-
在MKID对未知光脉冲进行光子数检测的过程中,由于商用理想单光子源产品的缺乏,采用的测试单光子源是激光输出的相干光经强线性衰减而得到的贋单光子源。因此,测试光源虽然已经很弱(平均光子数仅为数个光子),但其光子数仍服从泊松分布。
假定在很短时间dt内,探测到单个光子的概率是:
$\Delta P(t)=\alpha I(t) \mathrm{d} t $
(1) 式中,α是探测器的灵敏度,它取决于探测面积和入射光的光谱范围; I(t)是在某一时间光的辐射强度。显然,在这个很短时间内,即Δt→0时,没有光子被探测到的概率是:1-ΔP(t)。所以,探测不到光子的概率可以等效为P0(t)=1-ΔP(t), 等价于exp[-ΔP(t)]。
假设在不同时间间隔内光子计数事件是独立的,那么在时间间隔t0~t0+T内不发生光子计数的联合概率可以表示为:
$ \begin{aligned} &\prod\limits_{t_{0}}^{t_{0}+T}[1-\Delta P(t)] \approx \prod\limits_{t_{0}}^{t_{0}+T} \exp [-\Delta P(t)]= \\ &\exp \left[-\sum\limits_{t_{0}}^{t_{0}+T} \Delta P(t)\right]=\exp \left[-\int_{t_{0}}^{t_{0}+T} \mathrm{~d} P(t)\right] \end{aligned} $
(2) 则在t0~t0+T内发生0个光子计数的概率为:
$ P_{0}=\exp \left[-\int_{t_{0}}^{t_{0}+T} \alpha I(t) \mathrm{d} t\right] $
(3) 对应地,在t0~t0+T内得到1个光子计数的概率为:
$ P_{1}=\left[\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right] \exp \left[-\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right] $
(4) 同样, 得到2个光子计数的概率可以表示为:
$ P_{2}=\frac{\left[\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right]}{2}\left[\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right] \cdot\\ \exp \left[-\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right] $
(5) 由此可以推导出在t0~t0+T内发生n个光子计数的概率为:
$ P_{n}=\frac{\left[\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right]^{n}}{n !} \exp \left[-\alpha \int_{t_{0}}^{t_{0}+T} I(t) \mathrm{d} t\right] $
(6) 所以对于恒定辐射强度的光(相干光),(6)式可以表示为:
$ P_{n}=\frac{\langle\bar{n}\rangle^{n}}{n !} \exp [-\langle\bar{n}\rangle] $
(7) 式中, $\langle\bar{n}\rangle=\alpha I T $, 代表光脉冲的平均光子数。一段时间内探测器所测到的恒定光强脉冲光的光子数符合(7)式,这正是一种泊松分布[11]。
图 1是实验中所用的弱相干光脉冲响应测量系统示意图。主要包括信号产生器、810nm/1550nm激光光源、探测器样品盒、同相正交(in-phase quadrature, IQ)混频器、以及模数(analog/digital, A/D)转换器等。这里,微波发射源发出的微波信号被功分器分为两路:一路进入最低稳定温度为10mK~20mK的低温腔,通过衰减器衰减输入作为探测信号,实现探测光脉冲到MKID的吸收探测; 另外一路作为参考信号,直接进入IQ混频器,实现与探测信号的混频。两路信号经过IQ混频器、低通滤波器、和放大器后用数据采集卡进行A/D转换。探测的弱相干激光脉冲改变了探测器的电感,从而影响扫频微波信号的传输特性,并在数据采集卡中进行记录。然后,将采集卡上存储的数据进行分析处理。图中, L0表示微波的另一路信号,直接进入IQ混频器。
数据处理过程中,假定信号噪声是白噪声,因此, 可以使用基于数值平均的最优滤波算法[12],对信号取平均得到信号的幅值估计[13]:
$ A^{\prime}=\frac{\int_{-\infty}^{+\infty} \mathrm{d} f \cdot V(f) \cdot \tilde{S}^{*}(f)}{\int_{-\infty}^{+\infty} \mathrm{d} f \cdot|\tilde{S}(f)|^{2}} $
(8) 式中,A′是单次脉冲和模板匹配的最优幅值倍数,V(f)代表单次脉冲幅值函数的傅里叶变换,$ {\tilde S(f)}$是模板函数的傅里叶变换, *表示针对$ \tilde S$的傅里叶变换。脉冲光中光子数信号强度是通过统计所有脉冲的最优幅值倍数A的分布来表征的。所以,光子数强度估计值就可以通过统计A的分布情况来得到。图 2中给出了作者对20000次弱相干脉冲光进行探测所得到的光子数幅值统计分布。图中,横坐标表示一个相对数值,无单位, 蓝色手指峰部分为多次脉冲光子数幅值统计分辨,红色曲线为对每个光子数信号进行高斯拟合的分布图。需要说明的是,由于取高斯分布的精度限制,拟合光子数脉冲模板与实际对应的光子数脉冲并不一致,因此各光子数峰的峰值与下标光子脉冲高度并非完全一致。取数据拟合的高斯分布面积分布图(蓝色直方图)来进行重新标定,就可以得到20000个弱相干脉冲光平均的光子数分布图(黄色直方图)。可以看到,实测的多个弱光脉冲中的光子数分布与理论预计的标准泊松分布基本一致。这说明,照射到MKID探测器芯片上的光脉冲确实是弱相干光脉冲信号。由以上的泊松分布,可以推算测试的激光脉冲平均光子数为μ=1.55。
Figure 2. Statistical distribution image of photon number amplitude of 20000 weak coherent light pulse detection data using white noise model
具备对弱相干脉冲光的光子数分布进行分辨探测的单光子探测器,就称为光子数可分辨的单光子探测器,其重要的一个性能指标是光子能量分辨率。由于探测器存在光子响应信号的噪声,所以每个光子数峰都不会是严格的δ函数而是近似的高斯峰。因而,可以将每个光子数响应信号峰的半峰全宽来定义该信号峰的能量分辨率[14]。表 1中给出了标定的各光子数峰的半峰全宽ΔE。
表 1 Peak half-width value of each photon measured by the detector after the optical attenuation of 17dB
ΔE0/eV ΔE1/eV ΔE2/eV ΔE3/eV ΔE4/eV ΔE5/eV 0.1489 0.2992 0.3772 0.4382 0.4448 0.5113 测试光是波长为1550nm的相干光,其单光子能量为hν=0.8eV,所以从图中可以看出,该探测器可实现从0~5个光子的光子数分辨探测:这6个光子数峰的半峰全宽到小于单个光子的能量。如果信号光的平均光子数很大,分辨更多的光子数也是可能的。
基于光子数可分辨探测器的单脉冲光子数检测
Single pulse photon number detection based on photon number distinguishable detector
-
摘要: 为了对未知脉冲所含的光子数进行标定,针对不同光子数的光脉冲在超导环境下,微波动态电感探测器(MKID)作用时,测量系统输出的信号差异性,采用平均区间取值法和迭代法分别进行标定,并进行了理论分析和实验验证。结果表明,MKID能够在低温测量系统中对未知1550nm单脉冲光的光子数进行识别; 经过数据处理后得到平均光子数分别为1.98和1.81;其中平均区间取值法标定光子数过程较为简单,迭代法有待继续探索。这一结果对单脉冲光子数检测是有帮助的。
-
关键词:
- 量子光学 /
- 光子数可分辨探测器 /
- 微波动态电感单光子探测器 /
- 光子数标定 /
- 超导低温测量系统
Abstract: In order to calibrate the number of photons contained in the unknown pulse, the signal difference of the system output was measured when the light pulses with different photon numbers were under microwave kinetic inductance detector (MKID) effect in the superconducting environment. The average interval method and iterative method were adopted, and the theoretical analysis and experimental verification were carried out. The results show that, the number of photons of unknown 1550nm single-pulse light in the cryogenic measurement system can be identified by MKID. After data processing, the average photon numbers were 1.98 and 1.81 respectively. Among them, the process of calibrating the number of photons by the average interval method is relatively simple, and the iterative method needs to be further explored. It is helpful for single-pulse photon number detection. -
表 1 Peak half-width value of each photon measured by the detector after the optical attenuation of 17dB
ΔE0/eV ΔE1/eV ΔE2/eV ΔE3/eV ΔE4/eV ΔE5/eV 0.1489 0.2992 0.3772 0.4382 0.4448 0.5113 -
[1] HISKETT P A, LITA A E, HUGHES R J, et al. Long-distance quantum key distribution in optical fibre[J]. New Journal of Physics, 2006, 8(9): 193. doi: 10.1088/1367-2630/8/9/193 [2] KNILL E, LAFLAMME R, MILBURN G J. A scheme for efficient quantum computation with linear optics[J]. Nature, 2001, 409(6816): 46-52. doi: 10.1038/35051009 [3] ZWINKELS J C, IKONEN E, FOX N P, et al. Photometry, radio-metry and "the candela": Evolution in the classical and quantum world[J]. Metrologia, 2010, 47(5): 15-32. doi: 10.1088/0026-1394/47/5/R01 [4] ZHOU P J. Weak light detection based on low temperature superconducting technology[D]. Chengdu: Southwest Jiaotong University, 2014: 23-42 (in Chinese). [5] ENSS C. Cryogenic particle detection[M]. Berlin, Germany: Springer Heidelberg, 2005: 417-452. [6] ZHENG F, XU R, ZHU G, et al. Design of a polarization-insensitive superconducting nanowire single photon detector with high detection efficiency[J]. Scientific Reports, 2016, 6(1): 705-707. [7] ZHANG Q Y, DONG W H, HE G F, et al. Review on superconducting transition edge sensor based single photon detector[J]. Acta Physica Sinica, 2014, 63(20): 200303(in Chinese). doi: 10.7498/aps.63.200303 [8] DAY P K, LEDUC H G, MAZIN B A, et al. A broadband superconducting detector suitable for use in large arrays[J]. Nature, 2003, 425(6960): 817-821. doi: 10.1038/nature02037 [9] LI Ch G, WANG J, WU Y, et al. Research and application progress of superconductivity electronics in China[J]. Acta Physica Sinica, 2021, 70(1): 184-209(in Chinese). [10] ZHOU P J, WANG Y W, WEI L F. Thermal-sensitive superconducting coplanar waveguide resonator used for weak light detection[J]. Acta Physica Sinica, 2014, 63(7): 9-25(in Chinese). [11] LITA A E, MILLER A J, NAM S W. Counting near-infrared single-photons with 95% efficiency[J]. Optics Express, 2008, 16(5): 3032-3040. doi: 10.1364/OE.16.003032 [12] LOLLI L, TARALLI E, PORTESI C, et al. High intrinsic energy resolution photon number resolving detectors[J]. Applied Physics Letters, 2013, 103(4): 041107. doi: 10.1063/1.4815922 [13] YANG X Y, LI H, ZHANG W J, et al. Superconducting nanowire single photon detector with on-chip bandpass filter[J]. Optics Express, 2014, 22(13): 16267-16272. doi: 10.1364/OE.22.016267 [14] LI X, TAN J R, ZHENG K M, et al. Enhanced photon communication through Bayesian estimation with an SNSPD array[J]. Photonics Research, 2020, 8(5): 637-641. doi: 10.1364/PRJ.377900 [15] GENG Y, ZHANG W, LI P Z, et al. Improving energy detection e-fficiency of ti-based superconducting transition-edge sensors with optical cavity[J]. Journal of Low Temperature Physics, 2020, 199(3): 1-7. doi: 10.1007/s10909-020-02383-9 [16] GUO W, LIU X, WANG Y, et al. Counting near infrared photons with microwave kinetic inductance detectors[J]. Applied Physics Letters, 2017, 110(21): 212601. doi: 10.1063/1.4984134 [17] LIU X, GUO W, WANG Y, et al. Superconducting micro-resonator arrays with ideal frequency spacing[J]. Applied Physics Letters, 2017, 111(25): 252601. doi: 10.1063/1.5016190 [18] SZYMKOWIAK A E, KELLEY R L, MOSELEY S H, et al. Signal processing for microcalorimeters[J]. Journal of Low Temperature Physics, 1993, 93(3/4): 281-285. [19] PERNICE W H P, SCHUCK C, MINAEVA O, et al. High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits[J]. Nature Communications, 2012, 3(1): 135-174. [20] ANDERSON B D O, MOORE J B. Optimal filtering[M]. New York, USA: Prentice Hall, 1979: 417-421. [21] WALLS D F, MILBURN G. Quantum optics[M]. 2nd ed. Berlin, Germany: Springer-Verlag, 2008: 46-48. [22] IRWIN K D, NAM S W. A self-biasing cyrogenic particle utilizing electrothermal feedback and a SQUID readout[J]. IEEE Transactions on Applied Superconductivity, 1995, 5(2): 32-37. [23] WANG L L, LI J, YANG N, LI X. Identifying extra high frequency gravitational waves generated from oscillons with cuspy potentials using deep neural networks[J]. New Journal of Physics, 2019, 21(4): 043005. doi: 10.1088/1367-2630/ab1310 [24] ZHANG X D. Matrix analysis and application[M]. 2nd ed. Beijing: Tsinghua University Press, 2013: 502-508 (in Chinese) [25] ALPERT B K, HORANSKY R D, BENNETT D A, et al. Note: Operation of gamma-ray microcalorimeters at elevated count rates using filters with constraints[J]. Review of Scientific Instruments, 2013, 84(5): 056107. doi: 10.1063/1.4806802