## Abstract

Hyper-entangled photon pairs are very promising in the quantum information field for enhancing the channel capacity in communication and improving compatibility for networks. Here we report on the experimental generation of a hyper-entangled photon pair at a wavelength of 795 nm and 1475 nm via the spontaneous four-wave mixing process in a cold ^{85}Rb atomic ensemble. The photons in each pair generated are entangled in both the time-frequency and polarization degrees of freedom. Such hyper-entangled photon pairs with special wavelength have potential applications in high-dimensional quantum communication and quantum physics.

© 2017 Optical Society of America

## 1. Introduction

Entangled photon pairs are important resources for realizing many quantum information protocols such as quantum cryptography [1], quantum-state teleportation [2,3], quantum communication [4,5] and so on. Generally, photons can be entangled in one of some available degrees of freedom (DOF) such as time-frequency, polarization, time-bin or orbital angular momentum. Recently, hyperentanglement [6–10], which means particles are entangled in multiple DOFs simultaneously, has attracted more researchers’ attention. Compared to the entanglement in single DOF, hyperentanglement enlarges the Hilbert space for encoding, leading to high channel capacity and effective performance of quantum information process [11–13].

Generally, spontaneous parametric down conversion (SPDC) process in a nonlinear crystal [14] is a common way to prepare entangled photon pairs. However, the generated photon has so wide bandwidth (~THz) that it does not match the working bandwidth of the quantum memory, an indispensable building block for quantum repeaters. To solve this problem, a nonlinear crystal inserted in a cavity [15,16] is used to generate narrowband photon pairs. Besides, spontaneously four-wave mixing (SFWM) in an atomic system [17–20] is another way for generating narrowband photon pairs. In this work, we experimentally generate a hyper-entangled photon pair via SFWM in a cold atomic ensemble. The wavelengths of two photons are 795 nm and 1475 nm respectively, so one photon in telecom-band can be used as an information carrier for long distance transmission, the other one can couple with atoms effectively. Compared with photons entangled in single DOF, the generated hyper-entangled photon pairs can largely improve the channel capacity, which is significant for quantum communication. Thereby, the generated hyper-entangled photon pairs are suitable for building up a long-distance and a high-dimensional quantum communication network based on atomic quantum memories.

Firstly we will introduce the generation of photon pairs based on an atomic ensemble trapped in a two-dimensional magneto-optical trap (MOT) and verify the non-classical correlation of photon pairs. After that we will measure the two-photon interference and check the violation of Clauser-Horne-Shimony-Holt (CHSH) inequality to verify the polarization entanglement of photon pairs. Besides, we will use the quantum beat phenomenon to demonstrate the time-frequency entanglement of photon pairs. Finally we give a simple conclusion.

## 2. Experimental setup

We prepare a non-classical correlated photon pair through SFWM process via a diamond configuration in a cold ^{85}Rb atomic ensemble. The energy level diagram is shown in Fig. 1. A laser beam at 780 nm from an external-cavity diode laser (DL 100, Toptica) and the other laser beam at 1530 nm from another external-cavity diode laser (DL Prodesign, Toptica) act as pump 2 and pump 1 respectively. Pump 2 couples to the atomic transition of 5S_{1/2}(F = 3)→5P_{3/2}(F' = 4) and pump 1 couples to the transition of 5P_{3/2}(F' = 4)→4D_{3/2}(F” = 4). The wavelengths of generated signal 1 and signal 2 photons are 795 nm and 1475 nm respectively. Here, five states, denoted as |1>, |2>, |3>, |4> and |5>, correspond to ^{85}Rb atomic levels 5S_{1/2}(F = 3), 5P_{1/2}(F' = 2), 5P_{1/2}(F' = 3), 5P_{3/2}(F' = 4) and 4D_{3/2}(F” = 4) respectively. ${\Delta}_{1}\text{=}{\omega}_{\text{P2}}-{\omega}_{41}$ and ${\Delta}_{2}\text{=}{\omega}_{\text{P}1}-{\omega}_{54}$represent the detuning of pump 2 and pump 1 respectively, and $\Delta ={\Delta}_{1}+{\Delta}_{2}$ represents two–photon detuning. We denote the blue frequency detuning to be positive in this article. During the experiment, pump 3, which couples to the transition of 5S_{1/2}(F = 2)→5P_{3/2}(F' = 3), is kept open for optical pumping the atoms to the state of |1>.

Experimental setup is shown in Fig. 2. Pump 1 co-propagates with pump 2 through the MOT in a collinear configuration. We focus pump 2 and pump 1 on the center of ^{85}Rb atomic ensemble by two lenses with the focal lengths of 600 mm and 485 mm respectively. The powers of pump 2 and pump 1 are 40 µW and 2.8 mW respectively. At first, we perform four-waving mixing experiment to adjust experimental setup. The pump 1 is horizontally polarized after propagating through PBS and pump 2 is vertically polarized after propagating through PBS and HWP 1 (Here, HWP 1, HWP 2 and HWP 3 are half-wave plates. HWP 2 and HWP 3 are placed for demonstrating polarization entanglement of photon pairs). In order to reduce the noise, pump 2 propagates through a filter (Semrock LL01-780 type with bandwidth of 3 nm and 98% transmittance), and pump 1 propagates through a single-band bandpass filter (SemrockNIR01-1535/3 type with bandwidth of 3 nm and 98% transmittance). A bandpass interference filter placed before Detector 1 and a 22 degree-tilted filter (Semrock LL01-808 type with bandwidth of 3.1nm and 98% transmittance) set before Detector 2 are also used to reduce noise from pump lasers. Besides, a space grating and three fiber Bragg gratings set before Detector 1 are used to further reduce the noise. We use two single-mode fibers to collect signal 1 and signal 2 with the coupling efficiencies of 51.8% and 70.3% respectively. Then a time-to-digital converter is utilized to measure the time-correlated function of two signals after the signal 1 and signal 2 photons are detected by Detector 1(a free-running In-GaAs photon detector, ID Quantique ID220 FRSMF, 10% efficiency) and Detector 2 (avalanche diode, PerkinElmer SPCM-AQR-15-FC, 50% efficiency) respectively.

## 3. Experimental results and analysis

Generally, the classical light satisfies the Cauchy-Schwarz inequality [21] written below

Under this condition, the detected rate of photon pairs is 24 s^{−1} with ${C}_{\text{S1}}=3\times {10}^{2}{s}^{-1}$ and ${C}_{\text{S2}}=3\times {10}^{3}{s}^{-1}$ (Here, ${C}_{\text{S1}},{C}_{\text{S2}}$represents the count rate of signal 1 and signal 2 respectively). Considering the total detecting efficiency for signal 1 of ~10% × 40% (including 10% detecting efficiency and 40% collecting efficiency) and of ~50% × 60% for signal 2 (including 50% detecting efficiency and 60% collecting efficiency), the generating rate of photon pairs is 24 s^{−1}/(10% × 40% × 50% × 60%) = 2000 s^{−1}. Noise here is mostly eliminated as we have used DM, space grating, and fiber grating to reduce noise of signal 1 (total transmission: 77%, isolation: 80 dB). And for signal 2, we have used three clean-up filters with 85% transmission and 90 dB isolation.

After SFWM process in this atomic ensemble, the generated hyper-entangled state of signal 1 and signal 2 can be denoted as

*H*

_{S1}> (|

*V*

_{S1}>) represents horizontal (vertical) polarized state of signal 1, ${\omega}_{\text{S1}}$and${\omega}_{\text{S2}}$are the frequency of signal 1 and signal 2 respectively. ${\Delta}_{23}$ is energy level spacing between level |2> and level |3>. $\alpha $ and $\beta $are the coefficients.

In our experiment, polarization entangled state is denoted as$|{H}_{\text{S1}}\u3009|{V}_{\text{S2}}\u3009+\alpha |{V}_{\text{S1}}\u3009|{H}_{\text{S2}}\u3009$. According to the observed two-photon interference curves in the bases of horizontal/vertical and diagonal/anti-diagonal, shown in Fig. 4(a), we find the best fitted $\alpha \approx 1/2$. Here, $\alpha $is very sensitive to the two-photon detuning of pump beam according to experimental measurement, so it is very difficult to give a fixed value.

We investigate the polarization entanglement of photon pairs via plotting two interference curves when $\Delta $ = −50MHz as shown in Fig. 4(a). The coincidence counts between two photons are measured in different bases of signal 1 and signal 2. In the bases of |*V*> and (|*H*> + |*V*>)/2^{1/2} of signal 1, we obtain the red and black interference curves respectively by changing the relative phase angle ${\theta}_{\text{S2}}$ of signal 2.

The visibilities of interference curves in the bases of |*V*> and (|*H*> + |*V*>)/2^{1/2} are 96.1% ± 0.7% and 78.2% ± 1.8% respectively, which both exceed 70.7% [22], the benchmark of Bell’s inequality. Therefore, two photons are entangled in polarization. According to the visibilities, we calculate an average visibility V^{'} = 87.2% ± 0.9% and derive CHSH inequality parameter S = 2.47 ± 0.03 via the formula $S=2\sqrt{2}{V}^{\text{'}}$ [23]. Therefore, it further demonstrates the existence of entanglement between photons since the value of *S* is larger than 2, violating the CHSH inequality [24]. So, the high visibilities of interference curves shown in Fig. 4(a) and the added demonstration of violation of CHSH inequality surely demonstrate the polarization entanglement of generated photon pairs.

For the entanglement in time-frequency DOF, we measure the cross-correlation of photon pairs shown in Fig. 4(b) when the two-photon detuning is −50MHz. It is easy to see the quantum beat phenomenon which demonstrates the generated photon pairs are time-frequency entangled. A simple analysis is shown below to illustrate the demonstration of time-frequency entanglement.

In Fig. 1, when two-photon detuning is not equal to zero, two pump beams can partially couple to the atomic transition of 5S_{1/2}(F = 3)→4D_{3/2}(F” = 3). According to the transition selection rule, when the atoms stay on the level 4D_{3/2}(F” = 3), there are two transition paths shown in the dotted arrow part of Fig. 1, which results in the generation of signal 1 and signal 2 via two hyperfine levels |2> and levels |3>. According to the work in [25–27], the quantum beat phenomenon will appear if there are two different transition paths, which will finally results in entanglement. Here, because the transition matrix elements of two transition paths are totally unknown (we cannot find any datasheet containing the needed transition matrix element), the precise evaluation of $\beta $ is not achievable at present, and we have to estimate the value of $\beta $ through data fitting method.

After fitting the experimental data using${G}_{s1s2}^{2}\propto {e}^{-bx}(1+V\mathrm{cos}[2\pi vx])$ [25] in Fig. 4(b), we have the best fitted $v\approx 353.68$MHz which is very close to ideal situation: ${\Delta}_{23}$ = 361.58MHz [28]. This proves the existence of two transition paths leading to the generation of entanglement in time-frequency DOF. We also obtain the best fitted parameter *V* corresponding to the visibility of quantum beat is equals to 0.35 and estimate the value of $\beta $ to be 0.35 or 2.86. The reason why the value of $\beta $is not 1 mainly results from the different transition strength between two transition paths, which results in the low visibility of quantum beats in experiment. The obtained entanglement in time-frequency DOF is non-maximal. For non-maximal entanglement [29], it can verify the nonlocality of quantum mechanics without inequalities [30] in the logical arguments and reduce the detector efficiencies for loophole-free tests of Bell inequalities [31,32]. Moreover non-maximal entanglement can be transformed to maximal entanglement via entanglement concentration method, therefore the generated non-maximal entangled state may be a significant resource for quantum information processing and researching about quantum physics.

In the end of paper, we illustrate a phenomenon found in our work. When we change the detuning of pump 1 with a step of 10MHz, it is very interesting to see that quantum beat phenomenon sometimes appears and sometimes disappears as shown in Fig. 5.

The curves shown in Fig. 5(1)~5(13) are results describing the different quantum beat phenomenon with the changing of two-photon detuning in the range of + 70MHz~-50MHz. Generally, the quantum beat phenomenon becomes weaker when the two-photon detuning gradually approaches to 0, and it will be stronger when the two-photon detuning is gradually away from 0.

In the following part, we make a simple analysis to explain this phenomenon. When pump 1 and pump 2 resonantly couple level |1> to level |5>, nearly all atoms are excited to level |5>, which causes that signal 1 and signal 2 can only be generated via one transition path (level |5>→level |3>→level |1>) according to the transition selection rule. Therefore, quantum beat phenomenon can hardly be observed shown in Fig. 5(8). On the contrary, if two pump beams are gradually away from resonance between level |1> and level |5>, the coupling between two pump beams and the atomic transition 5S_{1/2}(F = 3)→4D_{3/2}(F” = 3) becomes stronger, then signal 1 and signal 2 can be generated via two transition paths, and quantum beat phenomenon can be observed.

## 4. Conclusion

In conclusion, we generate a hyper-entangled photon pair in a cold ^{85}Rb atomic ensemble. To demonstrate the polarization entanglement of photon pairs, we measure two-photon interference curves and check the violation of CHSH inequality. Besides, we use the quantum beat phenomenon observed in our experiment to verify the time-frequency entanglement of photon pairs. All results clearly demonstrate these photons with special wavelength are hyper-entangled. Our work is very useful in high-dimensional and long-distance quantum communication.

## Funding

National Natural Science Foundation of China (Grant Nos. 61275115, 61435011 and 61525504).

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