spacepolarisation nonseparable Bell states (Fig. 1a). As we will show, the ray trajectory and vectorial control for a single coherent paraxial beam gives us access to new controllable DoFs to realise highdimensional classical entanglement. Specifically, we can use the intrinsic DoFs such as the oscillating direction (\(\left\pm \right\rangle\)), ray location (\(\left1\right\rangle\) or \(\left2\right\rangle\)), polarisation (\(\leftR\right\rangle\) or \(\leftL\right\rangle\) for right or lefthanded circular polarisation) and OAM and subOAM (\(\left\pm \!\ell \right\rangle\) and \(\left\pm\! m\right\rangle\)) to realise multipartite highdimensional classically entangled light from a laser (Fig. 1b). As illustrated in Fig. 1, one can view conventional vector beams (Fig. 1a) as a subspace within this more general classically entangled state of vectorial structured light (Fig. 1b).
To demonstrate that these DoFs are fully controllable in a highdimensional multipartite space, we experimentally exploit a raywave structured laser beam in a tripartite eightdimensional state and controllably modulate it to produce, for the first time, the complete set of classical Greenberger–Horne–Zeilinger (GHZ) states in vector beams, the maximally entangled states in an eightdimensional Hilbert space, as the classical analogy to multipartite or multiparticle entanglement. Our work introduces new DoFs beyond those of the traditional vector beams and creates new states of nonseparable structured light not realised before.
Results
Concept
Spatial light mode control in lasers allows one to specify the desired modal amplitude, phase and polarisation^{15}. While the spatial modes usually refer to the eigenmodes of the paraxial wave equation, there is also a class of complex spatial wavepacket modes that possess a geometric interpretation with SU(2) symmetry, a general symmetry for describing paraxial structured beams with OAM evolution mapped on a Poincarélike sphere^{69}. This geometric mode has the formation of SU(2) coherent state, with the salient property that the distribution of the wave function is coupled with a classical movement trajectory^{64,70}. This means that the spatial wave pattern can also be treated as a cluster of geometric rays, a form of raywave duality^{65,71}, a notion we will shortly exploit.
In this paper, we wish to demonstrate an eightdimensional state from the laser in order to go on and create the GHZ states. To do this we build a raywave duality laser resonator as our source and select the new DoFs to be the oscillating direction, roundtrip location, and polarisation. Such raywave duality light can be generated from an offaxis pumped laser cavity with frequency degeneracy, where the ratio of the transverse and longitudinal mode frequency spacings (Δf_{T} and Δf_{L}) is a rational number Ω = Δf_{T}/Δf_{L} = P/Q (P and Q are coprime), related to the period of the ray trajectory oscillating in the cavity (Q determines the number of rays)^{64,65}. To this end, we use the degenerate state of \(\left{{\Omega }}=1/4\right\rangle\) to illustrate our method. This is achieved when the cavity length is precisely controlled as the half of the radius of curvature of the concave mirror in a planoconcave cavity, and the ray trajectory has a period of four roundtrips in the cavity (see more general cases in Supplementary Information A). Then we carefully apply offaxis pumping to subsequently excite a Wshaped trajectory mode whose output mode can be shown to be exactly the desired SU(2) coherent state^{72}. Although raylike, the output is a paraxial beam with a coherent spatial wave packet based on the raywave duality, enabling us to label the output in terms of the ray trajectories rather than the modes themselves.
A freespace planar raywave geometric mode with the highlighted ray trajectory of \(\left{{\Omega }}=1/4\right\rangle\) is shown in Fig. 2a, and the corresponding oscillation in the laser cavity is shown in Fig. 2b, with the forward (\(\left+\right\rangle\)) and backward (\(\left\right\rangle\)) oscillating states and the first (\(\left1\right\rangle\)) and second (\(\left2\right\rangle\)) roundtrip location states forming a completed oscillation with the potential to reach dimension d = 4. In Fig. 2a, the raylike propagation is revealed by the marked trajectories, and at z = 0 and ±z_{R} (z_{R} is the Rayleigh range), the wavelike behaviour is evident by the fringes in the beam as trajectories overlap. To understand how to create this from a laser, we highlight the generation step inside a laser in Fig. 2b. Here, the output four rays originating from two locations (\(\left1\right\rangle ,\left2\right\rangle\)) on the rear mirror comprise two Vshaped locations, and the pair of rays in a certain location state have two different directions (\(\left+\right\rangle ,\left\right\rangle\)), such that the output is spanned by the basis states \({{\mathcal{H}}}_{\text{ray}}\in \{\left+\right\rangle \left1\right\rangle ,\left\right\rangle \left1\right\rangle ,\left+\right\rangle \left2\right\rangle ,\left\right\rangle \left2\right\rangle \}\), a fourdimensional Hilbert space, while in the wave picture the emitted output from the laser is a single paraxial spatial mode. By using an astigmatic mode converter consisting of two cylindrical lenses^{73}, the planar trajectory mode can be transformed into a skewed trajectory carrying OAM, as shown in Fig. 2c, d for the conversion technique and the resulting raywave vortex beam, respectively. Alongside the simulation are the measured beam profiles at selected propagation distances outside the cavity, as shown in Fig. 2e. To experimentally verify the raywave laser operation, we can use a lens to image the intracavity mode in free space and capture the transverse patterns at various propagation distances^{72}. The result of this measurement for the \(\left{{\Omega }}=1/4\right\rangle\) state is shown in Fig. 3a. The results show the telltale signs of wavelike behaviour, as evident by the interference fringes with high visibility at z = 0 and z = z_{R}, while appearing to be independent ray trajectories. Thus the laser output is a fourdimensional nonseparable scalar state.
Since one of the DoFs is polarisation, our scheme requires us to engineer the cavity to achieve raydependent polarisation control without any intracavity elements. We achieve this by exploiting the fact that differing ray trajectories will impinge on the laser crystal at angles other than normal, as well as thermal effects as a function of the pump power (see Supplementary Information E). By deploying a ccut crystal that exhibits angledependent birefringence at nonnormal incident angles, we can mark our orbits with polarisation by simple adjustment of the pump light position and power, the results of which are shown in Fig. 3b. The visibility of the fringes decreases to zero as the pump power is increased (for a particular trajectory size) as a result of the orthogonal ray polarisation states. The power control of the gain results in all rays evolving from an entirely linear horizontal state to a vector state of diagonal (\(\leftD\right\rangle\)) and antidiagonal (\(\leftA\right\rangle\)) polarisations. The results confirm that the cavity can be forced into a raylike mode with different polarisations on lights of various ray orbits. By contrast, the prior vectorial raywave lasers allowed only the complete mode to be marked, where all rays were the same, limiting the state to 2D but multiple DoFs^{62}. We derive a general analytical form for our output state (see Supplementary Information B); for the example geometry of Figs. 2 and 3, a laser mode in an eightdimensional Hilbert space spanned by the basis states in \({{\mathcal{H}}}_{8}\in \left\{\left+\right\rangle \left1\right\rangle \leftD\right\rangle ,\left\right\rangle \left1\right\rangle \leftD\right\rangle ,\left+\right\rangle \left2\right\rangle \leftD\right\rangle \right.\), \(\left.\left\right\rangle \left2\right\rangle \leftD\right\rangle ,\left+\right\rangle \left1\right\rangle \leftA\right\rangle ,\left\right\rangle \left1\right\rangle \leftA\right\rangle ,\left+\right\rangle \left2\right\rangle \leftA\right\rangle ,\left\right\rangle \left2\right\rangle \leftA\right\rangle \right\}\). We show the full states experimentally in Fig. 3c.
Controlled generation of GHZ states
Next, we show that it is possible to controllably modulate this raywave vector beam to the desired states in a highdimensional space. By way of example, we create the classical version of the famous GHZ states^{74}, with the concept and setup shown in Fig. 4. In particular, we wish to create a complete set of tripartite GHZ states, namely, \(\left{{{\Phi }}}^{\pm }\right\rangle\), \(\left{{{\Psi }}}_{1}^{\pm }\right\rangle\), \(\left{{{\Psi }}}_{2}^{\pm }\right\rangle\) and \(\left{{{\Psi }}}_{3}^{\pm }\right\rangle\), which construct the complete bases in eightdimensional space, with full details given in Supplementary Information C.
We externally modulate the output state from the laser to engineer the amplitude and phase of each term independently, producing the general state
and converting it to each of the eight GHZ basis states. For example, the transformation
requires a modulation that sets all amplitudes to zero except \( {\alpha }_{1} = {\alpha }_{8} =\frac{1}{\sqrt{2}}\) and a relative phase shift between the two decomposed ray modes. The general setup to achieve this (and other modulations) is shown in Fig. 4. The main experimental arrangement includes the laser for creating the initial highdimensional state, followed by a tailoring step to convert it into specific desirable classes, here the GHZ states are generated as an example. Finally, the states are directed to two measurement devices: the vectorial nature of the prepared states is measured by a polarizer and camera (OAMstate measure), and a Bellstate measurement device, which we introduce here, is used for the tomographic projections (see Methods section). Figure 4a–c graphically illustrate the transformations required. Using the \(\left{{{\Phi }}}^{\pm }\right\rangle\) state as an example (see full details for all cases in Methods section and Supplementary Information D), we switch from a linear polarisation basis to a circular one with a quarterwave plate (QWP), eliminate light on the ray states \(\left+\right\rangle \left2\right\rangle\) and \(\left\right\rangle \left1\right\rangle\) by iris I_{1}, and then modulate the polarisation of the \(\left+\right\rangle \left1\right\rangle\) and \(\left\right\rangle \left2\right\rangle\) paths into diagonal and antidiagonal polarized states using programmed phases on a spatial light modulator (SLM). Similar transformations allow us to generate all GHZ states in the complete family of maximally entangled states. The controllable generation of all the GHZ states provides not only the ability to shape an ondemand highdimensional structured beam but also the verification that the general SU(2) geometric beam in our system is indeed expressed in an eightdimensional space, as the GHZ states form a complete basis in eight dimensions.
Notably, both the OAM state and planar state of the raywave structured beam play important roles in identifying highdimensionality. For the OAM state, the mode carrying twisted raywave structures have a more effective raylike effect, rather than the planar state, where the spatial twisted ray bundle does not interfere upon propagation, which enables us to clearly identify the polarisation on each individual ray. On the other hand, the planar ray trajectory has a wave interference region for the included rays, and the interferometric fringes allow us to identify the phase differences among the various rays. This is the basic principle to realise higherdimensional control of the raywave structured beam rather than a common beam without raywave duality.
Experimental images of the twisted rays, together with the theoretical predictions, are shown in Fig. 4d for the \(\left{{{\Phi }}}^{\pm }\right\rangle\) states as a function of the orientation of the polarizer in the OAMstate measurement step, showing excellent agreement. The intensity distribution is a twolobed structure, consistent with the corresponding GHZ state, while the evolution of the lobe intensities confirms the vectorial nature of the field. A reconstruction of the GHZstate mode propagation in freespace is shown in Fig. 4e. The results for all other states are shown in Fig. S2 of the Supplementary Information.
To quantitatively infer the fidelity of our classical GHZ states, we introduce a new tomography based on Bellstate projections, with the fidelity results for all eight states shown in Fig. 5a. Each was calculated from a density matrix of the state, with the results of this for the \(\left{{{\Phi }}}^{+}\right\rangle\) state (as the lowest fidelity example) shown in Fig. 5b. All other density matrix results can be found in Fig. S3 of the Supplementary Information. We find that the theoretical (inset) and measured density matrices are in very good agreement, with fidelities of >90%. We measure these values by a new tomography of classical light fields: the two lobes of the GHZstate beam overlap at the nonOAM planar state, projected onto the polarisation states and the visibility in the fringes is measured: a Bellstate projection (see details in Methods section and Supplementary Information D), as illustrated in Fig. 5c, with the details for each projection shown in Fig. 5d–g. This allows the state amplitudes and relative phases to be determined, and hence the GHZ states can be clearly distinguished. From the visibility of each projection in the tomographic measurement, the density matrix is inferred and the fidelity of each GHZ state is quantitatively calculated. For example, \(\left{{{\Phi }}}^{+}\right\rangle\) shows no fringes for the original state, a centrebright fringe after projecting onto the \(\leftH\right\rangle\) state i.e. for \(\left\langle H {{{\Phi }}}^{+}\right\rangle\), and a centredark fringe for \(\left\langle V {{{\Phi }}}^{+}\right\rangle\), similarly for all other examples of Fig. 5d–g.
Discussion
Here, we have demonstrated vectorial structured light that is nonseparable (classically entangled) in three DoFs and eight dimensions, beyond the existing laser limitation of two dimensions in polarisation structured vector beams. In addition to the arguably simple approaches for vectorially structured beams at the source, we design an external SLM modulation system to realise the complete control of ondemand states in higherdimensional space, using the GHZ states as an example. The powerful classical GHZ states were proposed for quantum simulation in a classical optical system^{57,58}, and some intriguing applications to quantumlike information protocols were realised^{59,60,61}. However, the prior classical GHZ states were realised by complicated multibeam optical system. It is still a challenge to control a full set of GHZ states in a paraxial structured beam. Here we reached this target and realised the first complete set of GHZ states in vectorial light beams. We introduced a new tomography approach for verifying highdimensional states, which we used to quantify the fidelity of our generated GHZ states. In our experimental demonstration, we used a cavity to operate a fourray SU(2) geometric vector beam in eight dimensions to simulate tripartite entanglement.
Importantly, our method is easily scalable to achieve higherdimensional (more than eight dimensions) control. In the above results, we used only raywave structured light in the degenerate state of \(\left{{\Omega }}=1/4\right\rangle\) to demonstrate our scheme. As such, the ray trajectory has an oscillation period of four roundtrips inside the cavity with two oscillation direction states and two roundtrip location states. For this reason, we can control this structured light in 2^{3} = 8dimensional space. This choice is not peculiar, we can also choose cases of other frequency ratios such as \(\left{{\Omega }}=1/6\right\rangle\), the raywave mode of which has ray trajectories with a period of 6 in the cavity, thus, there are 3 location states, allowing control in a 2 × 3 × 2 = 12dimensional space (see discussion on more general cases in Supplementary Information). Moreover, the raywave duality beam has the potential to realise even higherdimensional and multipartite states with extended DoFs. In addition to the three DoFs used above, we can also use the OAM to form nonseparable states, realising fourpartite entangled states. Additionally, due to the multisingularity property of raywave structured light, the subOAM on the subray region can also be used to extend to the fivepartite state, see more examples and detailed demonstrations in Supplementary Information H.
These classical analogues to multipartite highdimensional quantum states have already been suggested for tasks such as quantum channel error estimation and correction^{35}, superresolution imaging^{24}, metrology and sensing^{27}, optical communication^{30} and quantum decoherence studies under easily controlled conditions^{36}. Excitingly for fundamental physical studies, the interaction between spin and trajectory involved in the optical spin Hall effects^{75} and spin–orbit Hall effects^{76} can now be studied in new ways with these engineered spin marked trajectorylike modes. Finally, we also offer a complete theoretical framework for our vectorially structured light beams, providing a rich parameter space for extending control and dimensionality, and fostering new fundamental studies with structured light.
Methods
Creating GHZ states
The GHZ generation step is shown as part of Fig. 4. The relevant transformations for generating our GHZ states are executed in two consecutive steps, intensity modulation and dynamic phase (SLM) modulation. A graphical representation of the scheme is illustrated in Fig. 4a, showing the polarisation control (rows) and intensity modulation of the orbits (columns). The unpacked schematics for the SLM and intensity modulations are given in Fig. 4b, c. Each panel describes the required modulation to produce each of the desired GHZ states.
To achieve path (intensity) modulation, we use an iris strategically positioned at I_{1} or I_{2}. We unpack this technique in Fig. 4c, showing an example for modulation at location I_{1}. The four quadrants represent the full orbits, while the grey regions represent the filtered orbits. Due to the special spatial structure of the SU(2) geometric beam, the iris induces a pathdependent intensity modulation: when an iris with an appropriate aperture size is placed at the negative Rayleigh length position (I_{1} position), paths \(\left\right\rangle \left1\right\rangle\) and \(\left+\right\rangle \left2\right\rangle\) can be blocked, resulting in a diagonal intensity pattern in the corresponding vortex SU(2) beam (see Fig. 4c). When applied at the positive Rayleigh length position (I_{2} position), \(\left+\right\rangle \left1\right\rangle\) and \(\left\right\rangle \left2\right\rangle\) are blocked, resulting in an antidiagonal intensity pattern of the vortex SU(2) geometric beam.
For polarisation modulation, we first convert all rays to circular polarisation with a QWP and direct them towards the SLM with a controlled location. This is illustrated graphically in Fig. 4b, where the four orbits are marked with (anti)diagonal polarisation states, as shown by the arrows. The phase mask of the SLM is split into two parts, one to modulate the state \(\left1\right\rangle\) and the other to modulate the state \(\left2\right\rangle\). Since the beam is circularly polarised and the SLM is sensitive only to the horizontal component, the polarisation of the \(\left1\right\rangle\) and \(\left2\right\rangle\) states can be independently controlled by encoding phase retardations on each section of the SLM. When the encoded phasestep mask is set to π/2 and 3π/2 for the split screen, respectively, states \(\left1\right\rangle \leftD\right\rangle\) and \(\left2\right\rangle \leftA\right\rangle\) are produced. Likewise, when the mask is flipped (3π/2 and π/2), states \(\left1\right\rangle \leftA\right\rangle\) and \(\left2\right\rangle \leftD\right\rangle\) are produced. The combination of intensity and polarisation modulation results in four kinds of vector vortex beams corresponding to the four maximally entangled groups of threepartite GHZ states. Finally, the intermodal phase between states is easily controlled with a thin BK7 plate as a phase retarder, which is partially inserted into the path of one ray group to add a phase difference of 0 or π.
Bellstate projections
We describe the working principle of our Bellstate measurement approach, shown as part of Fig. 4, which comprises a focusing lens (F = 250 mm), polarizer and CCD camera. The camera can be placed at different regions within the focus of the lens.
Our measurement scheme exploits the properties of the multiparticle GHZ states. A GHZ state will be reduced to a twoparticle Bell state after a superposition measurement of one of the particles (see Supplementary Information D). For the classical GHZ states, superposition projections of one of the DoFs leaves the remaining DoFs to collapse to maximally entangled Bell states (see Supplementary Information D for the full explanation). Here the polarisation DoF is chosen as a candidate to realise the Bell projection.
After projections onto the \(\leftH\right\rangle\) and \(\leftV\right\rangle\) states, the \(\left{{{\Phi }}}^{\pm }\right\rangle\) or \(\left{{{\Psi }}}_{3}^{\pm }\right\rangle\) states are reduced to the \(\left{\psi }^{\pm }\right\rangle\) and \(\left{\psi }^{\mp }\right\rangle\) states, \(\left{{{\Psi }}}_{1}^{\pm }\right\rangle\) or \(\left{{{\Psi }}}_{2}^{\pm }\right\rangle\) to \(\left{\phi }^{\pm }\right\rangle\) and \(\left{\phi }^{\mp }\right\rangle\) (refer to in Supplementary Information D and E). The “+” and “−” signals in the Bell states can be distinguished by the complementary interferometric fringes of the corresponding phase difference of 0 and π between two SU(2) rayorbits. For measuring \(\left{{{\Phi }}}^{\pm }\right\rangle\) and \(\left{{{\Psi }}}_{3}^{\pm }\right\rangle\), the CCD camera can be located at the z = − z_{R} position, where \(\left1\right\rangle\) and \(\left+2\right\rangle\) orbits overlap. For \(\left{{{\Psi }}}_{1}^{\pm }\right\rangle\) and \(\left{{{\Psi }}}_{2}^{\pm }\right\rangle\), the CCD camera should be located at the z = z_{R} position where the \(\left+1\right\rangle\) and \(\left2\right\rangle\) orbits overlap. Without the polarisation projection, the pattern shows no fringes since the light on the corresponding two orbits is incoherent. After projection onto the \(\leftH\right\rangle\) or \(\leftV\right\rangle\) states, different interference fringes are observed for different reduced Bell states.
For the group \(\left{{{\Phi }}}^{\pm }\right\rangle\), the “±” cannot be distinguished by the intensity patterns. If we project the polarisation onto the \(\leftH\right\rangle\) state to observe the pattern of \(\left\langle H {{{\Phi }}}^{\pm }\right\rangle\), the pattern of the original state \(\left{{{\Phi }}}^{\pm }\right\rangle\) will be reduced to the Bell states \(\left{\psi }^{\pm }\right\rangle\) and two different patterns of complementary fringes will be observed, centrebright fringes for \(\left{\psi }^{+}\right\rangle\) (the original state should be \(\left{{{\Phi }}}^{+}\right\rangle\)) and centredark fringes for \(\left{\psi }^{}\right\rangle\) (the original state should be \(\left{{{\Phi }}}^{}\right\rangle\)). We can also use projected state \(\left\langle V {{{\Phi }}}^{\pm }\right\rangle\) to distinguish the “±”, such that \(\left\langle V {{{\Phi }}}^{+}\right\rangle\) should be centredark fringes corresponding to the Bell state \(\left{\psi }^{}\right\rangle\) and \(\left\langle V {{{\Phi }}}^{}\right\rangle\) should be centrebright fringes corresponding to the Bell state \(\left{\psi }^{+}\right\rangle\). In the experiment, we can use a BK7 thin plate to cover one of the two orbits and rotate it slightly to control the phase difference between them to control a phase difference of π, switching from the “+” to the “−” state. Other GHZ states can be generated in a similar way, fulfilling a completed set in eightdimensional Hilbert space.
OAM measurement of the SU(2) geometric vortex beam
In contrast to the conventional vortex beam, the transverse pattern of which is always circularly symmetric (i.e., its pattern rotated around the central axis at an arbitrary angle coincides with itself), the SU(2) geometric vortex beam with SU(2) rotational symmetry (its pattern rotated around the central axis at a prescribed angle coincides with itself) can be directly observed in natural space, as the spatial twisted multiray trajectory manifests the OAM. The spatial twisted trajectories of an SU(2) geometric vortex beam can be directly measured by the scanning of a CCD camera, without other techniques such as interferometry of the conventional OAM measurement. Thus, the steps for the OAM measurement for an SU(2) geometric vortex beam are as follows: we first identify the beam trajectory by scanning the spatial distribution with a CCD device along the zaxis after the astigmatic lens converter, at a range from beam focus to Rayleigh length. Through this process, we obtain the 3D distribution of the beam which is characterised by a twisted trajectory with fourraypaths. The fourlobe pattern located on the corners of a square shape angularly rotates along the zaxis with the increase in propagation distance. We can numerically compare the experimental twisted trajectory to a variety of OAM values and draw a correlation between the experimental and simulated results, as shown in Fig. S6 of the Supplementary Information. Additionally, the topological charge of the OAM can be quantitatively measured experimentally. To achieve this, we apply a triangular truncated aperture located at the vortex centre, and then observe the farfield diffraction pattern, that emerges as a pattern of optical lattices, and the value of topological charges in the measured SU(2) geometric vortex beam is directly related to the spot number in the diffracting optical lattices^{77}. See the detailed process of this measurement in Supplementary Information G.
Change history
19 March 2021
Format of ’Multipartite’ in the title and figure 2b has been updated.
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Acknowledgements
Marie S.Curie MULTIPLY Fellowship (GA713694); National Key Research and Development Program of China (2017YFB1104500); National Natural Science Foundation of China (61975087); Natural Science Foundation of Beijing Municipality (4172030); and Beijing Young Talents Support Project (2017000020124G044).
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Y.S. and A.F. proposed the idea, Y.S. conceived and performed experiments and the theory, Y.S. and I.N. proposed the projection method of GHZ states, Y.S. and X.Y. performed the numerical simulations, all authors contributed to data analysis and writing the manuscript, and the project was supervised by A.F. and M.G.
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Shen, Y., Nape, I., Yang, X. et al. Creation and control of highdimensional multipartite classically entangled light. Light Sci Appl 10, 50 (2021). https://doi.org/10.1038/s4137702100493x
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DOI: https://doi.org/10.1038/s4137702100493x
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