Optimizing the signal-to-noise ratio of biphoton distribution measurements

Matthew Reichert, Hugo Defienne, and Jason W. Fleischer

Phys. Rev. A 98, 013841 – Published 26 July 2018

Abstract

Single-photon-sensitive cameras can now be used as massively parallel coincidence counters for entangled photon pairs. This enables measurement of biphoton joint probability distributions with orders-of-magnitude greater dimensionality and faster acquisition speeds than traditional raster scanning of point detectors; to date, however, there has been no general formula available to optimize data collection. Here we analyze the dependence of such measurements on count rate, detector noise properties, and threshold levels. We derive expressions for the biphoton joint probability distribution and its signal-to-noise ratio (SNR), valid beyond the low-count regime up to detector saturation. The analysis gives operating parameters for global optimum SNR that may be specified prior to measurement. We find excellent agreement with experimental measurements within the range of validity and discuss discrepancies with the theoretical model for high thresholds. This work enables optimized measurement of the biphoton joint probability distribution in high-dimensional joint Hilbert spaces.

Received 25 January 2018
Revised 11 May 2018

DOI:https://doi.org/10.1103/PhysRevA.98.013841

Physics Subject Headings (PhySH)

Entanglement detection Noise Photon pairs & parametric down-conversion Photon statistics Quantum fluctuations & noise Quantum measurements Quantum optics Quantum states of light

Imaging & optical processing Photon counting

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Matthew Reichert^*, Hugo Defienne, and Jason W. Fleischer^†

Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA

^*matthew.c.reichert@gmail.com
^†jasonf@princeton.edu

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Vol. 98, Iss. 1 — July 2018

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Images

Figure 1
Normalized signal-to-noise ratio [ $SNR / (η Γ_{j | i} \sqrt{N})$ ] versus mean count rate (solid curves) plotted for several values of $p_{e l}$ (indicated by the numbers below each curve, in percent). Dashed line shows the trend of the maximum SNR of the $p_{e l}$ , given by $1 - 〈 C 〉$ .
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Figure 2
Typical conditional probability distributions of gray-level (gl) outputs from EMCCD camera given (solid black) zero, (dashed red) one, and (dotted blue) two input photoelectrons. Plots are based on Eqs. (25, 26, 27, 28), with parameters in Table 1. Vertical dotted line indicates typical threshold level of $T = 210 gl$ . Shaded regions represent $P (x > T | k)$ , the areas of which give $p_{e l}$ and $η_{EMCCD}$ for $k = 0$ and 1, respectively.
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Figure 3
Experimental measurement of SNR of entangled photon pairs. (a) Experimental schematic. A 400-nm laser diode pumps a BBO crystal and near-degenerate down-converted photons are filtered, and the far field is projected onto an EMCCD camera (Andor, iXon Ultra 897). (b) Conditional probability distribution of gray-level output given zero input photoelectrons. (Black open circles) Measured histogram of gray levels from $\sim 10^{5} 101 \times 101$ pixel frames collected with the shutter closed at 5-ms exposure time. (Red curve) Fit of Eqs. (25, 26, 27, 28) to determine EMCCD properties given in Table 1. Dotted vertical line shows threshold at $T = 210 gl$ . The EMCCD measures (c) the irradiance distribution and (d) $Γ_{i j}$ , shown projected onto the sum coordinates. A region of uniform irradiance, indicated by the central black boxed region in (c), is selected for SNR measurements. The signal and noise are taken as the area of the anticorrelation peak in (d) and the standard deviation of the fluctuations of the background far from the peak. Measurements (open circles) are repeated for many values of $〈 C 〉$ , and (e) the signal (black) and noise (red) are calculated and fit (solid curves) to theory. (f) Their ratio is taken to determine the SNR. Black dataset in (f) corresponds to signal and noise in (e). Also in (f), many lower thresholds were applied to the gray-scale images, resulting in increased $p_{e l}$ and $η_{EMCCD}$ , all showing agreement with theory. Numbers next to each curve indicate threshold level (see Fig. 2 and Table 1). (g) We observe disagreement that worsens for higher thresholds, where the approximations in treating the EMCCD as an SPCM break down.
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Figure 4
Comparison of (circles) measured optimum count rate ${〈 C 〉}_{opt}$ and corresponding maximum SNR $(SN R_{max})$ with (curves) theory. Curve in (a) is ${〈 C 〉}_{opt}$ from Eq. (23) plotted versus $p_{e l}$ , which shows good agreement for $p_{e l} > 0.01$ . In (b) the maximum SNR is plotted versus gray-level threshold. The red curve shows Eq. (22) with $〈 C 〉 = {〈 C 〉}_{opt}$ and the known dependence of $p_{e l}$ on threshold, i.e., $P (x > T | 0)$ . There is agreement between theory and experiment for all but the highest thresholds, which correspond to the lowest values of $p_{e l}$ .
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Figure 5
Discrepancy between thresholded EMCCD and SPCM. (a) Probability of a gray level above threshold, $P (x > T | k)$ , given (solid black) $k = zero$ , (dashed red) one, and (dotted blue) two input photoelectrons. Curves were calculated with Eqs. (24, 25, 26, 27, 28) with EMCCD parameters in Table 1. Dot-dashed maroon curve is $1 - {[1 - P (x > T | 1)]}^{2}$ which is implicitly assumed in the model. The difference between dashed blue and dot-dashed maroon curves, and those for larger $k$ , is the cause of the discrepancy between experimental results at high threshold and theory. (b) Comparison of simulations of (black squares) EMCCD and (maroon triangles) SPCM with (teal circles) experiment and (curve) theory. Simulation parameters were the same as experiment with EMCCD at $T = 280 gl$ and SPCM with the same value of $p_{e l} = 0.002$ , and scaled to the same amplitude as experiment.
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