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
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:high-order photon correlation/ single photon counting/ delay time
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2.1.理论模型
四个相同的SPCM组成的双HBT理论模型如图1所示. 当光子数分布为Pin(n)的光场$|\psi\rangle$入射时, 考虑系统的总效率$\eta $, 该效率包括传输效率、收集效率及探测系统的量子效率, 由分束器B0模拟, 其余部分不再受效率影响. 同时考虑背景噪声$|\beta\rangle$的影响, 背景噪声$|\beta\rangle$相对入射光场$|\psi\rangle$较弱, 主要来源于探测系统的暗计数和背景辐射, 其光子数分布服从泊松分布$P_{\rm in}(n)=\gamma^{n}{\rm exp}{(-\gamma)}/n!$, 其中$\gamma = {\left| \beta \right|^2}$. 随后光场经过三个50/50无损分束器B1, B2, B3, 到达四个单光子探测器D1, D2, D3, D4. 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 Figure1. 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.
在不同计数率和分辨时间条件下, 分别测量了热态和相干态的高阶光子关联. 图6表示探测系统的分辨时间固定210 ns时, 热态和相干态的三阶光子关联g(3)(图6(a))和四阶光子关联g(4)(图6(b))随计数率变化的测量结果. 红色圆形表示热态的高阶光子关联测量结果, 蓝色三角表示相干态高阶光子关联的结果. 在计数率较小或较大时, 热态的实验结果与理论预期偏差较大. 这是由于在计数率较低时, 受到杂散光和背景噪声的影响, 难以获取足够的光子群聚信息, 并且受背景噪声及采样样本数不足的影响较大, 使得测量统计波动增大, 导致标准差较大. 相反, 当计数率较大时, 多光子事件显著增多, 且SPCM不能分辨响应多光子, 造成对多光子事件的丢失, 从而造成计数率过小和过大这两种情况都不能准确地反映光场本质. 相干光的高阶光子关联从0增加至1后基本保持恒定. 由图6可知, 在计数率为80 kc/s时, 测量得到的光场三阶及四阶光子关联与理论值最为接近. 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 Figure6. 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为探测系统的计数率约在80 kc/s时, 热态和相干态的三阶光子关联g(3)(图7(a))和四阶光子关联g(4)(图7(b))随分辨时间变化的测量结果. 图中红色圆形表示热态的高阶光子关联, 蓝色三角表示相干态高阶光子关联的测量结果. 在分辨时间较小时(小于探测系统的死时间), 热光和相干光的高阶光子关联都为0, 随着分辨时间的增大, 在高于系统死时间后热光的高阶光子关联迅速增加到理论值附近. 随着分辨时间继续增大, 高阶光子关联都趋向于1. 当分辨时间远大于热光场的相干时间后, 热光场可以看作过渡到泊松分布的平稳光场. 由图7可知, 在分辨时间为210 ns时, 测量得到的光场三阶及四阶光子关联与理论值最为接近. 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 Figure7. 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.
23.3.不同延迟时间下热态的高阶光子关联 -->
3.3.不同延迟时间下热态的高阶光子关联
在探测系统的分辨时间为210 ns、计数率为80 kc/s的条件下, 测量分析了热态的高阶光子关联g(3)(τ1, τ2)和g(4)(τ1, τ2, τ3)随延迟时间变化的结果. 图8中蓝点为热态的g(3)(τ1, τ2)随延迟时间τ1变化的测量结果, 图8(a)对应τ2 = 0的情况, 当τ1 = 0即零延迟处, 峰值g(3)(0)为$5.98_{ - 0.018}^{ + 0.018}$, 相对理论值3! = 6的统计偏差为0.3%. 图8(b)中τ2 = –10 μs, 当τ1 = 0或τ1 = –10 μs时, g(3)的值均为2. 图8(c)中τ2 = 10 μs时情况与图8(b)中的结果类似, 峰值出现在τ1 = 0或τ1 = 10 μs. 同时根据实验条件参数, 对测量结果进行了理论拟合, 如图8中黑色实线所示. 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$5.98_{ - 0.018}^{ + 0.018}$ Figure8. 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 $5.98_{ - 0.018}^{ + 0.018}$.
为了更全面地研究热态的四阶光子关联随延迟时间的变化, 测量了全时延条件下热态的四阶光子关联结果, 如图9所示。图9(a)—图9(c)分别对应τ3 = 0 μs, τ3 = –10 μs和 τ3 = 10 μs. 图9(a)中峰值对应${\tau _1} = {\tau _2} = {\tau _3} = 0$的情况, 此时g(4)(0)的值为$23.8_{ - 0.19}^{ + 0.19}$, 相对理论值4!=24的统计偏差为0.8%. 在图9中, 当三个延迟时间各不相等且都不为零时, 四路光子在不同的时延到达, g(4)的值为1; 当有一个延迟时间为0或有两个延迟时间相等时, 两路光子同时到达, g(4)的值为2; 当两路光子同时到达, 且另两路光子在另一延迟时间同时到达, 则g(4)的值为4; 当两个延迟时间相等并且为0或三个延迟时间相等时, 三路光子同时到达, g(4)的值为6. 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$23.8_{ - 0.19}^{ + 0.19}$ Figure9. 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 $23.8_{ - 0.19}^{ + 0.19}$.
与三阶光子关联的结果类似, 图10为热态的g(4)(τ1, τ2, τ3)随延迟时间τ1变化的测量结果, 图10(a)—图10(c)对应τ3 = 0, 图10(d)—图10(f)对应τ3 = –10 μs, 图10(g)—图10(i)对应τ3 = 10 μs, 每个τ3的取值下分析了τ2 = 0, –10, 10 μs三种不同的情况. 图10中的黑色实线为相应实验条件下的理论结果. 图10(a)中峰值对应${\tau _1} = {\tau _2} = {\tau _3} = 0$的情况, 此时g(4)(0)的测量极大值为23.8. 由图10可以看到g(4)的峰值变化规律与图9的结果保持一致. 结果表明, 热态光场在零延迟时间附近光子的聚束效应最强, n个光子同时到达的概率约为n个光子在不同时间到达概率的 n! 倍. 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 Figure10. 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.