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利用双Hanbury Brown-Twiss探测系统理论分析并在实验上精确测量了不同光场的高阶光子关联g(n) (n > 2). 探测系统通过四个单光子计数模块, 探测分析光子时间关联的联合分布概率. 在理论上, 考虑实际探测系统背景噪声和系统效率的影响, 分析研究了热态、相干态、压缩真空态和Fock态的三阶及四阶光子关联的结果, 及其随光场入射光强、压缩参数及光子数的变化. 并在实验中研究了探测系统分辨时间和计数率对相干态和热态的三阶及四阶光子关联的影响. 在分辨时间为210 ns, 计数率为80 kc/s时, 准确测量得到在零延迟处热态的三阶及四阶光子关联, 相对理论值的统计偏差分别为0.3%和0.8%. 此外还测量得到了不同延迟时间下热态的高阶光子关联的结果. 实验表明综合对各种影响因素的分析可精确测量光场的高阶光子关联, 该方法在量子关联成像及光场特性分析中有着重要的应用.
High-order photon correlations of light fields are important for characterizing the quantum nature. Since Hanbury Brown and Twiss conducted the pioneering experiments in the 1950s, the HBT effect has inspired extensive research on high-order photon correlation in quantum optics, quantum information, and quantum imaging. The Single-photon counting module is one of the most widely used single-photon detectors. Due to its high detection efficiency and low dark counts in the visible and near-infrared region, it is reasonably chosen for basic research on quantum mechanics. Many researches have demonstrated that the maximum value of second-order photon correlation g(2)(τ) at zero delay (τ = 0) can be used to distinguish different light fields. Therefore, the HBT scheme containing two single photon detectors have been widely used in many advanced studies, such as space interference, ghost imaging, single photon detection with high efficiency, etc. However, higher-order photon correlations g(n) (n > 2) can reveal more measurable characteristics of light fields, such as information about the non-Gaussian scattering process, the skewness and kurtosis of photon number distribution, etc. When the extended HBT scheme is used to measure higher-order photon correlations, the experimental conditions including quantum efficiency and background noise greatly affect the photon correlation measurement. The influences of the counting rate and resolution time of the detection system on the measurements are also very important and cannot be ignored. Therefore, the comprehensive considering of various influence factors is necessary for accurately measuring the high-order photon correlations and also a challenge. In this paper, we present a method based on double Hanbury Brown-Twiss scheme for the accurate measuring of high-order photon correlations g(n) (n > 2). The system consists of four single photon counting modules and is used to detect and analyze the joint distribution probability of temporal photon correlation. Considering the effects of the background noise and overall efficiency, theoretically, we analyze the correlations of the third- and fourth-order photon with the incident light intensity, squeezing parameter and photon number respectively for thermal state, coherent state, squeezed vacuum state, and Fock state. Meanwhile, experimentally we study the influences of resolution time and counting rate on correlations of the coherent state and thermal state with third- and fourth-order photon. On condition that the resolution time is 210 ns and the counting rate is 80 kc/s, the correlations of third and fourth-order photon with the thermal state at zero time delay are accurately measured, and the relative statistical deviations of the measured vales from the theoretical values are 0.3% and 0.8%, respectively. In addition, the third- and fourth-order photon correlations of the thermal state at different delay times are also observed. It is demonstrated that the high-order photon correlations of light fields are measured accurately by comprehensively analyzing various influencing factors. This technique provides a promising and useful tool to investigate quantum correlated imaging and quantum coherence of light fields. -
Keywords:
- high-order photon correlation /
- single photon counting /
- delay time
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图 1 双HBT装置理论模型. B0, B1, B2, B3: 分束器; D1, D2, D3, D4: 探测器. 带括号的字母L, N, K等表示各路分光的光子数
Fig. 1. Theoretical model of double HBT scheme. B0, B1, B2, B3: Beamsplitter; D1, D2, D3, D4: Detector. The letters in parentheses L, N, K, et al, denote the photon numbers of splitting light paths, respectively.
图 2 单模热光场的 (a) 三阶光子关联
$g_{\rm{T}}^{(3)}$ 和 (b) 四阶光子关联$g_{\rm{T}}^{(4)}$ 随平均光子数$\alpha $ 的变化Fig. 2. (a) Third-order, and (b) fourth-order photon correlations
$g_{\rm{T}}^{(3)}$ and$g_{\rm{T}}^{(4)}$ of single-mode thermal state versus the mean photon number$\alpha $ .
图 3 压缩真空态的 (a) 二阶光子关联
$g_{{\rm{SVS}}}^{(2)}$ , (b) 三阶光子关联$g_{{\rm{SVS}}}^{(3)}$ 和(c) 四阶光子关联$g_{{\rm{SVS}}}^{(4)}$ 随压缩因子r的变化Fig. 3. (a) Second-order, (b) third-order, and (c) fourth-order photon correlations of single-mode squeezed vacuum state versus the squeezing parameter r.
图 4 Fock态的三阶(蓝色空心方块)和四阶(红色实心圆圈)光子关联随光子数的变化结果 (a)
$\gamma = 0.001$ ,$\eta = 0.5$ ; (b)$\gamma = 0.001$ ,$\eta = 0.1$ Fig. 4. The third-order (blue hollow square) and fourth-order (red solid circle) photon correlations of Fock state versus the photon number: (a)
$\gamma = 0.001$ ,$\eta = 0.5$ ; (b)$\gamma = 0.001$ ,$\eta = 0.1$ .
图 5 实验装置原理图(ISO, 隔离器; HWP, 半波片; PBS, 偏振分束器; M1, M2, 反射镜; L, 光学透镜; RGGD, 旋转的毛玻璃; BS1, BS2, BS3, 分束器; D1, D2, D3, D4, 单光子计数模块SPCM; DAS, 数据采集系统)
Fig. 5. Schematic illustration of the experimental setup (ISO, isolator; HWP, half-wave plate; PBS, polarized beam splitter; M1, M2, mirror; L, optical lens; RGGD, rotating ground glass disk; BS1, BS2, BS3, beam splitter; D1, D2, D3 and D4, single photon counting module, SPCM; DAS, data acquisition system).
图 6 热态和相干态的 (a) 三阶光子关联、(b) 四阶光子关联随计数率的测量结果, 探测器分辨时间为210 ns. 图中红色和蓝色实线分别为热态与相干态高阶光子关联的理想值
Fig. 6. Measured (a) third-order and (b) fourth-order photon correlations of thermal state and coherent state versus the counting rate for resolution time of 210 ns. The red and blue solid lines are the ideal results of the high-order photon correlations of thermal state and coherent state.
图 7 热态和相干态的 (a) 三阶光子关联、(b) 四阶光子关联随分辨时间的测量结果, 计数率为80 kc/s. 图中红色和蓝色实线分别为热态与相干态高阶光子关联的理想值
Fig. 7. Measured (a) third-order and (b) fourth-order photon correlations of thermal state and coherent state versus the resolution time for counting rate of 80 kc/s. The red and blue solid lines are the ideal results of the high-order photon correlations of thermal state and coherent state.
图 8 热态的三阶光子关联随延迟时间的变化, 分辨时间为210 ns, 计数率为80 kc/s. 蓝点表示实验结果, 黑色实线为理论拟合(a) τ2 = 0 μs; (b) τ2 = –10 μs; (c) τ2 = 10 μs. 其中图(a)零延迟处的峰值为
$5.98_{ - 0.018}^{ + 0.018}$ Fig. 8. Measured third-order photon correlation of thermal state versus delay times for resolution times of 210 ns and counting rate of 80 kc/s. The blue dots and black solid curves are the experimental and theoretical results, respectively: (a) τ2 = 0 μs; (b) τ2 = –10 μs; (c) τ2 = 10 μs. The peak value of g(3) in Fig. (a) is
$5.98_{ - 0.018}^{ + 0.018}$ .
图 9 (a) τ3 = 0 μs, (b) τ3 = –10 μs, (c) τ3 = 10 μs 时, 全时延条件下热态的四阶光子关联, 图(a)中零延迟处的峰值为
$23.8_{ - 0.19}^{ + 0.19}$ Fig. 9. The fourth-order photon correlations of thermal state at complete time delays for (a) τ3 = 0 μs, (b) τ3 = –10 μs, (c) τ3 = 10 μs. The peak value of g(4) in Fig. (a) is
$23.8_{ - 0.19}^{ + 0.19}$ .
图 10 热态的四阶光子关联g(4)(τ1, τ2, τ3)随延迟时间τ1变化的测量结果, 分辨时间为210 ns, 计数率为80 kc/s. 圆点表示实验结果, 黑色实线为理论拟合
Fig. 10. Measured fourth-order photon correlation of thermal state versus delay time τ1 for resolution times of 210 ns and counting rate of 80 kc/s. The dots are the experimental results, and the black solid curves are the theoretical fittings.
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[1] Mandel L, Wolf E 1995 Optical Correlation and Quantum Optics (Cambridge: Cambridge University Press) p573
[2] Hanbury Brown R, Twiss R Q 1956 Nature 177 4497
[3] Glauber R J 1963 Phys. Rev. 130 2529
Google Scholar
[4] Glauber R J 1963 Phys. Rev. Lett. 10 84
Google Scholar
[5] Eisaman M D, Fan J, Migdall A, Polyakov S V 2011 Rev. Sci. Instrum. 82 071101
Google Scholar
[6] Kimble H J 2008 Nature 453 1023
Google Scholar
[7] Andersen U L, Neergaard-Nielsen J S, van Loock P, Furusawa A 2015 Nat. Phys. 11 713
Google Scholar
[8] Reiserer A, Rempe G 2015 Rev. Mod. Phys. 87 1379
Google Scholar
[9] Banaszek K, Demkowicz-Dobrzanski R, Walmsley I A 2009 Nat. Photonics 3 673
Google Scholar
[10] Matthews J C F, Zhou X Q, Cable H, Shadbolt P J, Saunders D J, Durkin G A, Pryde G J, O'Brien J L 2016 npj Quantum Inf. 2 16023
Google Scholar
[11] Tang Y L, Yin H L, Chen S J, Liu Y, Zhang W J, Jiang X, Zhang L, Wang J, You L X, Guan J Y, Yang D X, Wang Z, Liang H, Zhang Z, Zhou N, Ma X F, Chen T Y, Zhang Q, Pan J W 2014 Phys. Rev. Lett. 113 190501
Google Scholar
[12] Wang C, Song X T, Yin Z Q, Wang S, Chen W, Zhang C M, Guo G C, Han Z F 2015 Phys. Rev. Lett. 115 160502
Google Scholar
[13] Comandar L C, Lucamarini M, Fröhlich B, Dynes J F, Sharpe A W, Tam S W B, Yuan Z L, Penty R V, Shields A J 2016 Nat. Photonics 10 312
Google Scholar
[14] Lodahl P, Mahmoodian S, Stobbe S 2015 Rev. Mod. Phys. 87 347
Google Scholar
[15] Li G, Tian Y, Wu W, Li S K, Li X Y, Liu Y X, Zhang P F, Zhang T C 2019 Phys. Rev. Lett. 123 253602
Google Scholar
[16] Bai J D, Liu S, Wang J Y, He J, Wang J M 2020 IEEE J. Sel. Top. Quantum Electron. 26 1600106
[17] Guo Y Q, Yang R C, Li G, Zhang P F, Zhang Y C, Wang J M, Zhang T C 2011 J. Phys. B: At. Mol. Opt. Phys. 44 205502
Google Scholar
[18] Guo Y Q, Guo X M, Li P, Shen H, Zhang J, Zhang T C 2018 Ann. Phys. (Berlin) 530 1800138
Google Scholar
[19] Kimble H J, Dagenais M, Mandel L 1977 Phys. Rev. Lett. 39 691
Google Scholar
[20] Davidovich L 1996 Rev. Mod. Phys. 68 127
Google Scholar
[21] Boitier F, Godard A, Rosencher E, Fabre C 2009 Nat. Phys. 5 267
Google Scholar
[22] Sun F W, Shen A, Dong Y, Chen X D, Guo G C 2017 Phys. Rev. A 96 023823
Google Scholar
[23] 兰豆豆, 郭晓敏, 彭春生, 姬玉林, 刘香莲, 李璞, 郭龑强 2017 66 120502
Google Scholar
Lan D D, Guo X M, Peng C S, Ji Y L, Liu X L, Li P, Guo Y Q 2017 Acta Phys. Sin. 66 120502
Google Scholar
[24] Guo Y Q, Peng C S, Ji Y L, Li P, Guo Y Y, Guo X M 2018 Opt. Express 26 5991
Google Scholar
[25] Glauber R J 1963 Phys. Rev. 131 2766
Google Scholar
[26] Smith T A, Shih Y 2018 Phys. Rev. Lett. 120 063606
Google Scholar
[27] Chen X H, Zhai Y H, Zhang D, Wu L A 2006 Opt. Lett. 31 2441
Google Scholar
[28] Gu X R, Huang K, Pan H F, Wu E, Zeng H Q 2012 Opt. Express 20 2399
Google Scholar
[29] Ryczkowski P, Barbier M, Friberg A T, Dudley J M, Genty G 2016 Nat. Photonics 10 167
Google Scholar
[30] 刘雪峰, 姚旭日, 李明飞, 俞文凯, 陈希浩, 孙志斌, 吴令安, 翟光杰 2013 62 184205
Google Scholar
Liu X F, Yao X R, Li M F, Yu W K, Chen X H, Sun Z B, Wu L A, Zhai G J 2013 Acta Phys. Sin. 62 184205
Google Scholar
[31] Xu D Q, Song X B, Li H G, Zhang D J, Wang H B, Xiong J, Wang K 2015 Appl. Phys. Lett. 106 171104
Google Scholar
[32] Yu H, Lu R H, Han S S, Xie H L, Du G H, Xiao T Q, Zhu D M 2016 Phys. Rev. Lett. 117 113901
Google Scholar
[33] Cao M T, Yang X, Wang J W, et al. 2016 Opt. Lett. 41 5349
Google Scholar
[34] Liu X L, Shi J H, Wu X Y, Zeng G H 2018 Sci. Rep. 8 5012
Google Scholar
[35] Yao X, Zhang W, Li H, You L X, Wang Z, Huang Y D 2018 Opt. Lett. 43 759
Google Scholar
[36] Sun S, Liu W T, Gu J H, Lin H Z, Jiang L, Xu Y K, Chen P X 2019 Opt. Lett. 44 5993
Google Scholar
[37] Lemieux P A, Durian D J 1999 J. Opt. Soc. Am. A 16 1651
[38] Horikiri T, Schwendimann P, Quattropani A, Höfling S, Forchel A, Yamamoto Y 2010 Phys. Rev. B 81 033307
Google Scholar
[39] Hodgman S S, Dall R G, Manning A G, Baldwin K G H, Truscott A G 2011 Science 331 1046
Google Scholar
[40] Hamsen C, Tolazzi K N, Wilk T, Rempe G 2017 Phys. Rev. Lett. 118 133604
Google Scholar
[41] Stevens M J, Glancy S, Nam S W, Mirin R P 2014 Opt. Express 22 3244
Google Scholar
[42] Zhou Y, Simon J, Liu J, Shih Y 2010 Phys. Rev. A 81 043831
Google Scholar
[43] 杨宏恩, 韦联福 2019 68 234202
Google Scholar
Yang H E, Wei L F 2019 Acta Phys. Sin. 68 234202
Google Scholar
[44] 李诗宇, 田剑锋, 杨晨, 左冠华, 张玉驰, 张天才 2018 67 234202
Google Scholar
Li S Y, Tian J F, Yang C, Zuo G H, Zhang Y C, Zhang T C 2018 Acta Phys. Sin. 67 234202
Google Scholar
[45] Li Y, Li G, Zhang Y C, Wang X Y, Zhang J, Wang J M, Zhang T C 2007 Phys. Rev. A 76 013829
Google Scholar
[46] Bachor H A, Ralph T C 2004 A Guide to Experiments in Quantum Optics (Berlin: Wiley) p267
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