High-Dimensional Entanglement of Photonic Angular Qudits

We propose a method for generation of entangled photonic states in high dimensions, the so-called qudits, by exploiting quantum correlations of Orbital Angular Momentum (OAM) entangled photons, produced via Spontaneous Parametric Down Conversion. Diffraction masks containing N angular slits placed in the path of twin photons define a qudit space of dimension N², spanned by the alternative pathways of OAM-entangled photons. We quantify the high-dimensional entanglement of path-entangled photons by the Concurrence, using an analytic expression valid for pure states. We report numerical results for the Concurrence as a function of the angular aperture size for the case of high-dimensional OAM entanglement and for the case of high-dimensional path entanglement, produced by N × M angular slits. Our results provide additional means for preparation and characterization of entangled quantum states in high-dimensions, a fundamental resource for quantum simulation and quantum information protocols.

1 Introduction

In recent years, Spontaneous Parametric Down Conversion (SPDC) has become a fundamental process for generation of entangled photonic states, allowing for preparation of quantum states entangled in several degrees of freedom, such as position and momentum, time and energy, polarization or angular position and Orbital Angular Momentum (OAM), thus providing for a key resource in fundamentals of quantum physics and quantum information. Quantum correlations of SPDC photons in a given domain give rise to interference phenomena resulting from two-photon coherence [1–8]. These phenomena are routinely used in fundamental tests of quantum physics [9–11] and are a key ingredient for the implementation of quantum communication and information protocols, including quantum teleportation and quantum cryptography [12–14]. Fourier relations link angular position and OAM of photons, leading to angular interference in the OAM-mode distribution, as photons diffract through angular apertures, resulting in two-photon quantum interference [15–22].

In this article, we quantify entanglement of such high-dimensional angular qudits, in a scheme in which OAM-entangled photons produced by SPDC are transmitted through multiple angular apertures, in the form of N × M angular slits in the path of signal and idler photons, which results in path entanglement in a space of dimension D = N × M. Using this scheme, we demonstrate high-dimensional entanglement based on angular-position correlations of down-converted photons. Our results suggest that violations of Bell’s inequalities in even higher dimensions could in principle be achieved. Moreover, in contrast to previous approaches [16, 17] relying on the Fourier limit only, our results shine light on the quantum interpretation, providing new insights. Adding to the novelty of our work, we consider the case of asymmetric angular slits N and M for signal and idler, which can lead to high-dimensional angular interference phenomena. We note that linear qudits have previously been proposed [24]. Here we propose and characterize angular qudits, which due to their shape can enable generation of entanglement in a larger Hilbert space than linear qudits. In order to quantify entanglement, we derive an analytic expression for calculation of the Concurrence, valid for pure states. The results reported here extend the notion of angular qudits to an arbitrary number of angular slits N × M, which not only demonstrates two-photon coherence effects in the angular domain but also provide additional means for preparation and characterization of entangled quantum states in a high-dimensional Hilbert space. This is a fundamental resource for quantum communication [23–40] and quantum information protocols [12, 16, 19, 41–57].

The article is organized as follows: In Section 2 we introduce the concept of high-dimensional path-entanglement of angular qudits [56]. Next, in Section 3 we present an overview of angular diffraction in the position basis [57] and we derive an analytic expression for the Concurrence of high-dimensional qudits, valid for pure states. In Section 4 we present numerical results in two specific scenarios, (IVA) high-dimensional OAM entanglement and (IVB) high-dimensional path entanglement. Finally, in Section 5 we outline the conclusions.

2 High-Dimensional Angular Qudits

Consider the experimental scheme depicted in Figure 1. A Gaussian pump beam produces signal (s) and idler (i) entangled twin photons by the non-linear process of SPDC. In the simplest scenario of a Gaussian pump beam with zero OAM (l = 0), phase matching conditions determine the two-photon down-converted state |ψ₀⟩, which can be expressed in the following form [56]:

$|{\psi }_{\text{0}}\rangle =\sum _{l=-\nu }^{\nu }{c}_l|l{\rangle }_s|-l{\rangle }_i,(1)$

where s and i label signal and idler photons, |l⟩ refers to the OAM eigen-mode of order l, and D = 2ν + 1 is the dimension of the OAM Hilbert space under consideration. Such OAM modes are characterized by an helical phase front typically expressed as e^ilϕ. For |ψ₀⟩ to represent a quantum state, the normalization condition imposes ${\sum }_{l=-\nu }^{\nu }|c_l{|}^2=1$. Subsequently, signal and idler photons are transmitted through N angular slits, as shown in Figure 1. The transmission functions A_j,n of the individual angular slits are given by [56]:

$A_{j,n}\left({\varphi }_j\right)=1\mathrm{i}\mathrm{f}n\beta -\alpha /2\le {\varphi }_j\le n\beta +\alpha /2\mathrm{e}\mathrm{l}\mathrm{s}\mathrm{e}0,(2)$

where n = 0, … , N − 1 is the angular slit label, α represents the aperture of the angular slits, and β represents the separation between consecutive angular slits. For the simplest case N = 2 slits, we recover the results reported in [49]. Considering N slits in both arms, there are in principle N² alternative pathways by which the down-converted photons can pass through the apertures and get detected in coincidence at single-photon avalanche detectors D_s and D_i. The N² alternative paths, here labelled by the sub-index q = 1, … , N², can be expressed as the outer product of the sub-spaces corresponding to each photon (s, i) passing through the slits n = 0, … , N − 1, respectively, in the following form [56]:

$\begin{array}{ccc}\hfill & \hfill & |s,0\rangle \otimes \left\{|i,0\rangle ,|i,1\rangle ,\dots ,|i,N-1\rangle \right\};\hfill \\ \hfill & \hfill & |s,1\rangle \otimes \left\{|i,0\rangle ,|i,1\rangle ,\dots ,|i,N-1\rangle \right\};\dots .\hfill \\ \hfill & \hfill & |s,N-1\rangle \otimes \left\{|i,0\rangle ,|i,1\rangle ,\dots ,|i,N-1\rangle \right\}.\hfill \end{array}(3)$

Spatial quantum correlations between the twin photons can be controlled so that only paths of the form |i, n⟩|s, n⟩ will have a significant contribution. We note there is a one to one mapping between the path-way basis and the OAM-mode basis. For a given path n, the two-photon state can be written in the OAM-mode basis as [49]:

$\begin{array}{ccc}\hfill |s,n\rangle |i,n\rangle & =\hfill & C\sum _l{c}_l\sum _{l^{\prime }}\frac{1}{2\pi }{\int }_{-\pi }^{\pi }d{\varphi }_s{A}_{s,n}\left({\varphi }_s\right)e^{i\left(l^{\prime }-l\right){\varphi }_s}|l^{\prime }\rangle \hfill \\ \hfill & \hfill & \times \sum _{l^{″}}\frac{1}{2\pi }{\int }_{-\pi }^{\pi }d{\varphi }_i{A}_{i,n}\left({\varphi }_i\right)e^{i\left(l^{″}+l\right){\varphi }_i}|l^{″}\rangle .\hfill \end{array}(4)$

In this notation, the single-photon quantum state with OAM of order l diffracted by the k-th slit, with OAM momentum l, can be expressed as [56]:

$|{\psi }_l^k{\rangle }_{s,i}=\frac{\alpha }{2\pi }\sum _{l^{\prime }}{e}^{-i\left(l^{\prime }-l\right)\beta k}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-l\right)}{2}\right]|l^{\prime }{\rangle }_{s,i},(5)$

where $\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}[\mathrm{x}]=\frac{\sin [x]}{x}$. In the most general case, the overlap between the diffracted states by slits k and j, with OAM labels m and l, results in:

$\begin{array}{cc}\hfill \langle {\psi }_m^j|{\psi }_l^k\rangle & ={\left(\frac{\alpha }{2\pi }\right)}^2\sum _{l^{\prime }}\sum _{l^{″}}{e}^{-i\left(l^{\prime }-l\right)\beta k}{e}^{i\left(l^{″}-m\right)\beta j}\times \hfill \\ \hfill & \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-l\right)}{2}\right]\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{″}-m\right)}{2}\right]\langle l^{″}|l^{\prime }\rangle ,\hfill \end{array}(6)$

where $⟨l^{\prime \prime }|l^{\prime }⟩={\delta }_{l^{\prime \prime },l^{\prime }}$ due to orthonormality of OAM modes.

FIGURE 1

FIGURE 1. Proposed experimental scheme for generation of high-dimensional OAM-entangled photon pairs via SPDC. Spatial Light Modulators (SLMs) containing (N, M) angular slits are placed in the path of signal and idler photons, generating high-dimensional angular qudits. Further details are in the text. The figure was modified from Ref. [56].

For the simple case of a single slit, we have j = k = 1, the mode overlap ⟨ψ_m|ψ_l⟩ can be expressed as:

$\begin{array}{cc}\hfill ⟨{\psi }_m|{\psi }_l⟩& ={\left(\frac{\alpha }{2\pi }\right)}^2\sum _{l^{\prime }}{e}^{i\left(l-m\right)\beta }\times \hfill \\ \hfill & \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-l\right)}{2}\right]\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-m\right)}{2}\right].\hfill \end{array}(7)$

3 High-Dimensional Spatial Mode Entanglement and Diffraction

In this Section we derive expressions for twin photons angular diffracted states in the position basis. As already anticipated in Section 2, we focus on D − dimensional (D = 2ν + 1) entangled biphoton pure states of the form [57].

$|{\psi }_0\rangle =\sum _{l=-\nu }^{\nu }{c}_l|l\rangle |-l\rangle ,(8)$

where

$|l\rangle =\int d^2\mathbf{r}{u}_l\left(\mathbf{r}\right)|\mathbf{r}\rangle (9)$

are single photon states in the spatial modes defined by the orthonormal transverse functions u_l(r), i.e., $\int d^2\mathbf{r}{u}_{l^{\prime }}^{*}(\mathbf{r})u_l(\mathbf{r})={\delta }_{l,l^{\prime }}$, and ${\sum }_{l=-\nu }^{\nu }|c_l{|}^2=1$ to ensure normalization [57, 58].

We are interested in how the entanglement of such states is modified by diffraction of the two photons on two independent opaque screens of different shapes and forms. Moreover, we focus on the case where both photons are detected after the diffracting screens. In this post-selection scenario, entanglement is only modified by the change of the spatial profile of the modes occupied by the photons due to diffraction on the screens. Under these assumptions, the quantum state of the diffracted photons can be written [57, 58].

$|\psi \rangle =\sum _{l=-\nu }^{\nu }{\tilde{c}}_l|{\psi }_l\rangle |{\psi }_{-l}\rangle ,(10)$

with

$|{\psi }_l\rangle =\int d^2\mathbf{r}{\psi }_l\left(\mathbf{r}\right)|\mathbf{r}\rangle ,(11)$

where the modes ψ_l(r) are the images of the modes u_l(r) that can be computed using standard diffraction theory [57], and ${\tilde{c}}_l=c_l/\sqrt{\mathcal{K}}$ with ν a renormalization constant needed to compensate for the non orthogonality of the diffracted modes ψ_l(r), which we can express as [57, 58].

$\mathcal{K}=\sum _{lk=-\nu }^{\nu }{c}_l^{\ast }{c}_k{b}_{lk}{b}_{-l-k},(12)$

with

$b_{lk}=\int d^2{\psi }_l^{\ast }\left(\mathbf{r}\right){\psi }_k\left(\mathbf{r}\right),(13)$

the mutual overlaps between pairs of diffracted modes [57]. We note that since the single photon states |ψ_l⟩ are fully determined by the modes they occupy, we also have that ⟨ψ_l|ψ_k⟩ = b_lk.

We now set to characterize the entanglement of such states developing a high-dimensional generalization of the method presented in [56, 57]. Following this path, we quantify the entanglement of the diffracted state using the Concurrence, that for pure states in arbitrary dimensions can be expressed as a function of the purity Tr [ρ²] of the reduced density matrix ρ = Tr₂ [|ψ⟩⟨ψ|] [57].

$C\left(|\psi \rangle \right)=\sqrt{2\left(1-\mathrm{Tr}\left[{\rho }^2\right]\right)}.(14)$

The reduced density matrix can be computed using the expression (11) for the quantum state of single photons in the diffracted modes and the mutual overlap (13)

$\rho =\int d^2{\mathbf{r}}_2\langle {\mathbf{r}}_2|\psi \rangle \langle \psi |{\mathbf{r}}_2\rangle =\sum _{kl=-\nu }^{\nu }{\tilde{c}}_l^{\ast }{\tilde{c}}_k{b}_{-l-k}|{\psi }_l\rangle \langle {\psi }_k|.(15)$

In a similar fashion, we can determine the purity of the reduced density matrix

$\begin{array}{ll}\hfill \mathrm{Tr}\left[{\rho }^2\right]& =\int d^2{\mathbf{r}}_1⟨{\mathbf{r}}_1|{\rho }^2|{\mathbf{r}}_1⟩\hfill \\ \hfill & =\sum _{l,k,p,q=-\nu }^{\nu }{\tilde{c}}_l^{\ast }{\tilde{c}}_k{\tilde{c}}_p^{*}{\tilde{c}}_q{b}_{-l-k}{b}_{-p-q}{b}_{kp}{b}_{lq},\hfill \\ \hfill & =\frac{{\sum }_{l,k,p,q=-\nu }^{\nu }{c}_l^{\ast }{c}_k{c}_p^{*}{c}_q{b}_{-l-k}{b}_{-p-q}{b}_{kp}{b}_{lq}}{{\left({\sum }_{lk=-\nu }^{\nu }{c}_l^{\ast }{c}_k{b}_{lk}{b}_{-l-k}\right)}^2}\hfill \end{array}(16)$

which fully determines the Concurrence Eq. 14. From Eq. 16, we see that the entanglement of diffracted biphoton states is fully characterized by the entanglement of the input state (encoded in the coefficients c_l), and by the diffraction induced overlaps of the states |ψ_l⟩ (quantified by the coefficients b_lk).

We note that the goal of the scheme presented in Figure 1 is to provide an alternative means for generation of entangled quantum states in high dimensions, using only two photons. It should be mentioned that the operations introduced by the SLMs are local and cannot increase the total amount of entanglement, which is given by the entanglement content present in the initial state generated by the SPDC process, as quantified by the Concurrence (Eq. 14). In particular, for a maximally entangled pure input state in D-dimensions, the Concurrence is given $C_D=\sqrt{2(1-1/D)}$ whose upper bound is $\sqrt{2}$, since the reduced density matrix of a bipartite maximally entangled state is a maximally mixed state, proportional to the Identity operator.

4 Examples

In the following, we will apply the theoretical framework presented in Sections 2, 3 to two particular examples. In particular, in Section 4A, we will consider high-dimensional OAM-entangled biphotons diffracted on screens containing a single angular aperture, while in Section 4B we will consider initial biphotons entangled in few OAM modes impinging on screens containing multiple apertures.

A. High-Dimensional OAM Entanglement on Angular Apertures

We now specialize to the study of maximally entangled states, i.e., we set $c_l=1/\sqrt{2\nu +1}\forall l$ in Eq. 8. Moreover, we assume the spatial modes carrying OAM to be fully characterized by their helical phase front so that we have

$|l\rangle =\frac{1}{\sqrt{2\pi }}{\int }_{-\pi }^{\pi }d\varphi e^{il\varphi }|\varphi \rangle ,(17)$

where $|\varphi \rangle =\int rdr{u}_l(\vec{r})|\vec{r}\rangle$. After transmission through an angular aperture of size α, |l⟩ becomes:

$|{\psi }_l\rangle =\frac{1}{\sqrt{\alpha }}{\int }_{-\alpha /2}^{\alpha }d\varphi e^{il\varphi }|\varphi \rangle .(18)$

Using Eq. 18, we can calculate the mutual overlaps (13) which results into

$b_{lk}=\frac{1}{\alpha }{\int }_{-\alpha /2}^{\alpha }d\varphi e^{i\left(k-l\right)\varphi }=\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\left(l-k\right)\alpha }{2}\right].(19)$

Eq. 19 implies that b_lk = b_−l−k, accordingly in this case the purity (16) can be rewritten as

$\mathrm{Tr}\left[{\rho }^2\right]=\frac{{\sum }_{lkpq=-\nu }^{\nu }{b}_{lk}{b}_{pq}{b}_{lp}{b}_{kq}}{{\left({\sum }_{lk=-\nu }^{\nu }{b}_{lk}^2\right)}^2}.(20)$

Combining Eqs 19, 20 into Eq. 14 we can compute the concurrence of the diffracted state, which is plotted in Figure 2.

FIGURE 2

FIGURE 2. Concurrence vs. angular aperture size for different OAM subpsace dimension (D = 2ν + 1) as indicated in the label, considering a single angular slit for both signal and idler photons (M = N = 1). Details are in the text.

B. High-Dimensional Path-Entanglement on Angular Apertures

We now consider an alternative scenario for generation of high-dimensional entanglement relying on path-entangled modes generated by multiple angular apertures in the path of signal and idler photons (s, i). For simplicity, we consider twin photons initially prepared in a OAM qubit state of the form:

$|{\psi }_0\rangle ={|l_0\rangle }_s|-l_0{\rangle }_i.(21)$

When masks with (N, M) angular slits for (s, i) photons are placed in each spatial mode (seen Figure 1), the biphoton state immediately after the apertures can be expressed as [36]:

$|{\psi }^{\prime }\rangle \propto \sum _{k=-\frac{\left(N-1\right)}{2}}^{\frac{\left(N-1\right)}{2}}\sum _{k^{\prime }=-\frac{\left(M-1\right)}{2}}^{\frac{\left(M-1\right)}{2}}\varphi \left(x_s,x_i,z\right)|{\psi }^k{\rangle }_s|{\psi }^{k^{\prime }}{\rangle }_i,(22)$

where x_s,i denote the position of signal and idler photons at the crystal plane (z). ϕ(x_s, x_i, z) indicates the biphoton amplitude at z, which can be regarded as a product of the pump transverse amplitude and phase matching function [36]. For simplicity, in what follows we consider the biphoton amplitude to be approximately constant. |ψ^k,k′⟩ describe the quantum state of signal and idler photons diffracted by angular apertures (k, k′), respectively.

In the notation introduced in Section 2, for an initial state of the form |l₀⟩_s| − l₀⟩_i, the quantum state of (s, i) photons diffracted by angular apertures (k, k′), respectively, can be expressed as:

$|{\psi }_{l_0}^k{\rangle }_s=\frac{\alpha }{2\pi }\sum _{l^{\prime }}{e}^{-i\left(l^{\prime }-l_0\right)\beta k}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-l_0\right)}{2}\right]|l^{\prime }{\rangle }_s,(23)$

$|{\psi }_{-l_0}^{k^{\prime }}{\rangle }_i=\frac{\alpha }{2\pi }\sum _{l^{″}}{e}^{-i\left(l^{″}+l_0\right)\beta k^{\prime }}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{″}+l_0\right)}{2}\right]|l^{″}{\rangle }_i.(24)$

The biphoton path-entangled state diffracted by N × M angular slits results in:

$\begin{array}{ccc}\hfill |{\psi }^{\prime }\rangle & =\hfill & \sum _{k=-\frac{\left(N-1\right)}{2}}^{\frac{\left(N-1\right)}{2}}\sum _{k^{\prime }=-\frac{\left(M-1\right)}{2}}^{\frac{\left(M-1\right)}{2}}{\left(\frac{\alpha }{2\pi }\right)}^2\times \hfill \\ \hfill & \hfill & \sum _{l^{\prime }}\sum _{l^{″}}{e}^{-i\left(l^{\prime }-l_0\right)\beta k}{e}^{-i\left(l^{″}+l_0\right)\beta k^{\prime }}\times \hfill \\ \hfill & \hfill & \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-l_0\right)}{2}\right]\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{″}+l_0\right)}{2}\right]|l^{\prime }{\rangle }_s|l^{″}{\rangle }_i.\hfill \\ \hfill & \hfill & \hfill \end{array}(25)$

A compact expression for the biphoton diffracted state can be cast in the form:

$|{\psi }^{\prime }\rangle =\sum _{l^{\prime }}\sum _{l^{″}}{c}_{l^{\prime },l^{″}}|l^{\prime }{\rangle }_s|l^{″}{\rangle }_i,(26)$

where the coefficients in the summation $c_{l^{\prime },l^{\prime \prime }}$ are given by:

$\begin{array}{ccc}\hfill c_{l^{\prime },l^{″}}& =\hfill & \sum _{k=\frac{-\left(N-1\right)}{2}}^{\frac{\left(N-1\right)}{2}}\sum _{k^{\prime }=\frac{-\left(M-1\right)}{2}}^{\frac{\left(M-1\right)}{2}}{\left(\frac{\alpha }{2\pi }\right)}^2\times \hfill \\ \hfill & \hfill & e^{-i\left(l^{\prime }-l_0\right)\beta k}{e}^{-i\left(l^{″}+l_0\right)\beta k^{\prime }}\times \hfill \\ \hfill & \hfill & \mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{\prime }-l_0\right)}{2}\right]\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\left[\frac{\alpha \left(l^{″}+l_0\right)}{2}\right].\hfill \end{array}(27)$

The Concurrence for the path-entangled biphoton states can be derived considering the generalized overlap (b_lm) of the form:

$b_{lm}=\sum _{k,k^{\prime }}{\int }_{-\pi }^{\pi }d\varphi e^{i\left(m-l\right)\varphi }{A}_k{A}_{k^{\prime }},(28)$

where A_k,k′ describe the angular aperture functions introduced in Section 2.

Figures 3–5 present numerical simulations of the Concurrence (C) vs. Aperture Size (in radians), for different angular slit dimensions M × N for signal and idler photons, respectively. For the case of N and M angular slits the maximum angular aperture size per slit is 2π/N and 2π/M, respectively. Taking this limit into consideration, we performed numerical simulations for aperture sizes in the range 2π/N_max, where N_max is the maximum number of slits between the values of N and M considered in the specific numerical simulations. All angular dimensions considered are experimentally feasible in view of the resolution of state-of-the-art Spatial Light Modulators (SLMs) [56]. Different curves in Figure 3 correspond to symmetric path-dimensions M × N given by 2 × 2, 3 × 3, 4 × 4, 5 × 5, and 6 × 6 angular silts, for aperture sizes in the range [0, 2π/N_max], with N_max = 6. Different curves in Figure 4 correspond to asymmetric path-dimensions M × N given by 2 × 2, 2 × 4, 2 × 5, 3 × 4, 3 × 5 and 4 × 5 angular silts, for aperture sizes in the range [0, 2π/N_max], with N_max = 5. Different curves in Figure 5 correspond to asymmetric path-dimensions M × N given by 5 × 6, 5 × 7, 5 × 8, 6 × 7, 6 × 8 and 7 × 8 angular silts, for aperture sizes in the range [0, 2π/N_max], with N_max = 8. Interestingly, within our approximation, for a sufficiently large number of angular apertures (M × N) it is possible to reach the same amount of entanglement as with high-dimensional OAM.

FIGURE 3

FIGURE 3. Concurrence vs. angular aperture size for different path dimensions (M × N) and aperture sizes in the range [0, 2π/N_max] with N_max = 6. Details are in the text.

FIGURE 4

FIGURE 4. Concurrence vs. angular aperture size for different path dimensions (M × N) and aperture sizes in the range [0, 2π/N_max] with N_max = 5. Details are in the text.

FIGURE 5

FIGURE 5. Concurrence vs. angular aperture size for different path dimensions (M × N) and aperture sizes in the range [0, 2π/N_max] with N_max = 8. Details are in the text.

Figure 2 indicates that for the case of OAM-entangled modes in D-dimensions, it is possible to reach the maximal amount of entanglement present in the initial state for a sufficiently large angular aperture size (nearly equal to 2π). On the other hand, a similar amount of entanglement can be achieved with path-entangled states using smaller angular apertures, by introducing a sufficiently large number of angular slits (Figures 3–5). Therefore, we can conclude that the advantage of high-dimensional path-entangled states is that they can enable to achieve the maximal entanglement present in the initial state at reduced angular aperture sizes, by increasing the number of angular slits.

Close inspection of Figures 2–5 reveals that the monotonicity in the Concurrence is directly linked to a symmetric slit configuration (M = N), as depicted in Figures 2, 3. While non-monotonicity can be ascribed to high-dimensional interference phenomena associated with an asymmetric slit configurations (M ≠ N), as depicted in Figures 4, 5. Therefore, a tunable slit configuration can be regarded as an alternative means for tailoring the entanglement content in the angular qudit state. This could prove a promising solution for instance for improving the performance of next generation function-integrated quantum circuits, among other relevant applications [50–57].

5 Experimental Implementation

The proposed experimental scheme envisioned to characterize high-dimensional entanglement using (N, M) angular slits is depicted in Figure 1. It is based on the experimental setup described in Ref. [49]. In this experimental scenario, a Gaussian pump laser beam with zero OAM is customarily prepared by spatial filtering using single-mode fibers. The pump power is typically in the range of 100 mW and typical operational wavelength is at λ = 413 nm. Degenerate down-converted photons at λ = 826 nm are created by the non-linear process of Spontaneous Parametric Down Conversion (SPDC), where a pump beam is normally incident on a non-linear crystal, for either type-I or type-II phase matching conditions. Conservation of OAM for the twin photons is granted by the SPDC process itself, for the given pump and phase matching conditions [49]. The novel element introduced in the setup is given by the angular masks containing (N, M) angular slits, for signal and idler photons, respectively. Such angular slits are programmed using state-of-the-art Spatial Light Modulators (SLMs). The OAM spectrum of the diffracted states can be analyzed in terms of OAM spiral harmonics, typically over a range from l = 12 to l = 12. This is routinely accomplished in the laboratory using diffraction holograms, which for ease of implementation can be programmed in the same SLM used for the angular masks (Figure 6). Such OAM projective measurements are registered via Coincidence Counts (R_si), in the (l_s, l_i) OAM basis. The Coincidence Count (CC) rate R_si of single-photon detectors D_s(i) gives the probability that signal(idler) photons are detected at single-photon detector D_s(i) in mode |ls(i)⟩, in a given fixed-time coincidence window. Specifically, the CC rate is given by R_si = ⟨ l_i|⟨l_s‖ρ‖l_i|⟩l_s| [49].

FIGURE 6

FIGURE 6. Angular masks used to create path-entanglement and diffraction holograms used to analyze the OAM spectrum. Both the angular mask and the diffraction hologram can be programmed using the same Spatial Light Modulator. Details are in the text.

An alternative, more exciting, detection approach would consist of implementing a detection scheme that could resolve the angular slits taken by the photons and enable to reconstruct the density matrix directly in the path-way basis. We envision that such complex detection schemes could in principle be implemented by means of N-arm interferometers in combination with spatial multiplexing techniques. More specific, spatial multiplexing could enable active stabilization of such complex interferometers, which is a key experimental challenge in the context [59].

As mentioned, SLMs are used both for preparation of angular masks, and for analysis of the transmitted state in the OAM basis via diffraction holograms. As it is well known in the literature, SLMs are programmable refractive elements which enable full control of the amplitude of the diffracted beams. In the standard technique, if the index of the analysis l-forked hologram is opposite to that of the incoming mode, planar wave-fronts with on-axis intensity are generated in the first diffraction order. The on-axis intensity can be coupled to single-mode fibers with high efficiency, and can be measured with single-photon detectors D_s,i, using a coincidence count circuit (Figure 1). The maximum number of angular slits that can be implemented in a realistic experimental situation will be fundamentally limited by the finite spatial and angular correlation width of signal and idler photons, as well as by the resolution of the SLM itself [56].

6 Discussion

We presented a method to generate entangled photonic states in high-dimensional quantum systems, the so-called qudits, by exploiting quantum correlations of OAM-entangled photons produced by the non-linear process of Spontaneous Parametric Down Conversion. Diffraction masks containing N angular slits in the path of twin photons define a qudit space of dimension N², spanned by the alternative pathways of the entangled photons. We quantify the entanglement of path-entangled photons by an explicit calculation of the Concurrence, valid for pure states. We reported numerical results for the Concurrence as a function of the angular aperture size for the case of high-dimensional OAM-entangled photons and for the case the case of high-dimensional entanglement produced by N × M angular slits. Interestingly, within our approximation, it is possible to reach the same amount of entanglement using either high-dimensional OAM-entangled photons or path-entangled photons. Our results shine light into the fundamental quantum aspects of two-photon angular interference, and provide alternative means for preparation of entangled quantum states in high-dimensions, a fundamental resource for quantum information and quantum simulation protocols [12, 16, 19, 25–57].

Data Availability Statement

Data supporting the conclusion of this article will be made available by the authors upon request.

Author Contributions

GP conceived the idea, developed the theory, performed numerical simulations, and wrote the manuscript.

Conflict of Interest

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

The authors gratefully acknowledge Giacomo Sorelli for the derivation of Eq. 15. GP acknowledges Sonja Franke-Arnold, Antonio Zelaquett-Khoury, and Leonardo Neves for helpful discussions. GP acknowledges financial support via PICT Startup.

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