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Optical Entanglement using a Positron Spectrometer
William Pannell and Jaden Ma
Abstract
We examine the EPR paradox of entanglement optically using a positron spectrometer as a source of entangled photons. We use this spectrometer to analyze the angular dependence of Compton scattering energy, and verify the theoretical model. The polarization dependence of the Klein-Nishina equation was determined using the coincidence function of a multi-channel analyzer to measure the relative rate of directional azimuthal scattering at a scattering angle of θ=80º. We found a ratio of perpendicular to parallel scattering of ρ=1.75±0.09, which is not in agreement with the theoretical value.
Introduction
The EPR Paper famously introduced the concept of quantum entanglement[1], by which measurements on one particle are able to instantaneously affect measurements made on another particle, in Einstein's argument about the necessity of a wider quantum theory. Though the argument against completeness has been dismissed by the work of Bell[2], the example of entanglement originally used in the EPR paper has remained as one of the most esoteric predictions of quantum mechanics. It is then possible to experimentally verify quantum theory itself by measuring the effect of entanglement at work in a quantum-mechanical system.
After the theoretical discovery of positrons by Dirac[3], it was realized that systems of photons produced by electron-positron annihilation could provide a simple method for testing the proposed behavior of entangled particles[4]. In work on the subject, Pryce and Ward provided the first full quantum description of such a system, and proposed an experimental setup by which the maximum ratio of the perpendicular event rate to the parallel event rate, ρ, could be measured using Compton scattering[5]. We verify the prediction of entanglement and the completeness of quantum mechanics using a positron spectrometer to determine the correlation in azimuthal scattering of photons created by electron-positron annihilation.
Theory
Electron-positron annihilation creates a pair of 511 keV photons in a polarization singlet state, which can be expressed by the normalized wavefunction
where the arrows represent orthogonal directions of linear polarization. The differential cross-section of Compton scattering given by the Klein-Nishina equation explicitly depends upon the photon's polarization, so that it is possible to measure the entangled polarization state by combining annihilation with Compton scattering as shown in Fig. 1.
Figure 1: Diagram of the experiment used to measure the entangled polarization state[5]. Both photons scatter with the same
scattering angle, θ, but different azimuthal scattering angles φ.
From the scattering cross-section, it is possible to show that when both photons scatter with an angle θ, the ratio of the rate at which the photons scatter with perpendicular azimuthal angles, or ∆φ=90º, to the rate at which the photons scatter with parallel azimuthal angles, or ∆φ=0º, is given by the expression
This expression is maximized near a scattering angle of θ=80º, at which point ρ=2.85. Compton scattering off of electrons reduces the energy of the photons, so that after the collision the photons will have an energy that is given by
where E' is the final photon energy, and E is the initial photon's energy. For annihilation radiation, each photon will have energy equal to the rest mass of the electron, so that this expression reduces to
Experimental Setup
To realize the theoretical setup shown in Fig. 1, we used a 10μCi 22Na crystal as a source of annihilation radiation. The source was placed at the center of a lead collimator, which was capped with two Aluminum scatterers. Photons emitted from the source are scattered by electrons within the Aluminum before being detected by four NaI photomultiplier tubes placed as shown in Fig. 2.
Figure 2: Diagram of experimental setup used in this experiment[6].
To maximize the expression for ρ, the PMTs were positioned to only measure photons that had scattered by θ=80º, and were placed at right angles to determine relative azimuthal angles of ∆φ=90º and ∆φ=0º. The signals from the PMTs were amplified before entering the logic circuit shown in Fig. 3. Unfortunately, one of our amplifiers did not function properly, leaving only three PMTs, or two pairs of PMTs, to take measurements.
Figure 3: Digital logic circuit used to measure coincidence rates between the two sides of the source.
The measurement of ρ was then divided into two parts. One of the pairs was used to take scalar counter measurements of coincidence rates, while the other pair was used to determine the energy spectrum of the scattered photons, and determine the coincidence rates by integrating the signal peak.
Single channel analyzers (SCAs) functioned as digital to analog converters for this circuit, which combined AND logic gates with scalar counters to measure the rate at which events were seen by PMTs on both sides of the source. This circuit was augmented with a multi-channel analyzer, which used its built in coincidence function to measure the energy spectrum of scattered photons, and determine the number of events within the signal peak region Using the equation for Compton scattering energies, the SCAs were also able to isolate signal events by only selecting events with energies within 60 keV of the energy 279 keV expected for θ=80º scattering.
In order to use the SCAs and the MCAs properly, both were calibrated to determine their relationship with incident photon energy. These calibrations used the known energy peaks of 133Ba, 137Cs, and 241Am sources to derive linear relationships between MCA bin number, SCA voltage and photon energy. An example of this calibration is shown in Fig. 4., for one of the scintillators.
Figure 4: MCA calibration for one of the scintillators used in this experiment.
Results
To verify the behavior of the PMTs, we first investigated the angular dependence of Compton scattering energies using the spectrum analysis provided by the MCA. Locating the scattering peak at each angle, we used the MCA's calibration to determine the final photon energy. The results are shown in Fig. 5, plotted alongside the theoretical prediction. The data matches the theoretical curve well, with a reduced chi-squared of χv=0.97 and a p-value of p=0.43, indicating that the PMTs functioned properly.
Figure 5: Energy vs angle measurements for Compton scattering.
Using the scalar counter circuit, we measured coincidence rates for both ∆φ=90º and ∆φ=0º by changing the relative position of the PMTs within scattering plane. The background in this measurement was determined by running the circuit without the source in place for each configuration. Due to extremely low event rates, each measurement was taken for a period of 24 hours. The results are shown in Fig. 6, where the background has already been subtracted.
Figure 6: Coincidence rates measured by the scalar counter.
Using the mean values for these distributions, the ratio of perpendicular to parallel azimuthal scattering was determined to be ρ=0.8±0.1, which is 20.2σ from the theoretical value.
Using the MCA, we investigated the energy spectrum of the scattered photons for both ∆φ=90º and ∆φ=0º. Due to extremely low count rates, each measurement was taken for a period of at least three days to permit sufficient statistics to develop. The results are shown in Fig. 7, normalized by time and plotted against background measurements taken without coincidence.
Figure 7: Energy spectra for ∆φ=90º, ∆φ=0º, and the background as determined by the MCA.
By subtracting the measured background from the signal measurements, we were able to determine the ratio of perpendicular to parallel scattering to be ρ=1.75±0.09, which is 12.2σ from the theoretical value.
Conclusion
We have analyzed the coincidence rates of perpendicular and parallel azimuthal scattering of annihilation radiation at a scattering angle of θ=80º using two different measurement techniques to find ratios of ρ=1.75±0.09 and ρ=0.8±0.1. We were unable to verify the prediction of entanglement, but did find a notable difference between perpendicular and parallel scattering in the MCA measurement of the energy spectrum. This disagreement was likely caused in part by extremely low event rates, which allowed statistical fluctuations to interfere.
Due to the extremely low signal rates, we investigated methods of increasing measurement efficiency. We found large efficiency losses in the PMTs, and high decay times, which inhibited the counter circuit's ability to make accurate coincidence measurements. Increasing the size of the source and the aluminum scatters resulted in a factor of eight increase in the signal peak. Future work can implement these alterations to the experimental setup and increase the duration of MCA measurements to improve measurements.
Acknowledgements
We would like to thank Dan Hennessy for the invaluable aid and guidance he provided during this project.
References
[1] A. Einstein, B. Podolsky and N. Rosen, Can Quantum-Mechanical Description of Reality Be Considered Complete? Physical Review 47, 777 (15 May 1935).
[2] J. S. Bell, On the Einstein-Podolsky-Rosen paradox, Physics 1, 195 (1964).
[3] P. A. M. Dirac, On the annihilation of electrons and protons, Camb. Phil. Soc. 26, 361(1930).
[4] F. J. Duarte, The origin of quantum entanglement experiments based on polarization measurements, Eur. Phys. J. H 37, 311 (2012)
[5] M. H. L. Pryce and J. C. Ward, Angular Correlation Effects with Annihilation Radiation, Nature 160, 435 (27 September 1947).
[6] J. Engbrecht and N. Hillson, A flexible positron spectrometer for the undergraduate laboratory, American Journal of Physics 86, 549 (2018).
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