Positronium is an atom which is half-matter and half-antimatter. Its energy structure is very well defined by the theory of quantum electrodynamics (QED) [1]. QED essentially describes how photons (light particles) and matter interact. If you imagine an electron in the vicinity of another electron, the classical picture says that the electric field of one electron exerts a repulsive force on the other electron, and vice versa. In the QED picture the two electrons interact by exchanging photons. Beyond electrons, the protons and neutrons in a nucleus are held together by the Strong force via the exchange of another type of particle called gluons.

The aim of precision spectroscopy is to carry out new measurements that will be compared to theoretical calculations. Measuring these energy structure provide a way of verifying the predictions made by theory. Atomic systems like hydrogen and helium are widely studied for QED testing, but the presence of heavy hadrons like protons introduces complications. One does not need to worry about such complications in positronium as there are no hadrons involved.

Within positronium there are many energy levels one could choose to measure. The separation of the triplet and singlet states of the *n*=1 level (i.e., the hyperfine structure interval), the 1S-2S interval, and the 2S-2P intervals are all excellent candidates [2]. The theoretical calculations for these three intervals have been very precisely calculated, and have also been previously measured. The last measurement of the fine structure intervals [3], however, is now over 25 years old and is much less precise than theory. Because Ps is very well defined by QED theory, any disagreements between calculations and measurements could be an indication of new physics. To be sensitive to new physics, the experiments have to done with precision comparable to calculations.

Recently we have measured the 2S-2P fine structure intervals of positronium [4]. There are three transitions within this branch and in this post, we’ll talk about the *ν*_{0} transition (2^{3}S_{1} – 2^{3}P_{0}) which is resonant around 18 GHz. This transition, including the other two, is illustrated in figure 1. Initially, the Ps atoms (which are formed in the 1S state) have to be excited to the 2S state. This can be done in several ways (direct 1S-2S transition with one photon is not allowed), and we will cover our method in detail in another blog post soon. For now, let’s assume that the atoms are already in the 2S state. These atoms then fly into a waveguide where the microwaves drive them to the 2P state (via stimulated emission) as shown in figure 2. The atom then emits a 243 nm photon and drops down to the 1S state, where it will annihilate into gamma-rays after 142 ns (remember the lifetime of Ps in the ground state?). If nothing happens in the waveguide, the 2^{3}S_{1} state atom will annihilate after 1 *μ*s.

We placed gamma-ray detectors (D1-D4) around the target chamber, as shown in figure 2, to monitor the annihilation signal. The detector signal was then used to quantify the microwave radiation induced signal, S_{γ}. We scanned over a frequency range to generate a lineshape that describes the transition; the centre describes the resonance frequency and the width is due to the lifetime of the excited state. A Lorentzian function was fitted to extract this information and for the example shown in figure 3, the centroid and line width are 18500.65 MHz and 60 MHz respectively. The centroid is slightly off from theory because the lineshape was measured in a magnetic field which introduces Zeeman shift to the centroid. The measured width is 60 MHz, slightly wider than the expected 50 MHz, and is due to the time taken to travel through the waveguide.

Similar lineshapes were measured over a range of magnetic fields in order to account for the Zeeman shift. These data are shown in figure 4. Extrapolating to zero with a quadratic function allows us to obtain the field free resonance frequency, free of Zeeman shifts. However, all of the measured points, including the extrapolated number, are offset from theoretical calculation (dashed curve) by about 3 MHz. There are a few systematic effects to consider and the largest of them is the Doppler shift arising from the laser and waveguide misalignment, which amounts to 215 kHz. Our result, compared with theory and previous measurements, is shown in figure 5 and disagrees with theory by 2.77 MHz (4.5 standard deviations). While the precision has improved by over a factor of 6, the disagreement with theory is significant.

Precision measurements can be vulnerable to interference effects and there are two main types of effects that can cause lineshape distortion and/or shifts in line centre. Whenever the radiation emitted from the excited state (2P state in our case) is monitored to generate the signal, the emitted radiation can interfere with the incident/driving radiation (microwaves in our case)[5]. This leads to shift in the resonance frequency, but we are not sensitive to this kind of effect as we monitor the gamma-rays instead of the 243 nm emitted radiation (figure 1). Another type of interference arises from the presence of neighbouring resonance states [6], such as the two other 2P states in the Ps fine structure. The further apart the states are, the lesser the interference effect is and we expect a shift of 200 kHz in our line shape. This is, however, over 10 times smaller than the observed shift, and therefore, cannot be the reason for the disagreement.

There are two more transitions in the fine structure we have measured and they reveal interesting new features which were not previously seen. These additional data will provide a broader picture that will help us explain the shift we see in this transition. We’ll discuss those results in the next blog post.

[1] Karshenboim, S.G., *Precision Study of Positronium: Testing Bound State QED Theory*. Int. J. Mod. Phys. A, **19** (2004)

[2] Rich, A., *Recent Experimental Advances in Positronium Research*. Rev. Mod. Phys., **53** (1981)

[3] Hagena, D., Ley, R., Weil, D., Werth, G., Arnold, W. and Schneider, H., *Precise Measurement of n=2 Positronium Fine-Structure Intervals*. Phys. Rev. Lett., **71** (1993)

[4] Gurung, L., Babij, T. J., Hogan, S. D. and Cassidy, D. B., *Precision Microwave Spectroscopy of the n=2 Positronium Fine Structure *. Phys. Rev. Lett., **125** (2020)

[5] Beyer, A., Maisenbacher, L., Matveev, A., Pohl, R., Khabarova, K., Grinin, A., Lamour, T., Yost, D.C., Ha ̈nsch, T.W., Kolachevsky, N. and Udem, T., *The Rydberg Constant and Proton Size From Atomic Hydrogen*. Science, **358** (2017)

[6] Horbatsch, M. and Hessels, E.A., *Shifts From a Distant Neighboring Resonance*. Phys. Rev. A, **82** (2010)