Light with a helical phase exp (imϕ), which carries orbital angular momentum (OAM), has attracted close attention in the last few decades and found numerous photonic applications, such as high density data storage [1, 2], optical imaging [3–8], holography [9, 10], astrophysics [11, 12], optical manipulation [13–15] and in the optical interferometer for the detection of gravitational waves [16]. In addition, the OAM of light serves, in a sense, as an “alphabet” that allows information to be encoded into the spatial wavefunction of light. The key motivation is that the OAM has potentially an unlimited number of states [17, 18]. In the classical domain, the OAM can increase the capacity of optical communication links ranging from implementation in fibre [19, 20], over-city links [21] and free-space [22], and can also operate in mm-wave [23]. Actually, each OAM mode can be considered as an individual quantum degree of freedom [24, 25]. Quantum mechanically, the OAM occurs in discrete values of mℏ per photon, where m is in principle an unbounded integer. The OAM opens many promising perspectives for quantum communication and computation [26, 27]. Zeilinger’s group firstly realized the OAM-entangled photon pair [28] and verified that the maximum dimension of the OAM entanglement can reach 10,000 [29], which shows the great potential and novel advantage in the high-dimensional quantum information such as teleportation [30] and quantum communication [31–33]. A spontaneous parametric down conversion (SPDC) process is the most common workhorse for the generation of the OAM-entangled photon pairs [34, 35]. Albeit convenient, this process exhibits several drawbacks. For example, photon pairs generated in this way have a non-uniform OAM distribution, which constrains the increase of dimension [26]. The entanglement property of the OAM states is determined by the overlap between the transverse modes of the pump and photon pairs because the transverse intensity profile of the OAM mode depends on its topological charge m. The dimensionality of the two-photon OAM states can be increased with the use of perfect optical vortex (POV) in the pump because the POV is a class of size-invariant modes [36, 37]. The versatile high-dimensional quantum states are generated by using the structured pump [38–40]. Recently, the entangled OAM photon pairs are generated in the multiple SPDC processes by the path identity [41].

In addition, the non-uniformity of OAM entanglement is also restricted by the longitudinal phase mismatching. High-order OAM modes have the larg phase mismatch, which leads to the low generation efficiency. This restriction can be improved by utilizing the microscale flat-optics quantum source [42]. Here we have achieved relatively flat high-dimensional OAM entanglement in an ultrathin film of lithium niobite (LN) via the SPDC, which is impossible in a single bulk source. Being free from the phase matching constraints, the ultrathin source can be fabricated of any materials with large second-order susceptibility. Our method provides a new platform of high-dimensional OAM entanglement on which to investigate fundamental quantum effects but it also has practical applications.

In the paraxial approximation, photons carrying OAM can be described by a Laguerre-Gaussian mode LG p m . The radial index p shows the number of radial zero crossings and the azimuthal index m is the topological winding number, corresponding to OAM carried by the mode, mℏ per photon. Figure 1A shows the intensity and phase distributions of the LG modes with different OAM. A typical SPDC process, which a Gaussian beam propagating along the x direction pumps a nonlinear crystal with a length of L, as shown in Figure 1B, produces a pair of highly correlated, lower-frequency photons, commonly termed signal and idler photons. The generated two-photon quantum state by the SPDC is given by [43]. ψ = ∬ d k s d k i Φ k s , k i a ̂ s † k s a ̂ i † k i 0 , (1) where 0 is the multimode vacuum state, while a ̂ s † k s and a ̂ i † k i are creation operators for the signal and idler modes with the transversal wave vectors k _s and k _i , respectively. Φ k s , k i represents the joint amplitude with the following structure Φ k s , k i ∝ F k s , k i sinc Δ k L / 2 , (2) where F k s , k i describes the mode function of the Gaussian pump beam at the input face of the nonlinear crystal. Here we consider only LG modes with p = 0, correspondingly we simplify as LG p m → LG m . One can decompose the quantum state ψ in the base of the eigenstates of the OAM operator under the OAM conservation as ψ = ∑ m C m , − m m s − m i (3) where m s and − m i correspond to the signal and idler modes, respectively. The coincidence amplitudes C _m,−m is written as C m , − m = ∬ d k s d k i Φ k s , k i LG m k s LG − m k i , (4) where LG m k is the Laguerre-Gaussian mode function in the spatial frequency domain at x = 0. The coincidence probability is P m , − m = C m , − m 2 , which gives the value of the joint detection probability for finding one photon in the signal mode m s and one photon in the idler mode − m i . It is clear that the coincidence probability P _m,−m is mainly determined by the phase matching. If the 0 s 0 i mode is assumed to be phase matched, the phase mismatching of high-order OAM modes can be calculated analytically as [42]. Δ k L / 2 ∝ − m π L 4 λ ω 0 2 z R 2 1 − 2 z R L arctan L 2 z R , (5) where z _R is the Rayleigh range of the mode 0 s 0 i and ω ₀ is the beam waist of the down-converted beam, respectively. For a type-0 degenerate phase matching process, λ _s = λ _i = 2λ _p is satisfied. We use a common value of λ = 2λ _p to replace λ _s and λ _i . According to Eq. 5, the phase mismatching increase with the topological charge m of the down-converted photon because the high-order modes have a larger divergence angle, which will decrease the longitudinal component of the wave vector, as shown in Figure 1B. Comparing with the mode 1 s − 1 i , the mode 5 s − 5 i suffers from the larger longitudinal phase mismatching and has the lower coincidence probability. In this case, the longer nonlinear crystal will further increase the difference of the coincidence probabilities between the different OAM modes, although it can improve the total efficiency of the photon pairs generated by the SPDC. We calculate the spiral spectrum of the high-dimensional OAM entanglement for the LN crystal with L = 0.6 mm, as shown in Figure 1D. The brightness of photons greatly decreases for the high-order OAM modes, which often limits the usefulness of the high-dimensional OAM entangled states in the quantum experiment. When the length of the nonlinear crystal decreases to L = 6 μm, as shown in Figure 1C, the difference of coincidence probabilities between the different OAM modes is dramatically reduced and the spiral spectrum becomes flat, as shown in Figure 1E. The ultrathin nonlinear crystal not only allows very large longitudinal mismatch, but also leads to a very broad spectrum of emitted photons, both in frequency and in angle [42].

As the experimental setup shown in Figure 2A, we use an x-cut ultrathin film of MgO-doped LN on a fused silica substrate as a nonlinear medium. The LN ultrathin slice had a thickness of ∼6 μm. The pump source is a femtosecond (fs) pulsed laser at a central wavelength of 405 nm, with a pulse duration of ∼140 fs and a repetition rate of ∼80 MHz to create photon pairs at a degenerate wavelength of 810 nm by the SPDC. Benefiting from the largest element d ₃₃ of the nonlinear susceptibility tensor, the polarization of the pump fs laser is oriented along the z axis of the ultrathin LN film. Under the condition of type-0 degenerate collinear phase matching, down-conversion photon pairs have the same polarization. The 10 nm narrow bandwidth interference filters (centred at the wavelength of 810 nm) are used to improve the spectral purity of the correlated photon pairs and to block the remaining pump.

To confirm that the SPDC was mediated by the d ₃₃ element of the nonlinear susceptibility tensor, with the pump, signal, and idler photons all polarized along the z axis, we have measured the polarization dependence of the coincidence count, as shown in Figure 2B. The coincidence count depends on the angle θ between the pump polarization and the y axis as sin² θ (blue squares, red fitting curve) and reaches the maximum value only when the signal and idler photons are polarized along the z axis. In contrast, for the emission polarized along the y axis, almost no real coincidences were observed (green circles). Then we use two separate q-plates to transform the incoming field to a fundamental Gaussian mode, which is the unique mode that can be coupled into the single mode fibers, for measuring the OAM of the photon pairs. In the collinear configuration, OAM is conserved in the SPDC, hence we expect the OAMs of the signal and idler photons to be anticorrelated, i.e., the coincidence count is high only when m _s = −m _i . Figure 3 shows the measured coincidence counts for different combinations of signal and idler photon modes. For each combination the polarization-dependent coincidence counts verifies that the correlated photon pairs are generated from the ultrathin LN by the SPDC. Figure 4 shows the spiral spectrum of the 11-dimensional OAM entanglement and it becomes relatively flat comparing with thick nonlinear crystal. The measured coincidence of the high-order OAM state is lower comparing with the theoretical simulation because the collection efficiency decreases with the increment of the OAM sate. In addition, we observe that the SPDC of ultrathin crystal has better robustness for the position of the pump. When the crystal shift 1 mm from the focus of the pump, the decrease of the maximum coincidence counts for difference mode of photon pairs is about 5%.

In conclusion, we have demonstrated the relatively flat high-dimensional OAM entanglement based on the SPDC in the ultrathin LN crystal which allows large longitudinal phase mismatching and decreases the difference of coincidence rate between the different OAM mode. This state might be required for quantum experiment. Although we use the highest nonlinear component available in LN, but the efficiency remained much lower than for phase-matched SPDC in a macroscopic crystal. The generation efficiency can be optimised by using the nonlinear material with the giant second-harmonic generation [45]. Recently, metasurfaces offer an ultracompact and versatile platform for enhancing nonlinear optical processes by designing Bound State in the Continuum (BIC) resonances, which support a high confinement of the optical field within the nonlinear material [46, 47]. Then SPDC efficiency can be dramatically increased when the signal and idler photons are supported by BIC resonances [48] to generate high-dimensional hyperentanglement in the frequency and OAM regimes for increasing the quantum information capacity.