Diffraction-induced entanglement loss of high-dimensional orbital angular-momentum states

The study of optical vortices carrying an orbital angular momentum (OAM) lℏ has attracted considerable attention in various research fields [1–3]; l denotes the angular-momentum quantum number. The photons incorporate an azimuth phase term $e^{il\theta}$, which leads to a helical wave front. One of the most attractive features of OAM modes is that they provide a state space with theoretically infinite dimensions [4]. Thus, a larger alphabet size provides a significant increase in channel capacity when applied to quantum communication [5,6].

The helical wavefront structure of OAM states implies a strong sensitivity of OAM-encoded information with respect to disturbances along the propagation path. Atmospheric turbulence is a typical example that introduces random distortion to the wave front and results in entanglement degradation [7–9]. Another deterministic mechanism is due to diffraction upon obstacles. Most discussions on wave diffraction assume that the incident beam is a Gaussian beam. Recently, the diffraction and interference of OAM beams were studied to explore their helical phase structure [10–13]. Most studies investigated the diffraction of single OAM photons by slits, triangular apertures, or apertures with other shapes to determine the magnitude of the OAM index l.

Here, we study the entanglement loss of high-dimensional entangled OAM states in far-field diffraction by a circular aperture. We derived an analytical expression for the entanglement of the diffracted high-dimensional entangled states that depends on mutual overlaps between diffracted modes. Using the fast Fourier transform algorithm, we present some examples of qubit and qutrit encoding cases. The entanglement loss depends on the parity of the OAM index such that odd modes are not affected by diffraction, while the loss increases with the OAM index for even modes. When both apertures are placed off the axis with a small displacement, the entanglement loss remains invariable.

Laguerre-Gaussian modes

The cylindrical beam, i.e., Laguerre-Gaussian (LG), refers to a family of beams that describes a light beam possessing OAM; it can be expressed as:

$$\begin{equation} LG_{p,l}=\frac{1}{\sqrt{2\pi}}R_{p,l}(r,z)e^{il\theta}, \end{equation} \tag{ 1 }$$

where p and l denote the radial and azimuthal mode indices, respectively; R_{p, l} denotes the radial part and is expressed as:

$$\begin{eqnarray} R_{p,l}=\,& \frac{C}{(1+Z^2/z^2_R)^{-1/2}}\bigg(\frac{\sqrt{2}r}{\omega(z)}\bigg)^{l}L_{p}^{l}\bigg(\frac{2r^2}{\omega^2(z)}\bigg)e^{-r^2/\omega^2(z)}\nonumber\\ & {}\times e^{-ikr^2/[2(z^2+z^2_R)]}e^{i(2p+l+1)tan^{-1}(z/z_R)}, \end{eqnarray} \tag{ 2 }$$

where C is the normalization constant, z_R is the Rayleigh range, $\omega(z)$ is the beam waist at z = 0, and $L^{l}_{p}$ is the associated Laguerre polynomial. LG modes constituted the first OAM-carrying light beam known [1] and are the most common choice for studying OAM entanglement.

Far-field diffraction

When the OAM beam represented by eq. (1) illuminates a circular aperture, the far-field diffracted light field can be obtained by the Fresnel-Kirchhoff integral:

$$\begin{eqnarray} U(x,y,z)=\,& \frac{e^{ikz}}{i\lambda z}e^{\frac{ik}{2z}(x^2+y^2)}\nonumber\\ & {}\times \iint U_0(x_0,y_0)t(x_0,y_0)\nonumber\\ & \times e^{-i\frac{2\pi}{\lambda z}(xx_0+yy_0)}\mathrm{d}x_0\mathrm{d}y_0, \end{eqnarray} \tag{ 3 }$$

where λ and k are the wavelength and wave number of the incident beam, respectively. U(x, y,z) is the diffracted field while U₀(x₀, y₀) is the incident field. In this case, the LG beam is modified as $LG(x,y)=LG(r,\theta)$ in either Cartesian or cylindrical polar coordinates [12]. The transmission function of the circular aperture is given by the following expression:

$$\begin{eqnarray} t(x,y)= \begin{cases} 1& \text{{[}ABD-MATH-5{]}},\\ &&0 \text{{[}ABD-MATH-6{]}}, \end{cases} \end{eqnarray} \tag{ 4 }$$

where r denotes the radius. We considered the setting in fig. 1, where the Fraunhofer diffraction field is obtained in the back focal plane of a lens. The diffracted patterns were numerically studied using the fast Fourier transform algorithm [14]. The intensity and phase distribution are presented in fig. 2, in comparison with those of the incident LG mode.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Diffraction distorts the incident beam and thus introduces crosstalk between different OAM bases. Therefore, the diffracted mode can be expressed as follows:

$$\begin{eqnarray} &&|\psi\rangle=\sum_{p,l}c_{p,l}(z)|p,l\rangle, \end{eqnarray} \tag{ 5 }$$

where the expansion coefficients are given by the inner products of the diffracted field and the undisturbed basis.

$$\begin{eqnarray} &&c_{p,l}(z)=\iint u_{p,l}^*(x,y,z)\psi(x,y,z) \mathrm{d}x\mathrm{d}y. \end{eqnarray} \tag{ 6 }$$

Note that the inner product in eq. (6) is invariant under propagation along the z-axis [15] according to the Plancherel theorem. Furthermore, the Fraunhofer diffraction pattern is obtained in the back focal plane of a lens with a focal length of f [16]. Therefore, the z dependence of the expansion coefficients can be omitted.

The scenario described above is next applied to the evolution of high-dimensional entangled states. A source generates a maximally entangled state expressed as follows:

$$\begin{eqnarray} &&|\Psi\rangle=\frac{1}{\sqrt{d}}\sum_{l\in \mathcal{S}}|l,-l\rangle, \end{eqnarray} \tag{ 7 }$$

where $\mathcal{S}$ is referred to as the encoding subspace, and $|\pm l\rangle$ is the shorthand notation for single photon states of LG modes $LG_{p,\pm l}$. We assume the symmetric case when each of the photons is diffracted as in fig. 1, and both apertures have identical radii.

Under the transformation encoded by eq. (4) for each component, the diffracted photon state reads as follows:

$$\begin{eqnarray} &&\Psi(\bm{r_1,r_2})=\frac{1}{\sqrt{B}}\sum_{l\in \mathcal{S}}\Psi_{l}(\bm{r_1})\Psi_{-l}(\bm{r_2}), \end{eqnarray} \tag{ 8 }$$

where $\Psi_l(r_i)$ represents the corresponding diffracted mode, and B is the normalization coefficient. The normalization constant depends on the mutual overlap between the diffracted modes with indices $\pm l$:

$$\begin{eqnarray} B&= \sum_{i\in \mathcal{S}}\sum_{j\in \mathcal{S}}b_{i,j}*b_{-i,-j}, \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} b_{i,j}&= \int \mathrm{d}^2r\Psi^*_i(\bm{r})\Psi_j(\bm{r}), \end{eqnarray} \tag{ 10 }$$

where b_{i, j} describes the mutual overlap between the diffracted modes $\Psi_i(\bm{r})$ and $\Psi_j(\bm{r})$.

Using the diffracted state in eq. (8), the entanglement is measured by the high-dimensional generalization of concurrence [17,18]:

$$\begin{eqnarray} &&C(|\Psi\rangle)=\sqrt{2(1-\mathrm{Tr}(\rho^2_1))}, \end{eqnarray} \tag{ 11 }$$

where $\rho_1=\mathrm{Tr}_2(|\Psi\rangle\langle\Psi|)$ is the reduced density matrix of one of the entangled photons. This is obtained from the following expression:

$$\begin{eqnarray} \rho_1(\bm{r}_1,\bm{r^{\prime}}_1)&= \int \mathrm{d}^2r_2\Psi^*(\bm{r}_1,\bm{r}_2)\Psi(\bm{r^{\prime}}_1,\bm{r^{\prime}}_2)\nonumber\\ &= \frac{1}{B}\sum_{i\in \mathcal{S}}\sum_{j\in \mathcal{S}}\Psi_i(\bm{r}_1)\Psi^*_j(\bm{r^{\prime}}_1)b_{-i,-j}. \end{eqnarray} \tag{ 12 }$$

Inserting eq. (12) into eq. (11) leads to an explicit expression for the concurrence of the diffracted state:

$$\begin{eqnarray} C(|\Psi\rangle)&= \sqrt{2(1-D)}, \end{eqnarray} \tag{ 13 }$$

where $D=\frac{1}{B^2}\sum_{i,j,l,m\in \mathcal{S}}b_{-i,-j}*b_{-l,-m}*b_{i,m}*b_{j,l}$. For a better comparison of entanglement evolution in different dimensions, the concurrence is divided by its dimension-dependent maximum.

In an encoding subspace $\mathcal{S}$ of dimension d, there are d² orthogonal generalized Bell states [19]. However, due to the Choi-Jamiołkoswki isomorphism [20], the behavior of all maximally entangled states of dimension d can be characterized by considering a single state given by eq. (7) for each encoding subspace. Therefore, it is sufficient to characterize the behavior of all maximally entangled states by considering a single state in the encoding subspace.

Entanglement evolution

Given that the intensity of LG modes is mainly distributed with a single ring of radius $\rho=w(0)\sqrt{2p+l+1}$ [21] that grows with l, the power transmitted by the aperture depends on the OAM indices. To compensate for this effect, the beam waists for different values of l are chosen to keep ρ and $r/\rho$ constant. Following the experimental setting in [22], the radius of the circular aperture was chosen as $r=250\ \mu \text{m}$ and ρ was chosen as $\rho=1.25\ \text{mm}$ so that $r/\rho=0.2$.

We started by analyzing an encoding subspace using two modes with the opposite OAM $(\mathcal{S}= \left\{ -l,l\right\})$. This scenario was studied in near-field diffraction [17]. The concurrence is given by $C(|\Psi\rangle)=(1-b^2)/(1+b^2)$, which only depends on one parameter, i.e., the mutual overlap between two diffracted modes with OAM, i.e., l and $-l$. Figure 3 shows the dependence of $C(|\Psi)$ on $l\leq 10$. The concurrence of the diffracted entangled state depends on the magnitude as well as the parity of l. Only for states with even l does the diffraction lead to a degradation of entanglement, while the entangled states with odd l are not affected by the aperture, regardless of the OAM indices. For even modes, the smaller the initial OAM index l, the greater the robustness of entanglement against diffraction. Another distinct mechanism of entanglement loss is atmospheric turbulence, which shows an opposite tendency such that states with larger l are less likely to be degraded [8,9]. The far-field diffracted LG beams can be expressed via the numerical integration of the Bessel function of the first kind [22]. To note that if the order is even, the Bessel function is even and if the order is odd, the Bessel function is odd. The mutual overlap between odd modes is cancelled due to the property of odd functions, so the entanglement of odd OAM states is not affected.

Zoom In Zoom Out Reset image size

The qutrit states are obtained by adding the fundamental Gaussian mode to the encoding subspace $\mathcal{S}= \left\{ -l,0,l\right\}$ with $l\leq 10$. A similar result is shown in fig. 4 in comparison with qubit encoding. The odd modes are maximally entangled after diffraction. The difference lies in the fact that a less significant degradation is caused by even modes in contrast to the corresponding qubit states.

Zoom In Zoom Out Reset image size

We then considered the misalignment error when the aperture is placed off the axis. Recent studies experimentally proved that OAM states on some basis have the ability to reconstruct in quantum entanglement after encountering an obstruction off the axis [23,24]; this is also known as the self-healing property. The results of the Fraunhofer diffraction are shown in fig. 5. The transmission function of the aperture becomes as follows:

$$\begin{eqnarray} t(x,y)= \begin{cases} 1& \text{{[}ABD-MATH-32{]}},\\ &&0 \text{{[}ABD-MATH-33{]}}, \end{cases} \end{eqnarray} \tag{ 14 }$$

where d is the shift with respect to the beam axis. Unlike the near-field case, in which minimal output concurrence is observed [17], here the evolution follows a tendency close to a step function, in which the concurrence remains constant in certain regions. A similar result is that qutrit encoding is less degraded by diffraction than qubit encoding when the same displacement is applied. In all cases, the concurrence remains invariable when $d/r\leq 0.2$, demonstrating that the entanglement is not affected by a small misalignment error.

Zoom In Zoom Out Reset image size

To provide a framework for further discussion, we make a comparison between present results obtained in the far-field diffraction with results obtained in the near-field diffraction [17]. In the near-field case, the entanglement loss depends on the magnitude of the OAM index l. In the far-field case, not only the magnitude but also the parity of l affect the entanglement loss. In the near-field case, the entanglement evolution demonstrates the self-healing property, i.e., the ability to reconstruct. In the far-field case, the entanglement evolution demonstrates the property of a step function, that is, to keep invariant in some regions. Reasons that caused such difference could be the topic of future studies of complex spatial modes as well as the diffraction of light.

We conclude this section discussing the procedure to verify that the numerical simulation described above produces correct results. The numerical results obtained by the fast Fourier transform algorithm are compared with experimental results in fig. 6. The diffraction patterns in the top row were extracted from [10], in which LG beams were scattered by the slit. The bottom row shows the simulated diffraction patterns based on the fast Fourier transform algorithm.

Zoom In Zoom Out Reset image size

The patterns in the first column indicate the typical single-slit diffraction with a typical Gaussian beam. However, for higher orders of OAM, the patterns are similar to those of Young's double-slit interference pattern. Note that there is a central maximum for even OAM modes and a central minimum for odd OAM modes. This is attributed to the phase difference between any two bright points along the slit, which is $2\pi$ for even modes and π for odd modes.

We observed excellent agreement between the numerical and experimental results. Possible discrepancies were due to the different parameters employed. Therefore, numerical simulations of eq. (3) produce reasonable far-field diffraction patterns.

We investigated the Fraunhofer diffraction of high-dimensional entangled OAM states from a circular aperture. An analytical formula for the concurrence was derived that depends on the mutual overlap between the diffracted modes. Using the fast Fourier transform algorithm, we numerically studied the diffracted two-dimensional and three-dimensional entangled OAM states. In both cases, the entangled states encoded in odd OAM modes are not affected by diffraction, whereas the entanglement loss for even OAM modes increases with the OAM indices l. We further considered the misalignment of the aperture. When the displacement $d\leq 0.2r_0$, the concurrence remains constant regardless of the OAM indices. Therefore, entangled OAM states are robust to small misalignment errors.

The authors acknowledge the support of the National Key Research and Development Program of China (Grant No. 2017YFA0303700), the Key Research and Development Program of Guangdong province (Grant No. 2018B030325002), the National Natural Science Foundation of China (Grant No. 11974205), and the Beijing Advanced Innovation Center for Future Chip.