# Feasibility studies of the polarization of photons beyond the optical wavelength regime with the J-PET detector

## Abstract

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 511 keV photons via Compton scattering. For scattering angles of about 82\(^\circ \) (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\(^\circ \) (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.

## 1 Introduction

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 [1] and recent quantum information theoretical considerations [2].

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 the leptonic sector since they provide a new class of operators [3]. In addition, investigation of multi-partite entanglement of annihilation photons becomes possible [4, 5, 6].

The photon is a transverse electromagnetic wave and Compton scattering occurs most likely in the plane perpendicular to the electric vector of the photon [1, 7]. Thus we can estimate the direction of its linear polarization \(\hat{\epsilon }\) by the product of photons’ momentum vectors before \(\left( \hat{k}~=~\frac{\mathbf {k}}{|k|}\right) \) and after \(\left( \hat{k'}~=~\frac{\mathbf {k'}}{|k'|}\right) \) the scattering [3], namely \(\hat{\epsilon } = \hat{k} \times \hat{k'}\). Note that we assumed here that the polarization vector is a real-dimensional vector, for more details in this Compton-context see Ref. [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 511 keV-photons originating from the \(e^+e^-\) annihilations into two photons. Subsequently, in Sect. 3 the capability of the determination of the relative angle between the polarization directions of a photon pair originating from the para-positronium decay \(\text{ p-Ps } \rightarrow 2\gamma \) is provided. Next, in Sect. 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 Sect. 5.

## 2 Determination of a single photon polarization via Compton scattering

*E*is the energy of initial photon, \(E^{\prime }\) is the energy of photon after scattering, \(\theta \) is the Compton scattering angle and \(\eta \) 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 \(\theta \) is close to zero or to 180\(^\circ \), the variation of cross section with \(\eta \) is not observable. A scattering at \(\eta ~=~90^\circ \) makes the last term maximal, however, the energy of the outgoing photon depends also on the Compton scattering angle \(\theta \), Eq. (2). Thus the visibility, i.e. the interference contrast of the oscillation in \(\eta \), is for the Compton scattering process a function of energy and scattering angle, namely

*E*and Compton scattering angle \(\theta \) can be understood as the probability density distribution of the angle \(\eta \):

## 3 Relative polarization of photons from positronium decay into 2\(\gamma \)

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 1 and 2 is equal to \(90^\circ \). 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. [2], but simulate the separable states, \(\vert HV\rangle \) and \(\vert VH\rangle \). 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. [2] 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 \(\hat{\epsilon }\) of a single photon can be estimated as a direction perpendicular to the scattering plane. Therefore, the relative angle between the polarization direction estimators (\(\angle (\hat{\epsilon }_1,\hat{\epsilon }_2)\)) is equal to the angle between scattering planes, denoted by \(\varphi \) in Fig. 7. Thus, this angle \(\varphi \) 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).

The lower panel presents results for the case when \(\theta _1~=~\theta _2~=~10^\circ \), where the resolution of determining \(\eta \) angle is much lower resulting in the nearly overlapping curves representing the two possible polarization states.

*A*,

*B*and \(\delta \) 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 \(\mathcal {V}^2~=~\frac{A}{2B+A}\).

A contour plot of \(\mathcal {V}^2\) is given in Fig. 10 and it shows that in case of back-to-back 511 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 \(82^\circ \).

## 4 Feasibility of \(Ps\rightarrow 2\gamma \) studies with J-PET

It is important to emphasize that events corresponding to a given pair of scattering angles (\(\theta _1, \theta _2\)) 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 (\(\theta _1, \theta _2\)) plot. This suppresses systematical errors due to the uncertainty in the detectors dimensions and geometrical misalignment.

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 utilizes the 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 511 keV) from the \(Ps\rightarrow 2\gamma \) annihilation are isotropically emitted. The relative angle between the polarizations of the two photons (at the moment of interaction) was fixed to \(90^\circ \), 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 Fig. 13 show distributions of relative angle between the scattering planes \(\varphi \) for data selected from the region: \((\theta _{1} - 81.66^{\circ })^2 + (\theta _{2} - 81.66^{\circ })^2 \le R^2\), choosing two exemplary radii of \(R~=~10^\circ \) and \(R~=~30^\circ \), where the high visibility is expected. Fig. 13 compares results obtained for the case of (i) the ideal detector with 100% efficiency and infinitely good angular resolution for \(R~=~10^\circ \) (black solid line) and \(R~=~30^\circ \) (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 \(R~=~30^\circ \) (red dashed line) and with an additional condition that the interaction points should be more distant than 12 cm (red dotted line). Thus in the event selection it was required that the distance “d” between the primary and secondary photon scatterings is larger than 12 cm. The last condition is applied in order to ensure good angular resolution (\(\sim 2^\circ \)) and good selection power for primary and secondary interactions. The expected interaction time resolution of 100 ps [10] corresponds to about 4.2 cm resolution for the measurement of the distance between the interaction points. Thus the requirement of \(d>12\) cm separation between interaction points should allow for assignments of primary and secondary interaction at the purity of \(3\sigma \).

The relative angle between the scattering planes \(\varphi \) 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 the area around the most optimal scattering angle the more enhanced is the maximum around \(\varphi ~=~90^\circ \). 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 \(d~>~12\) 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 50 keV (this requirement emulates the electronic thresholds of the J-PET detector).

Results presented in Figs. 12 and 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 \(\varphi \) is smooth and nearly constant. These features enable reliable corrections of the measured \(\varphi \) distributions for the efficiency.

As an estimator of the average relative angle between polarization direction of the back-to-back photons, a parameter \(\delta \) may be used, which can be determined by fitting equation (7) to the efficiency corrected distribution of angle \(\varphi \). Thus we have generated for different numbers of samples the distribution corrected for the efficiency expected for the J-PET design and deduced \(\sigma (\delta )\). The uncertainty of the parameter \(\delta \) 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 5000 registered event.

## 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 \(\hat{k} \times \hat{k'}\) 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 \(\sigma =40^\circ \) for \(\theta =81.66^\circ \), 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 [24, 25, 26], including the geometry and material composition of the J-PET detector showed that the efficiency for the measurement of the polarization of 511 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\(^\circ \) around \(\theta _1~=~\theta _2~=~ 81.66^\circ \)—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 the distance *d* between interaction larger than 12 cm and the energy deposit for each interaction larger than 50 keV the total detection efficiency amounts to about \(10^{-6}\). 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 \(R~=~30^\circ \) 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 \(10^{-6}\), we expect about ten events of interest (Fig. 2) per second when using the sodium \(^{22}Na\) source with activity of \(10 \times 10^6\) Bq surrounded with the XAD4 porous polymer [27]. This will in practice allow for obtaining statistics of about million of events within a few days of measurements.

Finally, we have shown that the angular resolution achievable 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 \(\sigma (\delta ) \approx 2^\circ \) 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 [2, 4] 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 [4] as well as tests of the discrete symmetries, parity *P*, time reversal *T* and charge-conjugation–parity *CP*, via operators \(\epsilon _i \cdot \mathbf {k_j}\), where the indices \(i,j=1,2,3\) refer to the labeled photons from the ortho-positronium decays. Such discrete symmetries tests, carried out with the J-PET detector [3], are complementary to so far performed experiments where the operators are constructed from spin observables (\(\mathbf {S}\)) of ortho-positronium and photon’s momentum vectors [28, 29]. Violation of the *T* or the *CP* invariance in purely leptonic systems has never been seen so far [30]. The experimental search is limited by effects due to the photon-photon interactions expected to mimic discrete symmetry violations at the level of 10\(^{-9}\) [31, 32] ^{1} Therefore, there is still a range of about six orders of magnitude with respect to the present experimental limits (currently experimental upper limits for *T*, *CP* and *CPT* violations are at the level of 10\(^{-3}\) [28, 29]) where phenomena beyond the Standard Model can be sought for. The J-PET detector offers therefore a new experimental methodology.

## Footnotes

## Notes

### Acknowledgements

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/SPUB/2017/1, 7150/E-338/M/2017 and 7150/E-338/M/2018, and the Austrian Science Fund FWF-P26783.

## References

- 1.O. Klein, Y. Nishina, Y. Z. Physik
**52**, 853 (1929)ADSCrossRefGoogle Scholar - 2.B. Hiemsayr, P. Moskal, Witnessing Entanglement in Compton Scattering Processes via Mutually Unbiased Bases, arXiv:1807.04934
- 3.P. Moskal, Acta Phys. Pol. B
**47**, 509 (2016)CrossRefGoogle Scholar - 4.B.C. Hiesmayr, P. Moskal, Sci. Rep.
**7**, 15349 (2017)ADSCrossRefGoogle Scholar - 5.M. Nowakowski, D. Bedoya Fierro, Acta Phys. Pol. B
**48**, 1955 (2017)ADSCrossRefGoogle Scholar - 6.A. Acín, J.I. Latorre, P. Pascual, Phys. Rev. A
**63**, 042107 (2001)ADSCrossRefGoogle Scholar - 7.R.D. Evans, Compton Effect,
*Corpuscles and Radiation in Matter II/Korpuskeln und Strahlung in Materie II*(Springer, Berlin Heidelberg, 1958), pp. 218–298Google Scholar - 8.P. Moskal, Nucl. Instrum. Methods A
**764**, 317 (2014)ADSCrossRefGoogle Scholar - 9.P. Moskal, Nucl. Instrum. Methods A
**775**, 54 (2015)ADSCrossRefGoogle Scholar - 10.P. Moskal et al., Phys. Med. Biol.
**61**, 2025 (2016)CrossRefGoogle Scholar - 11.S. Niedźwiecki, Acta Phys. Pol. B
**48**, 1567 (2017)ADSCrossRefGoogle Scholar - 12.M. Pałka et al., JINST
**12**, 08001 (2017)CrossRefGoogle Scholar - 13.G. Korcyl et al., Acta Phys. Pol.
**47**, 491 (2016)CrossRefGoogle Scholar - 14.G. Korcyl, IEEE Trans. Med. Imag.
**37**, 11 (2018)CrossRefGoogle Scholar - 15.W. Krzemień et al., Nukleonika
**60**, 745 (2015)CrossRefGoogle Scholar - 16.W. Krzemień, Acta Phys. Pol. A
**127**, 1491 (2015)CrossRefGoogle Scholar - 17.W. Krzemień, Acta Phys. Pol. B
**47**, 561 (2016)CrossRefGoogle Scholar - 18.L. Raczynski, Nucl. Instrum. Methods A
**764**, 186 (2014)ADSCrossRefGoogle Scholar - 19.L. Raczynski, Nucl. Instrum. Methods A
**786**, 105 (2015)ADSCrossRefGoogle Scholar - 20.L. Raczyński et al., Phys. Med. Biol.
**62**, 5076 (2017)CrossRefGoogle Scholar - 21.D. Kamińska, Eur. Phys. J. C
**76**, 445 (2016)ADSCrossRefGoogle Scholar - 22.M.D. Harpen, Med. Phys.
**31**, 57 (2004)ADSCrossRefGoogle Scholar - 23.P. Kowalski et al., Phys. Med. Biol.
**63**, 165008 (2018)CrossRefGoogle Scholar - 24.G. Santin, Nucl. Sci. Symp. Conf. Rec. 2003 IEEE
**4**, 2672 (2003)Google Scholar - 25.S. Jan et al., Phys. Med. Biol.
**49**, 4543 (2004)CrossRefGoogle Scholar - 26.S. Jan et al., Phys. Med. Biol.
**56**, 881 (2011)CrossRefGoogle Scholar - 27.B. Jasinska et al., Acta Phys. Polon.
**B47**, 453 (2016)CrossRefGoogle Scholar - 28.T. Yamazaki et al., Phys. Rev. Lett.
**104**, 083401 (2010)ADSCrossRefGoogle Scholar - 29.P.A. Vetter, S.J. Freedman, Phys. Rev. Lett.
**91**, 263401 (2003)ADSCrossRefGoogle Scholar - 30.V. Alan Kostelecký, N. Russell, Rev. Mod. Phys.
**83**, 11 (2011)ADSCrossRefGoogle Scholar - 31.B.K. Arbic, Phys. Rev. A
**37**, 3189 (1988)ADSCrossRefGoogle Scholar - 32.W. Bernreuther, Z. Phys. C
**41**, 143 (1988)CrossRefGoogle Scholar - 33.W. Bernreuther, O. Nachtmann, Z. Phys. C
**11**, 235 (1981)ADSCrossRefGoogle Scholar - 34.A. Pokraka, A. Czarnecki, Phys. Rev. D
**96**, 093002 (2017)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}