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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 Phys. Rev. Lett. 113 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
${\rm{LG}}_0^0$ mode and${\rm{LG}}_1^0$ mode. However, the optimal pump mode is rather complicated and difficult to experimentally generate, in the alternative scheme the${\rm{LG}}_1^0$ mode is used as the pump field to obtain the optimal entanglement. In the experiment, the${\rm{LG}}_1^0$ mode is produced by converting the HG11 mode with a π/2 converter, and here the HG11 mode is achieved by tailoring the fundamental mode with a four-quadrant phase mask and a filtering cavity. Then the${\rm{LG}}_0^0$ mode or${\rm{LG}}_1^0$ 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 HG01 and HG10. 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${\rm{LG}}_0^0$ 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${\rm{LG}}_1^0$ 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.-
Keywords:
- quantum optics /
- entanglement /
- optical parametric amplifier
[1] Furusawa A, Sørensen J L, Braunstein S L, Fuchs C A, Kimble H J, Polzik E S 1998 Science 282 706
Google Scholar
[2] Jing J T, Zhang J, Yan Y, Zhao F G, Xie C D, Peng K C 2003 Phys. Rev. Lett. 90 167903
Google Scholar
[3] Giovannetti V, Lloyd S, Maccone L 2004 Science 306 1330
Google Scholar
[4] Alexander R N, Wang P, Sridhar N, Chen M, Pfister O, Menicucci N C 2016 Phys. Rev. A 94 032327
Google Scholar
[5] Kwiat P G 1997 J. Mod. Opt. 44 2173
Google Scholar
[6] Barreiro J T, Wei T, Kwiat P G 2008 Nat. Phys. 4 282
Google Scholar
[7] Schuck C, Huber G, Kurtsiefer C, Weinfurter H 2006 Phys. Rev. Lett. 96 190501
Google Scholar
[8] Chen K, Li C M, Zhang Q, Chen Y A, Goebel A, Chen S, Mair A, Pan J W 2007 Phys. Rev. Lett. 99 120503
Google Scholar
[9] Gao W B, Xu P, Yao X C, Gühne O, Cabello A, Lu C Y, Peng C Z, Chen Z B, Pan J W 2010 Phys. Rev. Lett. 104 020501
Google Scholar
[10] Barreiro J T, Langford N K, Peters N A, Kwiat P G 2005 Phys. Rev. Lett. 95 260501
Google Scholar
[11] Wang X L, Luo Y H, Huang H L, Chen M C, Su Z E, Liu C, Chen C, Li W, Fang Y Q, Jiang X, Zhang J, Li L, Liu N L, Lu C Y, Pan J W 2018 Phys. Rev. Lett. 120 260502
Google Scholar
[12] Coutinho dos Santos B, Dechoum K, Khoury A Z 2009 Phys. Rev. Lett. 103 230503
Google Scholar
[13] Liu K, Guo J, Cai C X, Guo S F, Gao J R 2014 Phys. Rev. Lett. 113 170501
Google Scholar
[14] Yang Y, Li F L 2009 Phys. Rev. A 80 022315
Google Scholar
[15] Lee S Y, Ji S W, Kim H J, Nha H 2011 Phys. Rev. A 84 012302
Google Scholar
[16] Hu L Y, Liao Z Y, Zubairy M S 2017 Phys. Rev. A 95 012310
Google Scholar
[17] Yan Z H, Jia X J, Su X L, Duan Z Y, Xie C D, Peng K C 2012 Phys. Rev. A 85 040305
Google Scholar
[18] Xin J, Qi J, Jing J T 2017 Opt. Lett. 42 366
Google Scholar
[19] Liu K, Guo J, Cai C X, Zhang J X, Gao J R 2016 Opt. Lett. 41 5178
Google Scholar
[20] Lassen M, Delaubert V, Harb C C, Lam P K, Treps N, Bachor H 2006 J. Eur. Opt. Soc. 1 06003
Google Scholar
[21] Guo J, Cai C X, Ma L, Liu K, Sun H X, Gao J R 2017 Opt. Express 25 4985
Google Scholar
[22] Navarrete-Benlloch C, Roldán E, de Valcárcel G J 2008 Phys. Rev. Lett. 100 203601
Google Scholar
[23] Beijersbergen M W, Allen L, van der Veen H E L O, Woerdman J P 1993 Opt. Commun. 96 123
Google Scholar
[24] Martinelli M, Huguenin J A O, Nussenzveig P, Khoury A Z 2004 Phys. Rev. A 70 013812
Google Scholar
[25] Abramochkin E, Volostnikov V 1991 Opt. Commum. 83 123
Google Scholar
[26] Duan L M, Giedke G, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 84 2722
Google Scholar
[27] Simon R 2000 Phys. Rev. Lett. 84 2726
Google Scholar
[28] Lassen M, Delaubert V, Janousek J, Wagner K, Bachor H, Lam P K, Treps N, Buchhave P, Fabre C, Harb C C 2007 Phys. Rev. Lett. 98 083602
Google Scholar
[29] Taylor M A, Janousek J, Daria V, Knittel J, Hage B, Bachor H, Bowen W P 2013 Nat. Photon. 7 229
Google Scholar
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图 1 抽运模式分别为
${\rm{LG}}_0^0$ 模(绿色)、${\rm{LG}}_1^0$ 模(红色)以及最佳抽运模式${\rm{L}}{{\rm{G}}_{\rm{opt}}}$ 模(蓝色)三种情况下, 纠缠不可分度随归一化抽运功率$p_{{\rm{th}}}^{{\rm{00}} \to {\rm{00}}}$ 变化的理论曲线(所取参数为${\eta _{{\rm{esc}}}} = 1$ ,$\varOmega = 0$ )Figure 1. Theoretical inseparability against normalized pump power
$p_{{\rm{th}}}^{{\rm{00}} \to {\rm{00}}}$ for three pump modes,${\rm{LG}}_0^0$ (green solid line),${\rm{LG}}_1^0$ (red solid line) and the optimal pump mode${\rm{L}}{{\rm{G}}_{\rm{opt}}}$ (blue solid line) under ideal conditions. The parameters are${\eta _{{\rm{esc}}}} = 1$ ,$\varOmega = 0$ .
图 2 实验装置图, 其中, RC, 三镜环形腔; DBS, 双色分束器; HWP, 半波片; PBS, 偏振分束器; PZT, 压电陶瓷; DP, 道威棱镜; MC, 模式转换器; FQ-PM, 四象限相位片; KTP, KTiOPO4晶体;
${{\text{π}}/ {\rm{2}}}$ MC,${{\text{π}}/ {\rm{2}}}$ 模式转换器; SA, 频谱分析仪; BHD, 平衡零拍测量装置Figure 2. Experimental setup. RC, three-mirror ring cavity; DBS, dichroic beamsplitter; HWP, half wave plate; PBS, polarizing beamsplitter; PZT, piezoelectric transducer; DP, Dove Prism; MC, mode converter; FQ-PM, four-quadrant phase mask; KTP, KTiOPO4 crystal;
${{\text{π}}/ {\rm{2}}}$ MC,${{\text{π}}/ {\rm{2}}}$ mode converter; SA, spectrum analyzer; BHD, balanced homodyne detector.
图 3
${\rm{LG}}_0^0$ 模做抽运场的纠缠测量结果 (a1)${\rm{H}}{{\rm{G}}_{01}}$ 模的振幅关联噪声谱; (a2)${\rm{H}}{{\rm{G}}_{01}}$ 模的相位关联噪声谱; (b1)${\rm{H}}{{\rm{G}}_{10}}$ 模的振幅关联噪声谱; (b2)${\rm{H}}{{\rm{G}}_{10}}$ 模的相位关联噪声谱; 黑线(2), SNL; 图(a1)和(b1)中, 红线(1),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} + {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle,$ $\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} + {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; 蓝线(3),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} - {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} - {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; 图(a2)和(b2)中, 红线(1),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i}},{\rm{H01}}}} - } \right.} \right.$ $\left. {\left. {{{\hat P}_{{\rm{s}},{\rm{H01}}}}} \right)} \right\rangle, \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} - {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; 蓝线(3),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H01}}}} + {{\hat P}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} + {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ Figure 3. Measured quantum correlations using the
${\rm{LG}}_0^0$ mode: (a1) Amplitude correlation noise of${\rm{H}}{{\rm{G}}_{01}}$ modes; (a2) phase correlation noise of${\rm{H}}{{\rm{G}}_{01}}$ modes; (b1) amplitude correlation noise of${\rm{H}}{{\rm{G}}_{10}}$ modes; (b2) phase correlation noise of${\rm{H}}{{\rm{G}}_{10}}$ modes. (a1) and (b1) Trace1 (red line),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} + {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} + {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; Trace3 (blue line),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} - {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle,$ $\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} - {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ . (a2) and (b2) Trace1 (red line),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H01}}}} - {{\hat P}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} - {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; Trace3 (blue line),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H01}}}} + {{\hat P}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} + {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ . Trace2 (black line), SNL.
图 4
${\rm{LG}}_1^0$ 模做抽运场纠缠测量结果 (a1)${\rm{H}}{{\rm{G}}_{01}}$ 模的振幅关联噪声谱; (a2)${\rm{H}}{{\rm{G}}_{01}}$ 模的相位关联噪声谱; (b1)${\rm{H}}{{\rm{G}}_{10}}$ 模的振幅关联噪声谱; (b2)${\rm{H}}{{\rm{G}}_{10}}$ 模的相位关联噪声谱; 黑线(2), SNL; 图(a1)和(b1)中, 红线(1),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} + {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle ,$ $\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} + {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; 蓝线(3),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} - {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} - {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; 图(a2)和(b2)中, 红线(1),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i}},{\rm{H01}}}} - } \right.} \right.$ $\left. {\left. {{{\hat P}_{{\rm{s}},{\rm{H01}}}}} \right)} \right\rangle,\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} - {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; 蓝线(3),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H01}}}} + {{\hat P}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} + {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ Figure 4. Measured quantum correlations with
${\rm{LG}}_1^0$ pumping : (a1) Amplitude correlation noise of${\rm{H}}{{\rm{G}}_{01}}$ modes; (a2) phase correlation noise of${\rm{H}}{{\rm{G}}_{01}}$ modes; (b1) amplitude correlation noise of${\rm{H}}{{\rm{G}}_{10}}$ modes; (b2) phase correlation noise of${\rm{H}}{{\rm{G}}_{10}}$ modes. (a1) and (b1) Trace1 (red line),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} + {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} + {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; Trace3 (blue line),$\left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H01}}}} - {{\hat X}_{{\rm{s,H01}}}}} \right)} \right\rangle,$ $ \left\langle {{\Delta ^2}\left( {{{\hat X}_{{\rm{i,H10}}}} - {{\hat X}_{{\rm{s,H10}}}}} \right)} \right\rangle$ . (a2) and (b2) Trace1 (red line),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H01}}}} - {{\hat P}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} - {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ ; Trace3 (blue line),$\left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H01}}}} + {{\hat P}_{{\rm{s,H01}}}}} \right)} \right\rangle , \left\langle {{\Delta ^2}\left( {{{\hat P}_{{\rm{i,H10}}}} + {{\hat P}_{{\rm{s,H10}}}}} \right)} \right\rangle $ . Trace2 (black line), SNL.
图 5 不同抽运模式下, 轨道角动量纠缠和自旋角动量纠缠的不可分度 (a)轨道角动量纠缠; (b)自旋角动量纠缠; 蓝线(1)和红线(2)分别对应
${\rm{LG}}_0^0$ 模和${\rm{LG}}_1^0$ 模做抽运的结果; 不可分度低于2表示存在纠缠Figure 5. Experimental measurement of inseparability for the orbital angular momentum and spin angular momentum with different pump mode: (a) Orbital angular momentum; (b) spin angular momentum. Blue line (1) and red line (2) respectively represent the results using the
${\rm{LG}}_0^0$ mode and${\rm{LG}}_1^0$ mode. Values below 2 indicate entanglement.表 1 不同抽运模式下的耦合系数
Table 1. Coupling coefficient with different pump modes.
抽运模式 ${\rm{LG}}_0^0$ ${\rm{LG}}_1^0$ ${\rm{LG}}_2^0$ … ${1 / 3}{\rm{LG}}_0^0 + {2 / 3}{\rm{LG}}_1^0$ 耦合系数${\varGamma _{{\rm{0}}p}}$ ${1 / 2}$ ${1 / {\sqrt 2 }}$ 0 0 ${{\sqrt 3 } / 2}$ 
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[1] Furusawa A, Sørensen J L, Braunstein S L, Fuchs C A, Kimble H J, Polzik E S 1998 Science 282 706
Google Scholar
[2] Jing J T, Zhang J, Yan Y, Zhao F G, Xie C D, Peng K C 2003 Phys. Rev. Lett. 90 167903
Google Scholar
[3] Giovannetti V, Lloyd S, Maccone L 2004 Science 306 1330
Google Scholar
[4] Alexander R N, Wang P, Sridhar N, Chen M, Pfister O, Menicucci N C 2016 Phys. Rev. A 94 032327
Google Scholar
[5] Kwiat P G 1997 J. Mod. Opt. 44 2173
Google Scholar
[6] Barreiro J T, Wei T, Kwiat P G 2008 Nat. Phys. 4 282
Google Scholar
[7] Schuck C, Huber G, Kurtsiefer C, Weinfurter H 2006 Phys. Rev. Lett. 96 190501
Google Scholar
[8] Chen K, Li C M, Zhang Q, Chen Y A, Goebel A, Chen S, Mair A, Pan J W 2007 Phys. Rev. Lett. 99 120503
Google Scholar
[9] Gao W B, Xu P, Yao X C, Gühne O, Cabello A, Lu C Y, Peng C Z, Chen Z B, Pan J W 2010 Phys. Rev. Lett. 104 020501
Google Scholar
[10] Barreiro J T, Langford N K, Peters N A, Kwiat P G 2005 Phys. Rev. Lett. 95 260501
Google Scholar
[11] Wang X L, Luo Y H, Huang H L, Chen M C, Su Z E, Liu C, Chen C, Li W, Fang Y Q, Jiang X, Zhang J, Li L, Liu N L, Lu C Y, Pan J W 2018 Phys. Rev. Lett. 120 260502
Google Scholar
[12] Coutinho dos Santos B, Dechoum K, Khoury A Z 2009 Phys. Rev. Lett. 103 230503
Google Scholar
[13] Liu K, Guo J, Cai C X, Guo S F, Gao J R 2014 Phys. Rev. Lett. 113 170501
Google Scholar
[14] Yang Y, Li F L 2009 Phys. Rev. A 80 022315
Google Scholar
[15] Lee S Y, Ji S W, Kim H J, Nha H 2011 Phys. Rev. A 84 012302
Google Scholar
[16] Hu L Y, Liao Z Y, Zubairy M S 2017 Phys. Rev. A 95 012310
Google Scholar
[17] Yan Z H, Jia X J, Su X L, Duan Z Y, Xie C D, Peng K C 2012 Phys. Rev. A 85 040305
Google Scholar
[18] Xin J, Qi J, Jing J T 2017 Opt. Lett. 42 366
Google Scholar
[19] Liu K, Guo J, Cai C X, Zhang J X, Gao J R 2016 Opt. Lett. 41 5178
Google Scholar
[20] Lassen M, Delaubert V, Harb C C, Lam P K, Treps N, Bachor H 2006 J. Eur. Opt. Soc. 1 06003
Google Scholar
[21] Guo J, Cai C X, Ma L, Liu K, Sun H X, Gao J R 2017 Opt. Express 25 4985
Google Scholar
[22] Navarrete-Benlloch C, Roldán E, de Valcárcel G J 2008 Phys. Rev. Lett. 100 203601
Google Scholar
[23] Beijersbergen M W, Allen L, van der Veen H E L O, Woerdman J P 1993 Opt. Commun. 96 123
Google Scholar
[24] Martinelli M, Huguenin J A O, Nussenzveig P, Khoury A Z 2004 Phys. Rev. A 70 013812
Google Scholar
[25] Abramochkin E, Volostnikov V 1991 Opt. Commum. 83 123
Google Scholar
[26] Duan L M, Giedke G, Cirac J I, Zoller P 2000 Phys. Rev. Lett. 84 2722
Google Scholar
[27] Simon R 2000 Phys. Rev. Lett. 84 2726
Google Scholar
[28] Lassen M, Delaubert V, Janousek J, Wagner K, Bachor H, Lam P K, Treps N, Buchhave P, Fabre C, Harb C C 2007 Phys. Rev. Lett. 98 083602
Google Scholar
[29] Taylor M A, Janousek J, Daria V, Knittel J, Hage B, Bachor H, Bowen W P 2013 Nat. Photon. 7 229
Google Scholar
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