Feasibility studies of the polarization of photons beyond the optical wavelength regime with the J-PET detector
J-PET is a detector optimized for registration of photons from the electron-positron annihilation via plastic scintillators where photons interact predominantly via Compton scattering. Registration of both primary and scattered photons enables to determinate the linear polarization of the primary photon on the event by event basis with a certain probability. Here we present quantitative results on the feasibility of such polarization measurements of photons from the decay of positronium with the J-PET and explore the physical limitations for the resolution of the polarization determination of 511keV photons via Compton scattering. For scattering angles of about 82 (where the best contrast for polarization measurement is theoretically predicted) we find that the single event resolution for the determination of the polarization is about 40 (predominantly due to properties of the Compton effect). However, for samples larger than ten thousand events the J-PET is capable of determining relative average polarization of these photons with the precision of about few degrees. The obtained results open new perspectives for studies of various physics phenomena such as quantum entanglement and tests of discrete symmetries in decays of positronium and extend the energy range of polarization measurements by five orders of magnitude beyond the optical wavelength regime.
Keywords:Positronium J-PET Compton scattering Polarization
Polarization is with no doubt one of the most interesting physical properties photons exhibit. It has been utilized to show several of the most basic foundations of quantum mechanics, where mainly optical (low energetic - few eV) photons were generated. However, thus far there were no studies where the degree of polarization was explored in measurements of high energy photons (in the range of MeV) originating from annihilations of positronium atoms. Polarization of such photons cannot be determined with optical methods. Here we show how it can be estimated via Compton scattering based on the well-known Klein-Nishina formula Klein2013 () and recent quantum information theoretical considerations beatrix-arXiv:1807.04934 ().
Measurement of the polarization degree of freedom of photons from positronium decay may open new possibilities in testing the discrete symmetries (T, CP and CPT symmetry) in leptonic sector since they provide a new class of operators ACTA2016 (). In addition, investigation of multi-partite entanglement of annihilation photons becomes possible Beatrix-Science-Report2017 (); Nowakowski (); Acin ().
The photon is a transverse electromagnetic wave and Compton scattering occurs most likely in the plane perpendicular to the electric vector of the photon Klein2013 (); Evans1958 (). Thus we can estimate the direction of its linear polarization by the product of photons’ momentum vectors before () and after () the scattering ACTA2016 (), namely . Note that we assumed here that the polarization vector is a real-dimensional vector, for more details in this Compton-context see Ref. beatrix-arXiv:1807.04934 ().
The J-PET detector is built of plastic scintillator strips consisting mostly from carbon and hydrogen. Due to the low atomic number of these elements photons from the positronium annihilation interact in plastic scintillators predominantly via the Compton effect and a significant fraction of them may undergo two or even more subsequent scatterings in different strips. A picture of the present prototype of the detector is shown in Fig. 1. Its geometry and properties are described in details in the references NIM2014 (); NIM2015 (); PMB2016 (); ACTA2017 (). Therefore, here for completeness we mention only briefly its main characteristics.
J-PET at the present stage is built from three cylindrical layers including in total plastic scintillators strips with dimensions of . Light signals from each strip are converted to electrical signals by photomultipliers placed at opposite ends of the strip NIM2014 (). The position and time of the photons interacting in the detector material are determined based on the arrival time of light signals at both ends of the scintillator strips. The signals are probed in the voltage domain with the accuracy of about ps by a newly developed type of front-end electronics Paka2017 () and the data are collected by the novel trigger-less and reconfigurable data acquisition system Korcyl2016 (); Korcyl2018 (). For the data processing and simulations a dedicated software framework was developed Krzemien2015 (); Krzemien2015framework (); Krzemien2016 (). The hit-position and hit-time are reconstructed by the dedicated reconstruction methods based on the compressing sensing theory and the library of synchronized model signals Raczynski2014 (); Raczynski2015 (); Raczynski2017 (); NIM2015 (). Though the photons interact in the plastic scintillators predominantly via the Compton effect, the J-PET detector enables studies of positronium decays with angular resolutions of about 1 Daria-EPJ-2016 (). Presently a new innermost layer is being installed and commissioned. This fourth layer consists of plastic scintillator strips with dimensions of read out by matrices of silicon photomultipliers, which is expected to triple the efficiency for the single photon detection and improve the time resolution by about a factor of PMB2016 (). Future measurements will be carried out with the full equipped detector, therefore for simulations conducted in this article we assumed the full version, i.e. a four layer geometry whose cross section is shown in Fig. 2.
In this article we explore first the possibility of the determination of the polarization of annihilation photons in the case of an ideal detector system. Section 2 provides an estimate of the accuracy of polarization determination as a function of the scattering angle for keV–photons originating from the annihilations into two photons. Subsequently, in Section 3 the capability of the determination of the relative angle between the polarization directions of a photon pair originating from the para-positronium decay is provided. Next, in Section 4 the efficiency and angular resolution of the J-PET detector for studies of the relative polarizations angle for photons from positronium decay is presented. Finally, the obtained results and their implications for studies of quantum entanglement and discrete symmetries are summarized in Section 5.
2 Determination of a single photon polarization via Compton scattering
where is the energy of initial photon, is the energy of photon after scattering, is the Compton scattering angle and is the angle between scattering and polarization planes (for definition see also Fig. 3).
There are two important limits that can be seen from Eq. (1). If the scattering angle is close to zero or to 180, the variation of cross section with is not observable. A scattering at makes the last term maximal, however, the energy of the outgoing photon depends also on the Compton scattering angle , (Eq. (2)). Thus the visibility, the interference contrast of the oscillation in , the angle between the scattering and polarization planes, is a function of energy and scattering angle, i.e.
For keV–photons the optimal equals to , i.e. gives the maximal visibility for the variation of the azimuthal angle . In Fig. 4 we have plotted therefore the cross section for two cases: for arbitrarily chosen small angle and the optimal angle . From the Klein-Nishina formula (1) we deduce directly that the scattering cross section favors small Compton scattering angles over large ones (exemplified also in Fig. 4).
Furthermore, we observe an azimuthal asymmetry of the scattered radiation (see Fig. 4), namely we find that scatterings around are favored over those of and . This finds a simple physical interpretation: since an initial polarization vector component normal to the scattering plane does not need to change its orientation for the outgoing photon since it is still normal to the new momentum vector. Whereas, a polarization vector component in the scattering plane has to change by since the polarization vector needs to be normal to the new momentum vector.
Due to different values of the cross section for photons scattered under various Compton angles , in order to compare the likelihoods of scattering parallel and normal to polarization, we introduce the following normalization , which under fixed initial energy and Compton scattering angle can be understood as the probability density distribution of the angle :
where the normalization factor is defined as:
Fig. 4 and Fig. 5 clearly show that indeed for the scattering angles around the probability of the scattering has its maximum value when the scattering plane is perpendicular to the direction of the electric vector of the primary photon (). It is important to stress that it is the case independently of the scattering angle , even though for small angles the maximum is hardly visible. This observation implies that, as stated in the introduction, we can estimate the direction of the linear polarization of the Compton scattering photon by constructing a product of the momentum vectors of the photon between and after the scattering: ACTA2016 (). When using such a definition of , we may interpret the normalized Klein-Nishina differential cross section as a probability density distribution of the deviation between 90 and the real angle . Thus, for example the solid curve in Fig. 4, after normalization to unity could be seen as a physical limitation of the achievable resolution for the determination of the direction of the polarization of 511 keV photons scattered under .
Fig. 6 presents the physical limit for the resolution of determining the polarization direction for keV–photons when identifying above defined with the direction of photon’s polarization. The obtained values of were determined as a fit of a Gaussian function to the distributions of in the range of positive values of . As mentioned earlier, for forward () and backward () scattering the polarization direction cannot be determined via Compton scattering, which results in a standard deviation close to as expected for the uniform distribution.
3 Relative polarization of photons from positronium decay into 2
In the previous section we discussed the limitations of determining the direction of single photon polarization with respect to the scattering plane, when the measurement is based on the Compton scattering formula. In this section we will extend this discussion to explore the possibilities of measurements of the relative angle between polarization directions of maximally entangled photon pairs originating from the decay of para-positronium.
The Bose-symmetry and the parity conservation in the decay of para-positronium (for details see beatrix-arXiv:1807.04934 (); Harpen2003 ()) imply that the state of the resulting two photons is maximally entangled. In the linear polarization basis () with respect to one fixed coordinate system, the state can be written as
where and denote the corresponding horizontal and vertical polarized states perpendicular to the photons propagation direction. It is important to note, as indicated pictorially in Fig. 7, that both photons originating from the decay of para-positronium propagate in its rest-frame along the same axis. Equation (6) implies in addition that the polarization states of photons and are orthogonal to each other.
Thus we have to simulate events of two emitted photons assuming that for each event at the moment of Compton scattering (the measurement act) the relative angle between the polarization directions of photons and is equal to . Our overarching goal is to obtain the feasibility of deducing the correlations with the J-PET setup, therefore we do not invoke the predicted entanglement by simulating directly the joint scattering cross section, e.g. given in Ref. beatrix-arXiv:1807.04934 (), but simulate the separable states, and . Herewith, the theoretical predicted uncertainties of the Compton scattering process are taken into account (our goal) without invoking the theory based on the quantum numbers in the decay of the positronium (except orthogonal polarisation in the moment of scattering). Note that as discussed in details in Ref. beatrix-arXiv:1807.04934 () the entanglement would be recognizable experimentally by observation in mutually unbiased bases/settings, revealing the stronger correlations exhibited by entangled states compared to separable states. Moreover, as we outline later our final simulations differ purely by a factor that can be easily inserted to the final result.
In the previous section, it was shown that the polarization direction of a single photon can be estimated as a direction perpendicular to the scattering plane. Therefore, the relative angle between the polarization direction estimators () is equal to the angle between scattering planes, denoted by in Fig. 7. Thus, this angle may be treated as an estimator of the relative polarization directions when measured via Compton scattering. Every single measurement is limited by the resolution described by the Klein-Nishina formula 1 (as discussed in detail in the previous sections).
In Fig. 8 we present the distributions of Klein-Nishina cross sections for two orthogonal polarized photons in the form of radial plots. Upper plot shows the result for the case when both scatterings occurred under (best resolution for the polarization determination).
The lower panel presents results for the case when
, where the resolution of determining angle is much lower resulting in the nearly overlapping curves representing the two possible polarization states.
In order to quantify the effect we have performed Monte-Carlo simulations of events, assuming that polarizations of photons in each event are orthogonal and generating for each photon independently an angle according to the Klein-Nishina distribution (1). Next, for each event a relative angle between scattering planes () was calculated. The distribution of this angle , which we treat as an estimator of the measured relative angle between the polarization directions, is presented in Fig. 9. The solid line shows the result for the case of the highest visibility , Eq. (3), at and the dashed line for the (close to zero visibility). These studies imply that the determination of the polarization degrees of freedom in the decays of positronium, even assuming ideal detectors, will be only effective for scatterings angles close to the values of .
Here in order to calculate the visibility as a function of scatterings angles, for each pair of (within grid) we have simulated distribution and fitted the formula:
which describes very well the simulated distributions with , and being free parameters of the fit. One exemplary result of the fit is shown in Fig. 9 and shows that the theory predictions overlap well with the simulation. Based on the equations, (7) and (3), the visibility squared is calculated as .
A contour plot of is given in Fig. 10 and it shows that in case of back-to-back keV–photons, in order to measure effectively the angle between their relative polarization directions, the detector should be designed in a way of maximizing efficiency for the scatterings angles close to .
4 Feasibility of studies with J-PET
In order to study the feasibility of the measurement of with the subsequent Compton scattering of both gamma photons, as indicated in Fig. 2, we first simulated the distribution of scattering angles versus in the case of the ideal detector, assuming that each event is measured with the efficiency of 100%. The scattering angles and were generated independently with the probability density distribution corresponding to the Klein-Nishina formula (1). The result is shown in Fig. 11. As expected it is strongly picked at forward angles, where the visibility (shown in Fig. 10) is negligible. However, though the overarching aim of the J-PET detector was medical imaging NIM2014 (); NIM2015 (); PMB2016 (); ACTA2017 (); PMB2018 (), its idealized efficiency (not including suppression due to the hardware thresholds and software selection criteria) for the registration of secondary scattered photons is maximized quite close to the region of the highest visibility (Fig. 12).
It is important to emphasize that events corresponding to a given pair of scattering angles () are registered by many different combinations of the scintillator strips. Due to the axial symmetry of the detector, all strips in the same layer contribute equally to a given bin in the () plot. This suppresses systematical errors due to the uncertainty in the detectors dimensions and geometrical misalignment.
The simulations were performed using the GATE package. GATE (Geant4 Application for Tomographic Emission) is a Monte Carlo simulation platform developed by the OpenGATE collaboration Santin2003 (); Jan2004 (); Jan2011 () based on Geant4 software. It is dedicated to numerical simulations in medical imaging and radiotherapy. It utilizes an easy macro mechanism to configure experimental settings for Computed Tomography, Single Photon Emission Computed Tomography, Positron Emission Tomography as well as Optical Imaging (Bioluminescence and Fluorescence) or Radiotherapy.
In the simulations the full geometry of the J-PET detector and the composition of the detector material were taken into account. The interactions of gamma photons in the scintillators were simulated by GATE which uses Klein-Nishina formula (1). In the simulations we assumed that the source of positronium atoms is placed in the center of the detector and that the back-to-back gamma photons (each with energy of keV) from the annihilation are isotropically emitted. The relative angle between the polarizations of the two photons (at the moment of interaction) was fixed to , while the polarization direction of the single photon was distributed isotropically around the axis of photons propagation. Note that direction of the propagation axis varies from event to event, however, the axes are isotropically distributed. The histograms in Figure 13 show distributions of relative angle between the scattering planes for data selected from the region: , choosing two exemplary radii of and , where the high visibility is expected. Figure 13 compares results obtained for the case of (i) the ideal detector with 100% efficiency and infinitely good angular resolution for (black solid line) and (red solid line) with (ii) distribution of the relative angle between the scattering planes reconstructed based on the interaction positions simulated in the detector for (red dashed line) and with an additional condition that the interaction points should be more distant than cm (red dotted line). The last condition is applied in order to ensure good angular resolution () and good selection power for primary and secondary interactions. The expected interaction time resolution of ps PMB2016 () corresponds to about cm resolution for the measurement of the distance between the interaction points. Thus the requirement of cm separation between interaction points should allow for assignments of primary and secondary interaction at the purity of .
Relative angle between the scattering planes is an estimator of the relative angle between the polarization directions of the registered photons. As discussed in the introduction, the distribution of this angle indicates the uncertainty (resolution function) of determining relative angle between polarization directions on an event by event basis. The shape of these resolution functions (shown with solid lines in Fig. 13 for the ideal detector in two chosen regions of high visibility) are determined by the nature of the Compton scattering (Klein-Nishina formula). Comparing red and black distributions one observes, as expected, that the smaller is the area around the most optimal scattering angle more enhanced is the maximum around = 90. The additional modification of these distributions are due to the angular resolution and the specific geometry of the detector. Red-dashed line indicates histogram after requiring that the distance between the interactions is larger than cm and the blue-dotted histogram shows final expected results assuming in addition that the energy loss in the scintillators for each interaction must be larger than keV (this requirement emulates the electronic thresholds of the J-PET device).
Results presented in Fig. 12 and Fig. 13 indicate that J-PET covers the full angular phase space with no holes in the efficiency map. The efficiency for the measurement of relative angle is smooth and nearly constant. These features enable reliable corrections of the measured distributions for the efficiency.
As an estimator of the average relative angle between polarization direction of the back-to-back photons, a parameter may be used, which can be determined by fitting equation (7) to the efficiency corrected distribution of angle . Uncertainty of the parameter is decreasing with the number of registered events. Fig. 14 indicates that the uncertainty (standard deviation) of the average relative angle between the polarization of the back-to-back photons is equal to few degrees already for a sample of about registered event.
Last but not least let us comment on the theoretically predicted entanglement of the two gammas. Taking latter into account would result in a double as big total visibility, then observed in Fig. 9 and Fig. 13. This factor stems from the fact that photons are bosons and obey the Bose-Einstein symmetry.
5 Summary and perspectives
Measurements of optical photon’s polarization have a long successful history in physics, constituting the basis for investigations of phenomena connected with quantum entanglement of photons such as quantum teleportation or quantum cryptography. In this article we explored the possibility of estimating the polarization of high energetic photons originating from the decays of positronium atoms with the novel technology of the J-PET detector. For the first time, polarization studies become possible in this energy regime and, by that, studies of photonic entanglement five orders of magnitude beyond the optical wavelength regime.
J-PET is the first PET tomograph built from plastic scintillators in which annihilation photons are measured via Compton scattering. We have shown that the polarization of photon, at the moment when it scatters on electron via Compton effect, can be estimated on an event by event basis. We have studied possibilities of estimating the photon’s linear polarization at the moment of its interaction with the electron by the cross product of the momentum vectors before and after the scattering. Based on this definition it was shown that in case of two back-to-back photons, the relative angle between their polarization directions may be estimated by the relative angle between their scattering planes.
Our simulations indicated that, for the ideal detector, due to the nature of the Compton effect, the resolution (visibility of the polarization) strongly depends on the scattering angle, achieving a standard deviation of for , and worsening towards smaller and larger scatterings angles. For forward and backward scatterings the measurement of the polarization via Compton effect becomes impossible. Furthermore, simulations performed with the GATE programming package Santin2003 (); Jan2004 (); Jan2011 (), including the geometry and material composition of the J-PET detector showed that the efficiency for the measurement of the polarization of keV photons originating from the positronium decay is smooth and relatively high. In the region of high visibility (circle with the radius of R = 30 around - the highest visibility), the efficiency of the J-PET detector updated with a fourth layer (Fig. 2) amounts to about 0.2%. However, due to the small cross section in this angular range (see Fig. 11) and the additional selection criteria such as that the distance between interaction is larger than cm and the energy deposit for each interaction is larger than keV the total detection efficiency amounts to about . This efficiency was calculated as the ratio of number of events for which both two primary and Compton scattered photons were registered in the region of high visibility to the overall number of simulated para-positronium decays (Fig. 11). Thus assuming that for the four-layer J-PET (Fig. 2) the final total detection and selection efficiency will be equal to , we expect about ten events of interest (Fig. 2) per second when using the sodium source with activity of MBq surrounded with the XAD4 porous polymer JasinskaActa-2016 (). This will in practice allow for obtaining statistics from million of events within a few days of measurements.
Finally, we have shown that the angular resolution obtainable with the J-PET detector, for the determination of the relative mean angle between the linear polarization of the back-to-back propagating annihilation photons is equal to about for samples of 5000 or more collected events.
The results are encouraging and show that it is feasible to perform measurements of the quantum entanglement of photons from positronium annihilation Beatrix-Science-Report2017 (); beatrix-arXiv:1807.04934 () with the J-PET detector. In particular, determination of the polarization on an event-by-event basis will enable, for the first time, tests of entanglement in the polarization degrees of freedom of the three photons resulting from the decay of the ortho-positronium Beatrix-Science-Report2017 () as well as tests of the discrete symmetries, parity , time reversal and charge-conjugation–parity , via operators , where the indices refer to the labeled photons from the ortho-positronium decays. Such discrete symmetries tests, carried out with the J-PET detector ACTA2016 (), are complementary to so far performed experiments where the operators are constructed from spin observables () of ortho-positronium and photon’s momentum vectors Yamazaki2010 (); Vetter2003 (). Violation of the or the invariance in purely leptonic systems has never been seen so far Kosteleck2011 (). The experimental search is limited by effects due to the photon-photon interactions expected to mimic discrete symmetry violations at the level of 10 Arbic1988 (); Bernreuther1988 () 111The contribution from weak interaction to the positronium decays can be neglected with respect to photon-photon interaction. For example the branching ratio of violating Ps decays due to the Z and W bosons is expected at the level of Bernreuther1981 () and Pokraka2017 (), respectively.. Therefore, there is still a range of about six orders of magnitudes with respect to the present experimental limits (currently experimental upper limits for , and violations are at the level of 10 Yamazaki2010 (); Vetter2003 ()) where phenomena beyond the Standard Model can be sought for. The J-PET detector offers therefore a new experimental methodology.
The authors acknowledge technical and administrative support by A. Heczko, M. Kajetanowicz and W. Migdał. This work was supported by The Polish National Center for Research
and Development through grant INNOTECH-K1/IN1/64/159174/NCBR/12, the
Foundation for Polish Science through the MPD and TEAM/2017-4/39 programmes, the National Science Centre of Poland through grants no. 2016/21/B/ST2/01222, 2017/25/N/NZ1/00861,
the Ministry for Science and Higher Education through grants no. 6673/IA/SP/2016,
7150/E-338/M/2017 and 7150/E-338/M/2018, and the Austrian Science Fund FWF-P26783.
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