Towards quantum technologies with gamma photons

Abstract

In the context of the second quantum revolution, the ability to manipulate quantum systems is already used for various techniques and a growing number of technology demonstrators, mostly with low energy photons. In this frame, our intention is to extend quantum technologies to gamma photons. Our aim is to take advantage of resources brought by entanglement with higher energy particles, particularly electron-positron annihilation quanta. Tools for low frequency quantum experiments are not suitable for penetrant radiation, consequently we need to use effects typical to the keV-MeV energy range instead. High energy photon protocols would include fundamental properties testing, industrial imaging, quantum random number generators, quantum simulators, military applications and improvement of already existing medical procedures. In this paper we review some important steps in the study of annihilation photon correlations, we point out the experimental differences and necessities with respect to the energy increase in quantum photonic experiments and we describe the design of a quantum gamma device we propose for experiments meant to prove feasibility of gamma ray based protocols. The perspective behind our project is to evidence the possibility to communicate via entangled quanta through media which are not transparent for low energy photons.

1 Introduction

Quantum entanglement, or the spooky action at a distance, as it was called during the debate on the Einstein Podolsky Rosen (EPR) [1] paradox, is one of the most important resources we have for the 21st century technology development. This became clear with the Bell-CHSH inequality violation, Bell’s discovery being considered by many researchers as being the most profound in the history of science. Still, the debate continued till the loop-hole free test of Anton Zeillinger’s group [2] in 2015, with the enormous contributions of Alain Aspect and John Clauser, just to mention the last Nobel Prize laureates. This debate is now closed; nevertheless, it is important to address remaining questions, such as why entangled and decoherent photons seem to have the same behavior in some experiments. Also, some authors disagree regarding the entanglement of annihilation photons. We want to help ending this argument too, by building a tool which would help clarify the situation. Our intention is to integrate the experimental capabilities of the device we are building to the product list of a quantum start-up. A recent survey [3] revealed over 92% of the quantum start-ups were created between 2012 and 2022; they were categorized into six fields and our case addresses two of them: complementary technologies and quantum communication. Albeit all activities we describe in this paper can be performed with low energy photons, to the best of our knowledge no such tool has been demonstrated for penetrant radiation sofar. One very important thing to consider in our case is radiation safety, as annihilation sources must have activities close to the GBq level in order to obtain satisfactory data in terms of statistics. The simulations show our count rates can be good enough while respecting all radioprotection rules and regulations [4]. Our hypothesis is that annihilation photons are clearly entangled – as first of all they are produced in a pure state – and it is time to build devices such as the one we describe in this paper, which can help us take advantage of entanglement in various fields. Of course we need to review the state of the art prior to going into the details, but first let’s take a brief look at some key studies that progressively lead to the current stage, in their chronological order.

2 History and state of the art

2.1 A short review of key findings

In 1943, Snyder, Pasternack, and Hornbostel [5], then Pryce and Ward [6] (in 1947), derived the azimuthal variation for the intensity of Compton scattered photons in coincidence counting for a symmetric setup. Meanwhile, Wheeler [7] pointed out in 1946 the polarization planes for quanta originating from electron-positron annihilation are orthogonal to each other.

Bleurer and Bradt [8] confirmed the theoretical predictions after studying experimentally the variation of coincidence signals from Compton scatters following emission of annihilation quanta with Geiger-Muller (GM) counters in 1948. In 1950 Wu and Shaknov [9] replaced the GM counters by at that time newly developed scintillation counters and also validated the result with a more reliable setup. The ratio of perpendicular and parallel direction detected coincidences was expected to be 2.00; the experimental ratio they obtained was $2.04\, {+/-}\, 0.08$. ⁶⁴Cu sources were used in both cases.

Singh et al. [10] studied the azimuthal variation of the Compton scattered polarized gamma radiation in 1965. In 1974, Kasday, Ullman and Wu [11] extended the study of those coincident events to a wide range of angles, polar and azimuthal. Their results being in agreement with quantum mechanics, they came with the first gamma-ray based evidence against local hidden-variable theories, even if their experimental conditions were far from ideal. ⁶⁴Cu was used as a positron source in this case too.

In order to investigate fundamental properties, Clauser and Freedman [12] set the first experiment to test Bell’s inequality for low energy photons in 1972. This was far from the 1982 performance of Aspect [13], who conducted the first experiment that really evidenced violation of Bell’s inequality using Time-Varying Analyzers. In fact, some 20 years after the formulation of Bell’s inequality nothing really satisfactory was obtained in the laboratories, although Clauser, Horne, Shimony and Holt had put the inequality in an experiment-adapted form [14]. In 1988 Shi and Alley [15], as well as Ou and Mandel [16] used ultraviolet photon splitting to obtain entangled photon pairs; the latter would serve for measurements both on continuous variables as in the initial EPR thought experiment and discrete variables as in the experiments proposed by Bell. Then, as explained by Aspect [17], an experiment at the University of Innsbruck [18] has for the first time fully enforced Bell’s requirement for strict relativistic separation between measurements. To avoid the communication loophole, Bell stressed the importance of experiments “in which the settings are changed during the flight of the particles”. Another paper by Aspect [19] brings the synthesis on how the debate on the EPR paradox got to an end, namely closing two loopholes at once.

One loophole which cannot be closed in the nuclear emission case is fair sampling. This is mainly due to the relatively low detection efficiency for gamma-rays – which cannot be stopped, but only attenuated. For the moment we can only assume gamma photons have the same behavior as those from the visible, IR or UV spectrum. Consequently, we would consider the fair sampling loophole closed by analogy with electromagnetic radiation of lower frequencies.

In theory, there are quite a few ways to produce entangled photons in the gamma energy range, such as Bremsstrahlung, inverse Compton scattering, gamma cascade in a nuclear de-excitation scheme and particle-antiparticle annihilation process. While for the Bremsstrahlung entanglement would originate from the decelerated high energy electrons, in the inverse Compton scattering case the entanglement from a laser pulse would be partially preserved after collision with a high energy electron beam. Nevertheless, in both cases total entanglement of the particles involved can only decrease. For the nuclear de-excitation scheme, it is still unclear how to measure the correlations, as it is very laborious to predict scattering angles from within an unpolarized emitter, such as a target for heavy ion reactions in an accelerator. For the annihilation process, the only one accessible to us from the experimental perspective, we focus on $\mathrm{e}^{+}\mathrm{e}^{-}$ pair annihilation, as we can only use positrons from proton rich nuclei obtained following proton acceleration towards a target material in a cyclotron facility. The setup we propose (Fig. 3) is based on $\mathrm{e}^{+}\mathrm{e}^{-}$ annihilation within the source, meaning the material used as target for the cyclotron and subsequently as positron source is also the deceleration and annihilation environment where our 511 keV entangled photon pairs are produced. We are interested in those annihilation photons emitted at an 180° angle, which are entangled in polarization. The latter result from the simplest mechanism of $\mathrm{e}^{+}\mathrm{e}^{-}$ annihilation process, $\mathrm{f}_{\mathrm{d}}\, \lambda _{2\gamma }$ – the left path in Fig. 1.

Figure 1

Diagram of the e⁺e⁻ annihilation processes in matter, after Bass et al. [20]. f_d is the direct annihilation, f_Ps are the annihilations via positronium formation, P-Ps/O-Ps para and ortho-positronium, “u” stands for up, “d” for down, “po” for pick off, “0” for positronium self-annihilation and “conv” for ortho-para conversion reactions

In our case positrons are generated by a proton rich isotope, such as the ⁴⁸V we plan to use, resulting from ⁴⁸Ti after proton irradiation (proton bombardment on natural Ti target), of half-life 16 days. The V/Ti foil is a good choice for the experiments we further describe in Sect. 5, as among many radioisotopes available, ⁴⁸V is not hard to obtain: a thin foil is activated to the GB activity level within hours in a regular cyclotron (in order to enable good statistics obtention). The half-life is such as a few months after the experiment the source would be extinct – which is better for radiation safety reasons, and the Ti resulting from the decay has low toxicity.

For annihilation photons, it was shown as soon as 1996 [21] the correlations of the Compton scattered gamma photons resulting from annihilation are in accordance with quantum mechanics and reveal a serious violation as for Bell’s inequality. Of course, the correlation coefficients and also the S function were corrected according to their polarimeters – a matter of analyzing power. To the best of our knowledge, no recent CHSH experiment has been carried out since.

The Compton effect (described in detail in Sect. 4) was investigated as a potential quantum key distribution mechanism by Novak and Vencelj [22] in 2011. Palffy [23] described a nonlinear Compton effect with X-rays in 2015. Neyens et al. [24] described quantum optics with gamma rays as early as 2003, pointing out coherence and interference effects lead to interesting and sometimes unexpected results in terms of light-matter interaction. The US DOE published in 2018 a white paper called “Opportunities for Nuclear Physics and Quantum Information Science” [25]. DOE points out the convolution of those fields may lead to important benefits on one hand, and there exists a real interest in this synergy at government level on the other.

2.2 State of the art

Meanwhile, gamma detection systems evolved towards high-efficiency setups providing both remarkable energy resolution (like BGO suppressed HPGe detectors), and time resolution (for example in the case of LaBr₃(Ce) detectors). The use of an array of detectors, e.g. Rosphere [26] (which is actually made up of two half spheres on which 25 interchangeable detectors are fixed as it can be seen in Fig. 2) leads to high quality data.

Figure 2

The Rosphere array – NIPNE, Romania: left BGO detectors, right BGO and HPGe detectors

Still, for a CHSH experiment we would need an optimized correlation table with two parallel planes for the detectors, as the annihilation photons are emitted at 180°. Another way of testing the entanglement properties for annihilation photons is a miniaturized setup we propose, consisting of small CeBr₃ detectors incapsulated in a multi-layer lead shield in a cylindrical symmetry. The core of the setup would consist of the radioactive source holder and vertical collimator (also a shield in itself) and the scatterers situated 180° from each other on the vertical axis, with horizontal collimation channels at well defined (CHSH) angles for the photons which undergo Compton scatter before being directed towards the detectors (see Fig. 3).

Figure 3

Design of our miniaturized setup for experiments with entangled gamma photons: first and second cylindrical shields (Pb, 1 and 2) vertical collimator and shield (Pb, 3) in the center with horizontal collimators (Fe, 4) towards the detectors (5), radioactive source (6) – left; core of the setup (detail) with dimensions in mm – right

3 Applications of quantum entangled gamma quanta

3.1 Relationship with nuclear physics

The US DOE white paper, among an important list of subdomains which includes various subjects related to quantum sensing, quantum computers and related topics, has a few sections which are directly linked to developing procedures with high energy photons. DOE considers the MeV regime – or even higher energies – meant to help understanding quark-gluon structure [27], impact on $\mathrm{e}^{+} \mathrm{e}^{-}$ thermalization [28, 29], or heavy ion collisions [30, 31]; despite the energy range, the latter are not the purpose of our paper. The interesting part for our research evidenced in the DOE bulletin is dedicated to quantum simulators, as this could lead to the possibility of preparing systems and tuning the dynamics Hamiltonian or measuring quantum states. The simulator we intend to build (see Figs. 3 and 7) is meant to take advantage of the correlations between entangled annihilation photons and also use gamma-rays resulting from the de-excitation which follows the β emission. Qubits, the basics of quantum systems, are meant to be uninvolved with external interactions, while quantum sensors are meant to be easily responsive to specific interactions. This is another feature we intend to adapt further in our experiments, as gamma radiation is penetrant: any environment is to a certain extent transparent for those photons, meaning sensing and information transfer are possible.

In terms of measurement science, we deal with a particular metrology case of the Heisenberg-limit, as quantum correlations lower the shot noise limit [32] bringing sensitivity to a linear dependence with respect to the number of particles, instead of $\sqrt{N}$; this is also interesting for the simulator we are building and it is a great opportunity to test those different statistics, as we deal both with quantum correlated (annihilation) and classic (post $\beta ^{+}$ emission) de-excitation photons. Quantum information also has applications based on isotopes. Without reviewing the variety of those protocols, we cite a well-known gamma emitter, ¹³³Ba, which has the potential of being used as a qubit [33], overcoming long established limitations.

A technique of great interest to us, which uses particle correlations in order to generate images albeit there is no spatial resolution for the scattered particles from the object to be studied is ghost imaging [34] (GI). Imaging is achieved using entangled particles that do not necessarily have the same energy and sometimes are not even of the same type. Only one particle interacts with the object and subsequently with a bucket detector, while its entangled pair contributes to generating the image with a pixel detector. This protocol based on coincidence measurement led to infrared imaging with visible light [35] and recently the technique has been proven feasible with electrons in the keV/MeV range [25]; we are interested to try GI with gamma photons, after studying entanglement conservation in Compton processes. High energy quantum correlated photons are also extremely useful in medical investigations; in the following section we shall review the latest studies in this field, which paradoxically lead to different conclusions regarding the entanglement of annihilation photons.

3.2 Medical investigations and related topics

Caradona et al. [36] examined Compton cross sections of entangled gamma photons in order to optimize Compton PET systems capabilities: they provided cross sections for annihilation photons in order to review the most precise Compton scattering protocols for annihilation photons and to resolve contradictions between reported data sets in the range for which experimental cross sections have been obtained. Hismayer and Moskal [37] showed the pair of annihilation photons from an electron-positron pair is not the only possibility for investigating processes in human bodies: new generation tomographers will be able to observe 3 photons originating from ortho-positronium; this is also based on the entanglement features depending on the polarization degrees of freedom which rely on the angles between the photons. Genuine multipartite entanglement involves all degrees of freedom and subsists provided there was a definite spin eigenstate for the positronium. Entanglement properties survival may provide quantum information from human metabolic processes. Watts et al. [38] also insist on the benefits of PET imaging of radiotracers built by gathering detected hit positions of annihilation photons in the array of detectors. Additionally to the geometric information, those annihilation photons are in an entangled state, enhancing correlations between the interaction processes. The predicted entanglement was integrated in a simulation in order to be compared with data obtained from a PET experiment with cadmium zinc telluride.

The entangled photon simulation describes the correlations; in the future this will provide a straightforward method for quantifying and removing PET imaging background by using only the quantum entanglement information. Their work strengthens confidence in the use of entanglement and enhances again the debate on the usefulness of entanglement for high energy photons.

Sharma et al. [39] work on a “decoherence puzzle in measurements of photons originating from electron-positron annihilation” and state entanglement has not been proven experimentally. So far, experiments unanimously confirmed the correlations between linear polarizations of photons originating from electron-positron annihilation – they sustain the idea of photon entanglement in polarization. Still, some new experiments show back-to-back photons with different correlations for the polarization direction post photon scattering (decoherence) of one photon on an electron from the scattering material. One case shows the same correlation before and after decoherence, while in another experiment scattering one photon decreases this correlation significantly – which reopenes a debate for the gamma-ray range. This is quite difficult to accept over 40 years after things were cleared out in cornerstone papers like Clauser and Shimony’s [40], where they affirm “experimental results […] favor those of quantum mechanics”, and state so diplomatically “either one must totally abandon the realistic philosophy of most working scientists, or dramatically revise our concept of space-time”. Also, the 2022 Physics Nobel Prize was awarded to Alain Aspect, John Clauser and Anton Zeilinger “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science”. Although their results have cleared the way for new technology based upon quantum correlations, the above cited debate is still here.

Furthermore, Ivaskin et al. [41] also studied quantum entanglement of positron-electron annihilation photons. The high energy of annihilation photons allows “controlled formation of decoherent states and successive measurements of the same photon”. Since a comparison between polarization correlations in the entangled and separable states shows the angular distributions are the same for both quantum states and the correlation function in Bell’s inequality is also the same, their group states there is no actual evidence of the annihilation photons’ entanglement. Abdurashitov et al. [42] also built an experimental setup for entangled and decoherent annihilation photon study. The 511 keV entangled quanta resulting from $\mathrm{e}^{+}\mathrm{e}^{-}$ annihilation emitted at a 180° angle with respect to each other have mutually orthogonal polarizations; in order to measure them, two analogous arms of Compton polarimeters consisting in a plastic scintillation scatterer with an NaI(Tl) detector array for measuring the photons deviated at 90° were used. An intermediate scatterer inserted in one arm generated a tagged decoherence process before measuring in polarimeters. The Compton scattering of photons in entangled and decoherent states was compared: the modulation factor and analyzing power of Compton polarimeters were estimated using the angular distributions. The distributions were identical in the entangled and decoherent cases. The experiment showed identical kinematics in terms of Compton scattering, independently of the quantum state. Consequently, the authors reopen the debate whether annihilation photons are entangled or not. Thus, according to their positions on the Bloch sphere, they should be in a pure state, meaning maximally entangled Bell states as $\Psi ^{+/-}$ or $\Phi ^{+/-}$.

4 From visible to gamma photons

4.1 Main differences involved by increasing the energy

Quantum Technologies are nowadays present worldwide from sensing to communications, imaging, computing, and so on [43]. This became possible mostly for low energies, frequently visible photons; there are demonstrated and dedicated building blocks for all experimental requirements, from mirrors (standard, half-silvered, dichroic), wave plates (half or quarter), prisms (classic, Wollaston…), (polarizing) beam splitters, fiber optics, waveguides, gratings, lenses and related devices. The quantum resource to make use of is entanglement. The strongest entanglement between two or more particles occurs at the moment of pair (group) creation, for after that entanglement can only decrease.

The usual way to produce entangled photons is spontaneous parametric downconversion (SPDC) in a nonlinear crystal, where only one event out of 10⁷ generates an entangled pair of photons. For gamma energies we must use nuclear or nuclear-related phenomena. The advantage of electron-positron annihilation is that all gamma pairs resulting from this process are maximally entangled; nevertheless, there is a drawback, namely the radiation safety issues. Entangled gamma photons can be produced in mechanisms such as annihilation processes, nuclear de-excitation and inverse Compton scattering; we intend to extend our study to the latter in the Extreme Light Infrastructure.

There are two effects which can be used for gamma ray deflection. Both keep (some of) the pair’s entanglement. The first is Compton scattering, an interaction in competition with the photoelectric effect – which collapses the photon wave functions – and pair creation, which occurs above $2\mathrm{m}_{\mathrm{e}}\mathrm{c}^{2}=1022\text{ Mev}$ and is not involved in our experimental proposal. Consequently, among the three interactions referred to, the process of interest for studying photon polarization, subsequently photon entanglement, is Compton scattering of gamma rays on (in a very good approximation) unbound electrons, resulting in a lower frequency scattered photon and the associated recoil electron, as pictured in Fig. 4. Coincident detection of photon pairs generated by electron-positron annihilation processes which subsequently undergo Compton scattering at the Bell angles enables measuring the CHSH inequality violation.

Figure 4

Mechanism of Compton scattering (left) and energy transferred related to scattering angle (right), – after G. Gilmore, Practical Gamma-ray Spectrometry [44]

Compton scattering implies an energy shift depending on the scattering angle. The cross section is described by the well known Klein-Nishina [45] formula. Our gamma photons are initially entangled in polarization. Compton scattering partially polarizes the gamma rays, as the propagation axis changes. Consequently, the electric field has a null component along the propagation direction before scattering. In our first tests, the photons of interest were scattered at an angle $\theta = \pi /2$. Polarization for the scattered photons varies between 0 and P_max, as pictured below, detailed in Hiesmayr and Moskal (Fig. 5) [46].

Figure 5

Compton scattering for incoming linear polarized photons

The second process is called Delbrück effect; it is an elastic coherent process involving gamma quanta in the nuclear field, generally of heavy nuclei. This effect is not in competition with Compton scattering, but rather with Rayleigh scattering, Thomson scattering and Giant Dipole Resonance. Although Delbrück scattering generally occurs at very high energies, Wilson [47] pointed out in 1953 that it occurs also in the MeV range, which is a typical energy for gamma cascades in nuclear de-excitation, and close to the electron-positron annihilation energy. Such phenomena, as light by light scattering, are still under investigation nowadays [48]. Nevertheless, Delbrück scattering is now understood [49] even for lower energies (MeV), but not yet accessible to us in terms of resources. Our approach is currently limited to Monte Carlo (MC) simulations in order to see to what extent a protocol which uses both Delbrück and Compton effects would be feasible and useful.

Consequently, the Compton effect is the only one experimentally available to us for the time being. Theoretical and experimental studies of the Compton effect began 100 years ago, but Compton scattering experiments still bring in novelty, as demonstrated by Maruyana et al. [50] Significant efforts have been made in order to obtain tools for high energies similar to the low energy ones, still devices such as waveguides are available just for the X-ray domain now [51].

4.2 Testing prerequisites

Tests with positron emitting nuclides such as ⁴⁸V or ²²Na were carried out in the National Institute for Physics and Nuclear Engineering of Bucharest (IFIN-HH) in order to obtain spectroscopic information in setups similar to those used by Wu and to design the next experiments meant to prove the utility of quantum correlated annihilation radiation. Our first sources were too weak to ensure the required statistics, still this allowed testing the modern setup and fast time electronics, strengthening the precursors with spectroscopic information. Stronger positron sources are needed to continue this series of experiments with adapted shielding, as we deal with ionizing radiation – which means we have to respect all constraints and regulations related to radiation hazard.

Also, recording Compton scattered annihilation radiation spectra for several metal scatterers (including alloys), allows us now to validate along with Monte Carlo simulations which is the ideal metal/alloy for each type of experiment; there is an optimisation to perform with respect to material and electron density, gamma energy and relied photon absorbance in that material. Nevertheless, given the fact the scatterers would be very small, it is highly unlikely to have an alloy for a few cubic centimeters. Simulations showed us the ideal material for our current situation is iron, as the first Compton interaction has a high and almost constant probability in the 5 to 10 mm depth interval (Fig. 6), which makes it compatible with the dimensions of a miniaturized device.

Figure 6

Dependence of the first Compton scattering probability for 511 keV photons on the interaction depth in materials available for our demonstrator

Simulation of the classical unpolarized photon pairs’ path from the core of the setup to a (any) couple of detectors placed in front of the horizontally collimated iron channels (Fig. 3) leads to an expected 400 coincidence counts per day for each detector pair (one up and one down), at a given activity of 1 GBq, for an energy of 255.5 keV. However, the number of coincident pairs above background nearby the mentioned energy may be affected by non-uniformities within the central collimation structure, as from our prior experience lead bricks may contain small holes from the casting process. As this setup is miniaturized, an air hole or casting defect of a few mm can produce severe damage to our plans – we intend to slice and analyze the central collimator after the experiment.

The goal is to demonstrate the possibility of transferring information through media which are not accessible to commonly used photons, opening the way to quantum communication with penetrant radiation, which can apply to industry, space, or military-like protocols. Manipulating gamma photons would also enable simulations (as recording photons in different detectors is equivalent to random number generation in a base equal to the number of detectors) and imaging in non-transparent media. In all instances the key operation is information obtention or transfer. Gamma radiation has well-defined energies depending on nuclear structure (and scattering angles if scattering occurs). Detection simultaneity is ensured by the extremely short lifetimes of the related nuclear states in a cascade. Perfect simultaneity is obtained for annihilation quanta which subsequently follow identical optical paths in our experiment. Annihilation photons are entangled in polarization, which is our most precious resource for this project.

4.3 Pilot setup

Our basic setup meant to demonstrate quantum correlations with gamma photons is pictured in Fig. 7.

Figure 7

Pilot experimental setup. Sh are shielding and collimation elements (Pb), Sc are scatterers (symmetric, Al, Cu, Fe, alloys or active devices), D1 are the “direct” annihilation photon counters, D2 and D3 are the upper and lower detector rings (left). The dark lines for scatterers and detectors are plastic caps, transparent for gamma photons. Right: upside view* of the setup (up) and the point source confined in transparent material inserted in a thin Al foil (down). Mechanical constraints apply to such a setup, as detectors are very fragile, and given the source activity, the lead shields are very heavy. (*) the angles in this drawing are generic: for the experiment we need to set the CHSH ones

The key elements for this setup are a $\beta ^{+}/\gamma$ source, collimation and shielding devices, metal scatterers adapted to the Compton process and gamma detectors with related electronics. It has two detectors on the initial collimated direction (mainly for calibration/control purposes) and several other detectors in the perpendicular planes to the latter, as the scattering angle of interest for this series of experiments is $\pi /2$ in the azimuthal plane. The detectors are by default Hyper Pure Germanium ones, Compton suppressed (BGO suppression shield), Lanthanum Bromide ones – LaBr₃(Ce) – for better timing precision if necessary, or the smaller Cerium Bromide (CeBr₃) for the miniaturized case. The latter would optimize the detection and shielding solid angles, as the distance from source to detector entrance is also smaller. This allows reducing the setup scale and consequently lowering source activity and shielding thicknesses. The radioactive source emits positrons and gamma radiation in the de-excitation path following the $\beta ^{+}$ decay. An emitted $\beta ^{+}$ particle is stopped very quickly in the plastic layer or aluminum foil right in the center of the symmetric collimator (Fig. 7) and annihilates with an electron. This generates two annihilation photons of 511 keV energy propagating in opposite directions and entangled in polarization; those photons are in a singlet state.

In terms of electronics for detection, we are testing the CAEN DT5730 desktop digitizer that is an 8-channel, 14-bit, 500 MS/s FLASH ADC Waveform Digitizer. It offers a choice between a 2 Vpp or 0.5 Vpp input dynamic range. The DC offset is within a range of ±1 V (for 2 Vpp) or ±0.25 V (for 0.5 Vpp) using a 16-bit DAC (digital to analog converter) available on each channel. The resolution of the ADC (analog to digital converter) and the sampling rate render these digitizers highly suitable for moderate to fast signal detection systems, such as those using inorganic scintillators (LaBr3) connected to Photomultiplier Tubes (PMTs). The algorithms used are either the trapezoidal shaping filter (Pulse Height Analysis) or the mathematical integration (Pulse Shape Discrimination), depending on the type of the detector used for measurements. Precise timestamps are required in order to make all the necessary correlations so the significant events can be extracted from the unwanted ones.

5 Experimental proposal

5.1 QRNG

This setup can be used as a simple communication device for emitting and detecting correlated photons, which are recorded in a narrow time gate. If one chooses to label the detectors and record the order in which photons are detected, then we have a classic random number generator, provided the relative efficiencies of each detector are well known. If additionally we analyze photon polarization – which is initially entangled for annihilation quanta – by Compton scattering, we move towards a quantum random number generator (QRNG). This can be achieved by assigning a couple of collimation channels (one up and one down the setup, which detect a pair of entangled photons post Compton scattering) to a digit. After performing the efficiency correction for each couple of channels we can run the experiment in order to generate numbers in base b, where b is the number of detector pairs involved.

5.2 Bayesian simulator

This device can serve as a Bayesian simulator, depending on which detectors, gates and triggers are used. The most important characteristic is that we can detect two entangled quanta resulting from the annihilation and one classic photon following beta disintegration; this leads to a Monty Hall (MH) problem, a restricted choice scheme based on Bertrand’s Box Paradox [52]. The latter is in fact no paradox, but just a counter intuitive problem which is best treated with Bayes’ theorem [53]. In other words, if time gates and triggers are set in order to record simultaneous detection and we also take photon energy into account, we obtain a simulator for events with different statistics: the energies corresponding to the nuclear de-excitation obey classic statistics (i); those which originate from the annihilation process obey quantum statistics (ii). For (i) the energies are available from the nuclear structure data, while for (ii) the energies are calculated as a function of the scattering angle with the Klein-Nishina formula. Let us suppose we deal with a ²²Na source that undergoes $\beta ^{+}$ decay, resulting in ²²Ne, which subsequently emits gamma rays of 1274.5 keV to ground level in a time orders of magnitude shorter than the detector resolution. The 511 keV annihilation quanta and the de-excitation gamma ray reproduce a Monty Hall (MH) problem. This is also achievable in a setup like the one we propose. There have been some attempts to treat a quantum MH problem [54–57], but nothing was carried out experimentally in such a frame. The MH problem led to several configurations in game theory and applications. We do not intend to study any game pattern, but instead to develop a tool to analyze biased versus unbiased information, using quantum correlated gamma photons. The whole study depends on which energies (consequently photon and statistic type) we set the gates.

5.3 Imaging

If at least one of the detectors recording coincidence annihilation quanta is replaced by a segmented one (clover, then pixel) which detects this radiation directly and an object is placed between the source and a bulk (single crystal) detector, coincidence analysis would open the way towards a ghost imaging [58] (GI) device. Gamma rays are penetrant radiation, so this protocol will be further oriented towards developing industrial and military-like applications. Quantum imaging had major breakthroughs, like imaging with undetected photons [59], or interaction-free imaging [60]. GI has already been demonstrated with entangled photons [61], and also with particles of different energies and types as described in Sect. 3.1. We think of applying the same principle to medical imaging or any application where the use of different, lower energies would enable decreasing the ionizing radiation dose.

5.4 Fundamental approach

If the detectors in the perpendicular plane (Fig. 7) to the collimation axis are set at the Bell-CHSH angles, measurements of the Compton scattered photons provide the data necessary to calculate if the correlations violate the Bell-CHSH inequality [62, 63]. The latter is a fundamental result and to the best of our knowledge it has not been calculated for gamma photons with the exception of the above cited [21], although there is a growing interest for even higher energies [64]. Consequently, this experiment would be useful for strengthening the result obtained by Osuch et al.

5.5 Statistics and metrology

A statistical advantage is pointed out in the work of Giovannetti et al. [65] Classically, the central limit theorem implies the statistical error reduction is proportional to the square root of the number of repetitions. Quantum metrology uses entanglement to yield higher statistical precision than classical approaches. Using this property would allow studying an experimental MH problem and then generalize the (un)biased information impact. Our setup is a multivariate tool, as we can choose detector order, time and energy gates, scattering angles and efficiency ratios. The latter depend upon the solid angle from the source towards the sensitive volume of the detector. Consequently, this setup can be turned into a simulator for which randomness is guaranteed by the properties of radioactive decay and variation of the detection parameters depend on the experimenter’s choice. We intend to set an experiment which would help showing entangled gamma quanta obey different (lower error) statistics; for example, by playing a MH game, the three photon system described above for ²²Na or ⁴⁸V would point out the differences from the classic case. Also, we would like to contribute in clarifying the contradictions about entanglement that generated the last arguments discussed in the medical investigations and related topics section.

6 Outline and conclusion

From the most recent publications it is straightforward entangled gamma photons are useful in various fields; consequently, our ongoing technology demonstrator will be beneficial for achieving milestones on the way towards increasing the energy range for controllable entangled photons.

Given the theoretical background, the literature review and related results, our available prerequisites and preliminary data, the available infrastructure in terms of experimental setup and related devices, we propose an experimental activity plan based on a step by step approach, from fundamental properties testing to various applications. After validation of our setup together with its shielding and coincidence electronics, the first step is to compare experimental Compton scattering profiles to the simulations, in order to assess the ideal scattering alloy for each incident photon energy. Then we need to record the experimental angular correlation for the Compton scattered annihilation radiation; follows the fundamental testing of Bell-CHSH inequality violation, measuring at the Bell angles and evaluating to what extent Compton scatterers can be considered a partial polarizing beam splitter. Also, we would attempt an entangled gamma spectroscopy experiment, as there is a wide range of reactions available at IFIN-HH accelerators.

For the applications section, we are looking forward to using our setup for a QRNG, investigating how to adapt X-ray waveguides to higher energies or testing a Compton based ghost imaging detection system by replacing at least one of the detectors from our basic setup with a segmented detector. Testing a Compton MH quantum game with three photons (two annihilation photons and a gamma-ray that follows the $\beta ^{+}$ disintegration) is achievable with our basic setup. The idea behind this is in fact not to bring radiation hazard experiments towards gaming, but to obtain a Bayesian type experiment, as all MH-like applications can be. Correlating simulations of the Delbrück process performed by the related ELI-NP group (Bucharest) with the experimental results from Göttingen and BINP (Novosibirsk) would open the way for a Compton – Delbrück application with entangled quanta. Then correlating the resulting Compton – Delbrück scattering matrix with recent advances in this field would also be interesting, even if the cross sections are not comparable in the same energy range, for at least those two effects do not compete each other and it’s been demonstrated photons of different energies can be strongly entangled. We conclude the presented plan is suitable for the pilot we described in order to obtain a demonstrator. The latter can be further adapted according to the particular need of communicating with entangled penetrant radiation.