Analysis and measurement of high-order photon correlations of light fields

Guo Yan-Qiang; Wang Li-Jing; Wang Yu; Fang Xin; Zhao Tong; Guo Xiao-Min

doi:10.7498/aps.69.20200325

Abstract

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:

Authors and contacts

1.
Key Laboratory of Advanced Transducers and Intelligent Control System, Ministry of Education, College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China
2.
State Key Laboratory of Cryptology, Beijing 100878, China
3.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Shanxi University, Taiyuan 030006, China

Corresponding author: Guo Xiao-Min, guoxiaomin@tyut.edu.cn

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References

[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
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[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
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[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
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[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
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Cited By

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

Figure 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.

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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

Figure 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 $.

DownLoad: Full-Size Img PowerPoint

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

Figure 3. (a) Second-order, (b) third-order, and (c) fourth-order photon correlations of single-mode squeezed vacuum state versus the squeezing parameter r.

DownLoad: Full-Size Img PowerPoint

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) $\gamma = 0.001$, $\eta = 0.1$

Figure 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) $\gamma = 0.001$, $\eta = 0.1$.

DownLoad: Full-Size Img PowerPoint

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)

Figure 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).

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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

Figure 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.

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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

Figure 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.

DownLoad: Full-Size Img PowerPoint

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$5.98_{ - 0.018}^{ + 0.018}$

Figure 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 $5.98_{ - 0.018}^{ + 0.018}$.

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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$23.8_{ - 0.19}^{ + 0.19}$

Figure 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 $23.8_{ - 0.19}^{ + 0.19}$.

DownLoad: Full-Size Img PowerPoint

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

Figure 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.

DownLoad: Full-Size Img PowerPoint

[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 物理学报 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 物理学报 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 物理学报 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 物理学报 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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Vol. 69, Issue 17 - 2020-09-05

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