## Abstract

Using orbital angular momentum (OAM) conservation in second-order nonlinear interaction processes, we create a classical to quantum optical network link in the OAM degrees of freedom of light via sum frequency generation, followed by spontaneous parametric down-conversion. Coherent OAM-carrying beams at 1550nm are up-converted to 525.5-nm OAM-carrying beams in the first crystal, and are used to pump a second crystal to generate non-degenerate OAM entangled photon pairs at 795nm and 1550nm. By switching the OAM carried by the classical part, OAM correlation in the quantum part is shifted. High-level OAM entanglements in two-dimensional subspaces are verified.

© 2015 Optical Society of America

## 1. Introduction

The field of optical communications consists of two quite different schemes, classical optical communications and quantum optical communications. These schemes differ in terms of the principles of their physical realization, such as the methods used for encoding, decoding and detection [1]. Information transmission capacity and security are the two most important indexes used to estimate the quality of an optical communication network. To enhance the transmission capacity of a single communication channel, multiplexing of the different degrees of freedom of light is used. In the early stages, time multiplexing, wavelength multiplexing and polarization multiplexing were used to dramatically increase communication speeds. More recently, multiplexing of the spatial degrees of freedom of light has been demonstrated in both free space [2, 3] and fiber-based [4] optical communications for orbital angular momentum (OAM)-carrying light beams, and the transmission rates have reached the Tbit/s level. Almost since the birth of quantum mechanics,quantum optical communications have been under developmentin pursuit of unconditional security in optical communications. Based on the basic principles of quantum mechanics, quantum communication offers unconditional security in principle. Encoding information in the OAM degrees of freedom of photons can enhance both the information-carrying capacity and noise tolerance in quantum key distribution [5–7].

Since it was first introduced by Allen et al. in [8], OAM-carrying light has been widely studied in many fields, including optical trapping and manipulation [9, 10], metrology [11–13], testing of the basic principles of quantum mechanics [14–17] and optical communications [2–4, 7]. The total OAM of a light beam is conserved in nonlinear optical interaction processes (e.g., second order [18–25], third order [26–28] and higher orders [29]). The conservation of OAM in nonlinear interaction processes is very useful for frequency conversion of OAM-carrying beams [18–25] and creation of OAM entangled photon pairs in two-dimensional subspaces [30–34] or in higher dimensions [16, 35]. Because of this conservation law, it is possible to create a link between a classical optical network and a quantum optical network encoded in terms of the OAM degrees of freedom of light. This link acts as a compatible interface to join a classical network and a quantum network to a hybrid network, which will then enable information transfer between two completely different communication systems.

In this article, we report the construction of a hybrid optical network by linking a classical network with a quantum network using the OAM degrees of freedom of light beams based on a process of sum frequency generation (SFG) followed by spontaneous parametric down-conversion (SPDC) in nonlinear crystals. A coherent OAM-carrying beam at the telecommunications wavelength of 1550nm is efficiently up-converted into a 525.5-nm OAM-carrying beam in the first periodically poled KTiOPO_{4} crystal (PPKTP1); then, the up-converted OAM-carrying beam is used to pump crystal PPKTP2 to generate non-degenerate OAM entangled photon pairs at 795nm and 1550nm. Because the total OAM is conserved in this cascade process, the OAM correlation of the quantum part is shifted by switching the OAM carried by the classical part. The dependence of the quantum OAM correlation on the OAM carried by the classical beam implies that information is transferred from the classical part to the quantum part in the OAM degrees of freedom of the light beams. In addition, high levels of OAM entanglements in the two-dimensional subspaces are verified via various methods, including interference, CHSH (Clauser-Horne-Shimony-Holt) inequality and quantum state tomography methods.

## 2. Experimental setup

The experiment consists of two parts, where the first part involves efficient up-conversion of an OAM-carrying light beam based on SFG in an external cavity, and the second part is SPDC using the up-converted OAM beams to generate non-degenerate OAM entangled photon pairs (see the simplified diagram at the top of Fig. 1). High-efficiency frequency conversion of OAM-carrying light beams in periodically poled nonlinear crystals has previously been studied in detail by the authorsin [19–22]. In the experiment presented here (Fig. 1), the external cavity is the same as that used in [22], and the cavity design parameters are described in detail in that paper. The type-I SFG and SPDC crystals were manufactured by Raicol Crystals, and all crystals have dimensions of 1 mm × 2mm × 10mm. The poling period of each crystal is 9.375µm, both end faces of each crystal are anti-reflection coated for wavelengths of 525.5 nm, 795nm and 1560nm, and the measured quasi-phasing matching temperatures of PPKTP1 and PPKTP2 are 39.4°C and 43.0°C,respectively. The power of the 1550-nm beam is 12mW (Toptica Prodesign laser), and that of the 795-nm beam is 700mW (Ti:sapphire laser, MBR110). The powers of the up-converted 525.5-nm beam for OAM indexes of 0 and −1 are 0.7mW and 0.15mW,respectively, after removing the two pump beams using filter F1(Thorlabs, FBH520-40,FESH1000). The up-converted OAM-carrying beam is then imaged into another crystal, PPKTP2, using a 150-mm lens (L1) for SPDC with a magnification of about 2, and thus the beam diameter at the center of PPKTP2 is approximately 130 µm. The PPKTP2 crystal emits non-degenerate photon pairs at 795nm and 1550nm. These two photons are separated by a dichromatic mirror, and they are then imaged separately to two spatial light modulators, A and B (SLMA/SLMB, PLUTO; active area of 15.36 × 8.64 mm^{2}, pixel pitch size of 8 μm, and total number of pixels of 1920 × 1080) using 150-mm lenses (L2, L3) with magnifications of 15. The beam diameters at SLMA and SLMB are approximately 2mm. After transformation by SLMA and SLMB, the photons are then collected into single mode fibers (SMF1, SMF2) using 4-f imaging systems with similar lens groups (L4/L5 and L6/L7). The collected photons are detected using a Si avalanche detector and a free running InGaAs avalanche detector (IDQ 220), and the output signals of these detectors are sent to a coincidence device (Timeharp 260, Pico, with a coincidence window of 3.2ns).

## 3. Experimental results

The generated non-degenerate two-photon state that is correlated in the OAM can be expressed as [36]:

Here, *s* and *i* stand for the signal and the idler respectively; $|{m}_{s},{\lambda}_{s}\u3009$ and $|{l}_{p}-{m}_{s},{\lambda}_{i}\u3009$ denote the OAM eigenmodes of the signal and the idler, respectively; the OAMs carried by the signal (${m}_{s}$) and the idler (${l}_{p}-{m}_{s}$) sum to give the OAM carried by the pump beam${l}_{p}$; ${\left|{c}_{{m}_{s}}\right|}^{2}$ is the probability of generationof a photon pair with OAM of${m}_{s}$, ${l}_{p}-{m}_{s}$; in the special case where ${l}_{p}=0$, we can infer that ${c}_{{m}_{s}}={c}_{-{m}_{s}}$ from the symmetry of the SPDC process.

To show that the total OAM is conserved in the cascade processes of SFG and SPDC and that the OAM information carried by the classical coherent light beams at 1550nm is transferred to the signal and idler photons at 1550nmand 795nm, respectively,we perform OAM correlation measurements for different signal and idler OAMs. Experimental results for OAM numbers of 0 and −1 carried by the 1550-nm coherent beam are shown in Fig. 2. Figure 2(a) shows a three-dimensional bar chart for the signal and idler OAM ranges from −5 to 5 for a pump beam OAM of 0. Because of misalignment, the coincidences of the nearby OAMs are larger than those of the non-adjacent OAMs. The maximum coincidence counts in 10 s are 31475 for the signal and idler OAMs of 0, and the smallest off-diagonal coincidence counts are approximately 100. Figure 2(b) shows that the corresponding coincidence counts for the signal and idler OAM sum to 0. We can infer that the spiral bandwidth of this two-color source is approximately 5.To increase the spiral bandwidth, we can use a larger beam waist, a shorter crystal length [37, 38] or change the phase-matching conditions by tuning the crystal temperature [39]. Figures 2(b) and 2(c) show the results for the pump OAM of −1, where the coincidence counts when the signal and idler OAMs sum to −1 are much higher than the other uncorrelated cases. The inset images in Figs. 2(a) and 2(c) show the intensity distributions of the 525.5-nm SFG beams. The slight spatial dependence of the conversion efficiency will lead to mode impurity in the SFG beam, and this mode impurity will lead to crosstalk from other OAM channels;however, the mode impurity in the present experiments is very small, and this crosstalk can simply be ignored. All these results are in good agreement with Eq. (1), which implies that the total OAM is conserved in the cascade processes, and that the OAM information flows from the classical part to the quantum part.

Equation (1) also implies that the signal and idler OAMs are entangled in a high-dimensional Hilbert space. To investigate the entanglement of the signal and idler photons, we simplify the measurement conditions to two-dimensional subspaces. For OAM entanglement in two-dimensional subspaces, we can use a Blochsphere to describe the state in a manner that is analogous to the polarization states. An equal superposition of $|l\u3009$ and $|-l\u3009$ with arbitrary phase is represented by a point along the equator of the sphere [40–42]. In a fashion that is analogous to the polarization states, we define the superposition statesas follows:

These states are represented by points along the equator of the sphere. The states $|\theta \u3009$ are also called sector states, and have 2*l* sectors of alternating phases.

To quantify the entanglement, we must demonstrate that the correlations between the signal and the idler are persistent for the superposition states. To verify this condition, we detect the photons in the sector states that were defined in Eq. (2) when oriented at different angles ${\theta}_{A}$ and ${\theta}_{B}$. Based on Eqs. (1) and (2), the coincidence rate for detection of one photon in $|{\theta}_{A}\u3009$ and the other in $|{\theta}_{B}\u3009$ is given by

The above states are normalized with respect to the chosen subspaces and are entangled in the OAM. By changing the phase masks on the SLMs, entangled states can be prepared with any given *l*. The experimental results in the two-dimensional subspace given by $l=\pm 1,\pm 2$ are shown in Figs. 3(a) and 3(b). By fixing the orientation of the hologram in SLMB at angle ${\theta}_{B}=0,\pi /4l$, we measure the coincidence count in 10 s by changing the angle from 0 to $\pi /l$. The angle period is $\pi /l$ and is dependent on *l*. The visibilities of the sinusoidal fringes are 89.36% and 84.11%, and 96.89% and 97.32% for ${\theta}_{B}=0,\pi /4l$ (*l* = 1, 2), respectively. All the visibilities are higher than 71%, which is sufficient to violate the CHSH inequality and imply the presence of the two-dimensional entanglement. Therefore,we further characterize the entanglement by measuring the CHSH inequality *S* parameter [43, 44]. The definition of *S* in our experiment for the angles ${\theta}_{A}$ and ${\theta}_{B}$ for the phase masks on the two SLMs is

The inequality is violated for values of $\left|S\right|$ that are greater than 2. $E({\theta}_{A},{\theta}_{B})$ is calculated from the coincidence counts at specific sets of the hologram on the two SLMs:

For the entanglement state $|\Phi \u3009{}_{l}$, the maximal violationof the inequality occurs,e.g., when ${\theta}_{A}=0,$${\theta}_{B}=\pi /8l,{{\theta}^{\prime}}_{A}=\pi /4l$ and ${{\theta}^{\prime}}_{B}=3\pi /8l$. The measured *S* parameters for $|\Phi \u3009{}_{1}$ and $|\Phi \u3009{}_{2}$ are 2.28 ± 0.016 and 2.82 ± 0.0021 with 17 and 390 standard deviations, respectively.

To describe an entangled statecompletely, reconstruction of the density matrix is necessary. To reconstruct the density matrix of $|\Phi \u3009{}_{l}$, we follow quantum tomography methods that are analogous to the two-bit polarization entanglement states [45]. We define $|L\u3009=|l\u3009$ and $|R\u3009=|-l\u3009$. At least 16 projection measurements on the projection basis $|L\u3009$,$|R\u3009$,$1/\sqrt{2}(|L\u3009+|R\u3009)$ and $1/\sqrt{2}(|L\u3009-i|R\u3009)$ are required. As an example, the results for the state $|\Phi \u3009{}_{2}$ are shown in Figs. 3(c) and 3(d). The real and imaginary parts of the density matrix that were reconstructed using the maximum-likelihood method are shown in Figs. 3(c) and 3(d). The fidelity $F={\u3008\Phi |}_{2}{\rho}_{e}{|\Phi \u3009}_{2}$ of the experimentally reconstructed density matrix with respect to the theoretical density matrix is 0.84 ± 0.0062. This discrepancy in the fidelity is caused by slightly different coupling efficiencies between the states $|L\u3009$ and $|R\u3009$ and the states $1/\sqrt{2}(|L\u3009+|R\u3009)$ and $1/\sqrt{2}(|L\u3009-i|R\u3009)$. We use an OAM index of 2 as an example to illustrate the holograms that were used for each measurement in the experiments. The holograms that were used are listed in Fig. 4. Figures 4(a) and 4(b) show the holograms for the coincidence measurements shown in Fig. 1. Figures 4(c) and 4(d) show the definitions of the angles ${\theta}_{A}$ and ${\theta}_{B}$, which are used in the measurements shown in Figs. 3(a) and 3(b). Figures 4(e)–4(h) are four orthogonal projection bases for the quantum tomography measurements shown in Figs. 3(c) and 3(d).

## 4. Summary and discussion

We have demonstrated proof of principle experiments for construction of a link between a classical telecommunications network and a quantum network usingthe OAM degrees of freedom of light based on cascade second-order nonlinear frequency conversion. An OAM-carrying beam at the telecommunications wavelength of 1550nm is up-converted into an OAM-carrying beam at 525.5nm using cavity-enhanced SFG, and then the up-converted OAM-carrying beam is used to pump a second crystal that is used for SDPC to generate non-degenerate two-color OAM entangled signal and idler photons at 795nm and 1550nm, respectively. The dependence of the OAM correlation on the classical beam is observed at 1550nm; when the OAM carried by the classical beam is switched, then the OAM correlation between the signal and the idler is also changed because the total OAM is conserved in the cascade processes. Therefore, the OAM information carried by the classical beam is transferred to the signal and idler photons. We also demonstrate high-level entanglement of the signal and idler photons in the two-dimensional OAM subspace. This two-color OAM entangled source is highly promisingfor quantum communications, because one of the photon wavelengths corresponds to the atom quantum memory of Rb^{85}(where the bandwidth can be narrowed using a cavity), while the other wavelength is in a telecommunications band that is suitable for long distance transmission. Our work here will be helpful in fabrication of a compatible interface between a classical optical communication network and a quantum communication network based on OAM degrees of freedom; this hybrid network will offer both high capacity and superior security.

## Acknowledgments

This work was supported by the National Fundamental Research Program of China (Grant No. 2011CBA00200) and the National Natural Science Foundation of China (Grant Nos. 11174271, 61275115, and 61435011).

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