Entanglement is at the heart of quantum technologies. The experimental preparation of entanglement states is a basic task in the field of quantum information. High-dimensional quantum entanglement has stronger nonlocality, larger information storage capacity, and better security performance [1–3]. The Hilbert space can be increased through the independent coding of different degrees of freedom (DoFs) of the photons, including path [4], frequency [5], and transverse-spatial modes [6–9]. Researchers have realized an equivalent of 20 qubits encoding in only two photons with two-qudit gates, taking advantage of the high dimensionality in two particular DoFs of a single photon—time and frequency [10]. Eighteen qubits encoded in three DoFs of six entangled photons have been demonstrated [11]. This is achieved by different combinations of bulky optical elements such as a beam displacer, polarization element, and spiral phase plate. With the development of high-performance compact and integrated optics [12], chip-scale generation of quantum entanglement is of great interest.

The metasurface has been suggested as a new platform for quantum optics. It offers the ability to enhance the functions and integration capabilities of quantum optical manipulations [13,14]. The metasurface is composed of nanoresonators with specially varying dimensions and orientations. It has enabled innovative flat-optics devices that often outperform the capabilities of bulk components [15–23]. Recently, metasurfaces have enabled multiphoton quantum state measurement and reconstruction [24], path encoded high-dimensional quantum source generation and multiphoton quantum sources [25], and spin angular momentum (SAM) and orbital angular momentum (OAM) entanglement of single photons [26]. Spatial entanglement and disentanglement of a two-photon state at a dielectric metasurface have been studied [27]. The concept of space-time quantum metasurfaces has been introduced [28]. The metasurface has also been used in the field of quantum edge detection [29]. Recent progress in dynamic control of quantum states using non-linear metasurfaces further enriches their functions [30]. These works have shown that the metasurface has great potential in quantum optical technologies. However, the above-mentioned works mainly deal with one DoF or two DoFs of photons. It is reasonably expected to achieve high-dimensional entanglement through multiple DoF manipulation based on metasurfaces.

Here, taking advantage of the flexibility of a single dielectric metasurface, we show high-dimensional entanglement generation through joint modulation of the path, SAM, and OAM of photons. In our work, we send one of the entangled photons to interact with the metasurface for multi-dimensional information encoding. We experimentally demonstrate the nonlocal correlations between different DoFs by correlated imaging. We also measure the fidelity of the generated quantum state. Our results show that the metasurface can provide a novel integration platform for high-capacity optical communication applications.

The designed dielectric metasurface is a Pancharatnam–Berry (PB) phase-based fork-polarization grating (FPG) [27,31–33]. The PB phase is a geometric phase achieved by space-variant polarization manipulations. The FPG is essentially an inhomogeneous wave plate, like a combination of polarization gratings and a q-plate. Figure 1(a) shows the schematic diagram of the device. The metasurface is constructed from elliptic cylinders. Here the elliptic cylinders serve as wave plate elements, and the orientation of the fast axes depends on their spatial position. The function θ(x,y) describes the spatially variant distribution of the fast axes: θ(x,y)=πx/P+qφ,(1)where P is the period of the grating, and q is the charge value of the orientations with φ=arctan(y/x). This structure generates an extra geometric phase shift that is double the element’s orientation angle: ϕ(x,y)=±(2πx/P+2qφ),(2)where m=2q is the generated topological charge value of the FPG structure.

This metasurface-based FPG can generate and control the propagation of the spin-polarized optical vortex with simultaneously coupled SAM-OAM-path modes. Gaussian mode photons pass through the FPG and diffract into three orders: ±1st-order OAM modes and zeroth-order Gaussian mode. The +1st- and −1st-order OAM modes have opposite topological charges and are always circularly polarized and orthogonal to each other. The zeroth-order photons are in a Gaussian mode with the same polarization as the input beam. If the input light is circularly polarized, it will diffract into the +1st/−1st order with orthogonal circular polarization. Thus, the FPG can simultaneously encode SAM, OAM, and path information into photons. Figure 1(b) shows a scanning electron micrograph of the metasurface. The diameter of the metasurface is 28 μm. The fabrication process is described in Appendix A. We design the FPG to deflect the incident light into two different output directions with angles of ±10°. We define the polarization conversion efficiency (PCE) as the ratio of the total energy of the left-handed (L) and right-handed (R) circularly polarized modes transmitted through the metasurface to the energy of the incident mode. The numerically simulated PCE of the designed FPG at 810 nm is 83.6%. The optical characterizations of the FPG metasurface are given in Appendix B.

In Fig. 1(c), we show the experimental scheme carried out by the device. The general idea is to start with two-photon polarization entanglement 12(|R⟩|R⟩+|L⟩|L⟩). Then we send one photon to Alice and input the other photon (Bob’s photon) to the meta-FPG device to encode OAM, SAM, and path information. We only consider the +1st and −1st diffraction orders of the device in our present work. This device performs the transformation |R⟩→FPG|L⟩|−2q⟩|b⟩ and |L⟩→FPG|R⟩|2q⟩|a⟩, where the left and right circular polarizations are converted to output states with opposite spin and ±2qℏ OAM in different optical paths. Hence, the created entangled biphoton state shared between Alice and Bob reads as follows: ψFPG=12(|L⟩|−2q⟩|b⟩|R⟩+|R⟩|2q⟩|a⟩|L⟩),(3)where q equals 0.5, and a and b represent the corresponding paths of the +1st and −1st diffraction orders. If we denote R, 2q, a as logic 0, and L, −2q, b as logic 1, we arrive at a four-qubit high-dimensional state ψFPG=12(|1⟩|1⟩|1⟩|0⟩+|0⟩|0⟩|0⟩|1⟩).

In our experiment, polarization-entangled photons are generated via the spontaneous parametric downconversion process based on a Sagnac interferometer scheme [34]. We use a 10-mm-long PPKTP crystal (Raicol Crystals) with a grating period of 10 μm for frequency-degenerate type-II quasi-phase-matched collinear SPDC. The experimental setup to generate photon pairs is depicted in Fig. 2. We use a continuous-wave diode laser at 405 nm as the pump laser. The crystal is antireflection coated at 405 and 810 nm. The temperature of PPKTP is set at 27.8°C with the stability of ±0.1°C by a heating oven. The generated SPDC photon pairs are collimated by a lens with a focal length of 200 mm. For frequency and spatial filtering, we implement a 3 nm bandpass filter and couple the generated photons into single-mode fibers. A half-wave plate set at 45° is inserted in one arm. We prepared the photon pairs in the state of 12(|H⟩|H⟩−|V⟩|V⟩), which is equivalent to 12(|R⟩|R⟩+|L⟩|L⟩). The average twofold coincidence count rate is about 90 kHz, measured with a 5 ns time window. We perform quantum state tomography to reconstruct the density matrix ρ of this state (Appendix C) [35]. The fidelity of an arbitrary quantum state ρ with respect to the desired pure state ρ0 can be written as F(ρ0,ρ)=Tr(ρ1/2ρ0ρ1/2). According to our tomography result, the fidelity of this state is 93%.

The signal photons at Bob’s side are focused on the device by a combination of lenses and an objective. After the photons pass through the FPG device, they are encoded with the path, SAM, and OAM information. The measured PCE is 40%. The relatively lower PCE is mainly due to the fabrication imperfection and the limited size of the device, as well as the imperfect alignments of the experimental setup. To directly verify the successful generation of the quantum state described in Eq. (1), we record the coincidence images by post-selection of different polarizations of the idler photon at Alice’s [36–38].

The results are shown in Fig. 3. We use Alice’s photon to trigger the intensified charge-coupled device (ICCD) for the detection of Bob’s photons. In our experiment, a 35 m optical fiber delay line is used to compensate for the electric delay (70 ns) and the insertion delay of ICCD, ensuring that the signal photons reach the camera at the same time when the camera is triggered. We set the gate width of ICCD to 5 ns, and the images are obtained by summing over 60 images with an exposure time of 2 s. We chose the polarization of Alice’s photon to be R and L separately, and then the ICCD recorded the path, OAM, and SAM information of Bob’s photon. To examine the OAM topological charge of each diffracted beam, we recorded the photon distribution in the focal plane of a cylindrical lens. The number of the resulting tilted dark bands determines the absolute value of the topological charge, whereas the tilt direction gives its sign [39]. Besides, we verified the polarization states by using a polarization filter (PF). The PF is a combination of a quarter-wave plate (QWP) and a polarizer (not shown in the experiment setup), which allows only L to pass. We show the results in the right column in Fig. 3. When the trigger state of Alice’s photons is R, Bob’s photons with the polarization component L are recorded; while there is no obvious signal on the ICCD plane when the trigger state is L, indicating the polarization state of Bob’s photons is R. Our results verify that the path, OAM, and SAM of Bob’s photon are entangled with Alice’s photon. The trigger-dependent images of the path, topological charge, and polarization information of Bob’s photon are a direct observable signature of high-dimensional quantum entanglement.

We also perform the quantum state tomography to further verify the entanglement. Here, instead of reconstructing a density matrix with 16×16 elements, we convert the generated state involving three DoFs of photons to a polarization entanglement state. We measured the fidelity of the resulting polarization entanglement. First, the two paths of the signal photons were combined interferometrically using a 27 mm long birefringent calcite beam displacer [40], after the beams passed through a quarter-wave plate (QWP). The function of the QWP is to transfer the R/L polarization state into the H/V polarization. The interference visibility was optimized with a suitable path compensation, and the measured maximum visibility reaches 93%. Note that if the path difference between H polarization and V polarization is longer than the coherence length, the state will lead to an entangled mixed state [41]. After the beam displacer, the path information has been projected into the polarization information. Then, we converted the OAM information into Gaussian mode by using another QWP and a q-plate. This QWP transfers linear polarizations to circular polarizations again, and the q-plate performs the transformation |L⟩|−2q⟩→q−plate|R⟩|0⟩ and |R⟩|2q⟩→q−plate|L⟩|0⟩. Finally, we coupled the Gaussian mode photons into a single-mode fiber for further analysis. The two-photon quantum state is now converted to the original polarization entanglement state. Figure 4(a) shows the quantum state tomography result, and Fig. 4(b) shows the experimental setup. According to the measured results, the fidelity of the reconstructed state is 82%. The reconstructed density matrix reflects the good quality of the high-dimensional entanglement state.

Overall, to verify the entanglement generated by the FPG metasurface, we conducted the coincidence imaging of high-dimensionally entangled quantum states and also performed the quantum tomography of these states. Our results show the nonlocal correlation and quantum entanglement of the generated photons. We only encode one of the entangled two photons here. If two metasurfaces are used separately to modulate Alice’s and Bob’s photons, one can easily obtain a six-qubit quantum state 12(|1⟩⊗6+|0⟩⊗6). Moreover, with a different design of the metasurface, one can encode higher topological charges of the OAM mode or the number of the paths [31,42]; thus, various high-dimensional entanglement schemes can be achieved. Recent research reports multichannel distribution and transformation of entangled photons with two metasurfaces [43]. The function of the two metasurfaces in their work is to simultaneously distribute polarization-entangled photons to spatially separated multiple paths. Here, high-dimensional entanglement generation is achieved through the joint modulation of the path, SAM, and OAM of photons, which is associated with more DoFs and offers more flexibility.

In this work, we have developed a scheme for generating high-dimensional entanglement with an FPG metasurface. With the flexible light manipulation capability of the metasurface, we can encode path, OAM, and SAM information onto the entangled photons. This idea can be extended to generate and manipulate diverse entanglement states by using multifunctional metamaterials. Our work shows the feasibility of metasurfaces for the manipulations of different DoFs of quantum states. As the Hilbert space can be increased by expanding the dimensionality within each DoF of the photon, metasurfaces offer an excellent platform for high-dimensional quantum information processing. This scheme has the potential to achieve on-chip quantum state manipulation, which is promising in an overall integrated high-capacity quantum communication system [44,45].

Acknowledgment. We thank Prof. Jian-Ping Ding, Dr. Jian-Ji Liu, and Jin-Tao Pan for their helpful discussions.