Electro-optic analyzer of angular momentum hyperentanglement

Characterizing a high-dimensional entanglement is fundamental in quantum information applications. Here, we propose a theoretical scheme to analyze and characterize the angular momentum hyperentanglement that two photons are entangled simultaneously in spin and orbital angular momentum. Based on the electro-optic sampling with a proposed hyper-entanglement analyzer and the simple matrix operation using Cramer rule, our simulations show that it is possible to retrieve effectively both the information about the degree of polarization entanglement and the spiral spectrum of high-dimensional orbital angular momentum entanglement.


Results
The proposed setup includes two stages, namely, hyperentanglement preparation and analysis, shown in Fig. 1. An ultraviolet (UV) beam (frequency ω 2 ) is incident on a thin nonlinear crystal cut for collinear type-II phase-matching, where degenerate photon pairs (frequency ω) with orthogonal polarizations are produced via SPDC 18 . A dispersion prism is used to separate down-converted photons from the pump ones. Then pairs of down-converted photons are further separated probabilistically by a non-polarizing beam splitter (50:50 BS) with 50-50% reflection and transmission coefficients. . To show the ability of our scheme in characterizing polarization entanglement, we prepare more general states with variable degree of entanglement. We employ a partial polarizer, consisting of a series of coated glass slabs tilted in approximately their Brewster's angle, to make such nonmaximally entangled states 19 . Assume that the transmission probability for horizontal and vertical polarizations of the partial polarizer are denoted by T H and T V , respectively. If we place such a partial polarizer in one path, then we are able to transfer |Ψ 〉 + into nonmaximally entangled states after polarization filtering, ( ) H is the tunable degree of entanglement. Meanwhile, quantum entanglement is also established well in the spatial degree of freedom. In general, if we assume that the UV pump beam is Gaussian, then the wave function in terms of OAM can be written in a more general form, In the analysis stage, the key elements are our proposed hyperentanglement analyzers. Let us first briefly introduce its function of spin-dependent and electrically controllable OAM generation, see ref 21 for more details. Such an analyzer consists of two nominally conjugated spiral phase plate (SPP) made of z-cut electro-optic ZnTe crystal, and both incident and exist faces are coated with transparent electrodes for applying an electric field to induce the Pockels effect. As the crystalline coordinates of two SPP are designed to have a relative orientation of 90°, so the horizontal and vertical polarization after traversing the analyzer will acquire optical vortices of opposite helix. And the topological charge Q is linearly dependent on the applied voltage U, namely, Q = αU, where α γ λ = / n 0 3 63 (n 0 and γ 63 are refractive index and electro-optic coefficient at wavelength λ). At the single photon level, the function of our electro-optic (EO) analyzer can be described by an operator, where Q can take in principle any integer or non-integer values by adjusting a suitable voltage conveniently. Of particular interest is that fractional Q can be decomposed into a coherent superposition of theoretically infinite OAM eigenstates  , namely, Recently, we used such an EO analyzer to demonstrate a quantum protocol of hybrid teleportation 22 . Here we further exploit its possibility for engineering and characterizing high-dimensional angular momentum hyperentanglement. To this end, we incorporate two EO analyzer biased with the same voltage U, each in one path. After interaction with two EO analyzers as described by equation (2), the hyperentangled state of equation (1) becomes, As mentioned above, the horizontal and vertical polarizations, H and V, yield opposite optical vortices, + Q and − Q, respectively. Reasonably, we can infer that the information of polarization entanglement can be transferred and contained in the OAM entanglement. Thus the labels of H and V in equation (3) can be discarded, and two-photon state is simplified to, In contrast, if the vortex generation is not spin-dependent (e.g. using traditional spiral phase plates 23,24 ), namely, Obviously, the information about the polarization entanglement has been lost. The emerging photons from our EO analyzers are collected by a lens and focused into the single-mode fiber (SMF) which supports exclusively the fundamental Gaussian mode with zero OAM. Inversely, SMF in connection with single-photon detector (D) projects the incoming photons into the OAM state of =  0 . The output of both single-photon detectors are fed to a coincidence counting circuit (&), which records a joint detection probability, which denotes the joint probability amplitude. The equation (6) forms the key result in present work. One of its characteristic features is the nontrivial role played by fractional Q. If Q is integer, ] 0, then only those OAM modes of = ±  Q can be detected, as can be inferred from equation (6). In contrast, if Q is fractional, ] 0, then each  mode will contribute towards the coincidence counts via ,−   C . The ability of fractional Q to access many OAM modes can be understood from its multidimensional nature. As mentioned above, a fractional vortex corresponds to a multidimensional vector state residing in the infinite OAM Hilbert space. In this regard, our work can be connected to those reporting the so-called infinite dimensional entanglement exploited by half-integer spiral phase plates 12,23,24 . Besides, if we further make Q vary continually in a certain range (by adjusting U), then we can image that the corresponding vector state traces a multidimensional curve in that space.
Another characteristic feature of equation (6) is the dependence of joint probability P, equivalently P, on the parameters ε and ,−   C , which suggests inversely that it is possible for us to retrieve ε and ,−   C by measuring and analyzing the coincidence rates. Based on equation (6), we plot a group of coincidence curves for different degrees of polarization entanglement in Fig. 2, where ε = , . , 0 0 5 1 can be easily obtained by adjusting the partial polarizer introduced in Fig. 1 (a). Besides, under the thin crystal approximation, the expression of spiral spectrum can be analytically given by 11 ,  As illustrated in Fig. 2, each coincidence curve is symmetrical, as a result of the symmetry of spiral spectrum given by equation (7). Besides, from a careful investigation on equation (6), we find that parameter ε can be directly retrieved from the following equation, These results are in good agreement with predictions given by equation (8), as shown in Fig. 3, where the red circles and the blue line are calculated based on equations (6) and (8), respectively. A little surprisingly, regardless of no use of any polarizers in the measurement stage, our EO analyzers are capable to characterize the polarization entanglement, owing to quantum information transferred from polarization to OAM.
Furthermore, we can retrieve ,−   C based on equation (6), and illustrate a typical example of spiral spectrum in Fig. 4, which is retrieved by using an 41-point electro-optic sampling, where ε = .
0 5 (upper row) and ε = 1 (lower row) are trivially considered. Fig. 4 (a,d) indicate the sampling points of applied voltages (in terms of Q, as Q = αU). By means of electro-optic sampling and the matrix operations using Cramer's rule 25 , we show the retrieved (solid bars) and original (empty bars) spiral spectra (− ≤ ≤  20 20) in Fig. 4 (b,e). After making comparisons between the retrieved and original spectra, we illustrate the relative errors of retrieved spiral spectrum Fig. 4 (c,f). And we find that ,−   C of the lower OAM modes () are well retrieved, but with an increasing  , ,−   C will have a larger deviation from the expected, which is obviously due to the very limited sampling points selected in Fig. 4 (a). As is well known, the entangled spiral spectrum generated by SPDC is generally of limited bandwidth; photon pairs with a smaller OAM are more frequent than the higher-order ones, namely, > So it can be expected mathematically that a more precise spiral spectrum containing more OAM modes could be retrieved by our EO analyzer, only if the number of electro-optic sampling points increases enough.

Discussion
In an actual experiment, there are several reasons that can reduce the signal-to-noise ratio, such as the imperfections in the SPPs orientation and the misalignment of the SPPs. For the SPPs with non-integer topological charges, there is a radial phase discontinuity that characterizes its orientation. In our simulations, we have assumed that the SPPs in two arms are both placed radically along the x-axis direction. However, it is noted that a slight mismatch of the SPP orientations, β, will cause the change of non-integer OAM states. This can be understood well by decomposing the non-integer vortex into a superposition of integer OAM states, namely, One can see that this imperfection will bring a phase shift of β ( )  i exp to each OAM mode 26 . Thus the joint probability of Eq. (6), and therefore, the performance of retrieve algorithm of Eqs. (9) and (10) will be both affected. Besides, as the OAM state measurements are highly symmetric, the good alignment of the SPPs in the optical paths is highly desirable. It was found that both lateral displacement and angular deflection will cause the coupling between different OAM and therefore broaden the OAM spectrum of a nonlocal vortex 27 . Accordingly, the measurement state with non-integer SPPs will be changed also in our case. In other words, both the imperfect orientation and misalignment of SPPs will affect the performance of our scheme, such as a reduction of the signal-to-noise ratio. These practical factors certainly deserve our very careful attention in the further experimental study. In summary, we have demonstrated a theoretical scheme to analyze and retrieve the information of two-photon spin-orbit hyperentanglement. Our method is based on electro-optic sampling using a proposed hyperentanglement analyzer and simple matrix operation using Cramer's rule. It is noted that an arbitrary pump beam will give birth to a more complicated OAM spectrum that is characterized by its complex elements, ,   C 1 2 , where  1 and  2 are arbitrary integers 28 . This case also deserves our further investigation. From the experimental point of view, it is noted that a variety of high-quality SPPs have been made with the state-of-the-art micro-machining and a molding process 29 . Besides, spiral phase mirrors were also produced by direct machining with a diamond tool 30 , and this technique can be suitably adopted to make such an OAM generator. So we think that our proposed scheme of electro-optic hyper-entanglement analyzer is experimentally feasible with current lab technology, and may find potential application in designing a fast electro-optic switching for OAM-based quantum information processing.

Methods
For spiral spectrum analysis, we can retrieve ,−   C in the following procedure: 1) Pick the joint probability amplitude P k for each sampling point = / Q k K k , where = , ,  k K 0 1 (K + 1 is total number of sampling points). 2) Using these collected data of P k vs. Q k , we can construct a system of K + 1 linear equations based on equation (6), as follows: . According to Cramer rule, we know if K = 2L and A is nonsingular, then the linear system of equation (9) has a unique solution, x = A −1 b, where A −1 denotes the inverse of A. In other words, a desired spiral spectrum with  ranging from − L to L can be retrieved using electro-optic sampling by our EO analyzers.