Enhancement of continuous-variable hyperentanglement by optimizing pump mode
Authors: Li Juan , Li Jia-Ming , Cai Chun-Xiao , Sun Heng-Xin , Liu Kui , Gao Jiang-Rui ,   State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Opto-Electronics, Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China
Published: December 2020

In recent years, more and more researchers have paid attention to the hyperentanglement, because it plays a very important role in the quantum information and quantum communication. Continuous-variable hyperentangled state with orbital angular momentum and spin angular momentum has a promising application in the parallel processing of continuous-variable multi-channel quantum information and multiparameters quantum metrology. Recently Liu et al. (2014 <i>Phys. Rev. Lett.</i> <b>113</b> 170501) have produced a quantum correlation of about 1.00 dB for the continuous-variable hyperentangled state by a type-II non-degenerate optical parametric amplifier. The generation of continuous-variable hyperentangled state is affected by the mode matching between the pump field and the down-conversion field, since the hyperentanglement contains spatial high-order transverse mode entanglement. In the present paper, we first theoretically analyze the relationship between the pump and the two down-conversion modes and demonstrate the dependence of the inseparability on normalized pump power for the different pump modes. Hence, we find that the optimal pump mode is the superposition of <inline-formula><tex-math id="M3000">\begin{document}${\rm{LG}}_0^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3000.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3000.png"/></alternatives></inline-formula> mode and <inline-formula><tex-math id="M3001">\begin{document}${\rm{LG}}_1^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3001.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3001.png"/></alternatives></inline-formula> mode. However, the optimal pump mode is rather complicated and difficult to experimentally generate, in the alternative scheme the <inline-formula><tex-math id="M3002">\begin{document}${\rm{LG}}_1^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3002.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3002.png"/></alternatives></inline-formula> mode is used as the pump field to obtain the optimal entanglement. In the experiment, the <inline-formula><tex-math id="M3003">\begin{document}${\rm{LG}}_1^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3003.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3003.png"/></alternatives></inline-formula> mode is produced by converting the HG<sub>11</sub> mode with a π/2 converter, and here the HG<sub>11</sub> mode is achieved by tailoring the fundamental mode with a four-quadrant phase mask and a filtering cavity. Then the <inline-formula><tex-math id="M304">\begin{document}${\rm{LG}}_0^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M304.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M304.png"/></alternatives></inline-formula> mode or <inline-formula><tex-math id="M3005">\begin{document}${\rm{LG}}_1^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3005.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3005.png"/></alternatives></inline-formula> mode is used as the pump field to drive the non-degenerate optical parametric amplifier operating in spatial multimode. When the non-degenerate optical parametric amplifier is operated in the de-amplification, the hyperentanglement with orbital angular momentum and spin angular momentum is produced. The output entangled beams pass through polarization beam splitter and are analyzed by using the balanced homodyne detection systems with the local oscillator operating in the HG<sub>01</sub> and HG<sub>10</sub>. The noise of the phase quadrature or the amplitude quadrature is obtained, when the relative phase between the local oscillator and the signal beam is locked to π/2 or 0. Then the quantum correlations of orbital angular momentum and spin angular momentum can be deduced. The experimental results show that the continuous-variable hyperentanglement of light with a quantum correlation of (4.00 ± 0.02) dB is produced. Compared with the results of Liu et al. obtained by using the <inline-formula><tex-math id="M3006">\begin{document}${\rm{LG}}_0^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3006.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3006.png"/></alternatives></inline-formula> mode, the inseparability of orbital angular momentum and spin angular momentum entanglement are enhanced by approximately 96.2% and 96.3%, respectively, through using the <inline-formula><tex-math id="M3007">\begin{document}${\rm{LG}}_1^0$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3007.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="3-20181625-e-zhengbs-revised_M3007.png"/></alternatives></inline-formula> mode. Such a continuous-variable hyperentanglement may have promising applications in high-dimensional quantum information and multi-dimensional quantum measurement, and this approach is potentially extended to a discrete variable domain.