A previous article of ours (found here) described how we measured the transition energy between two quantum states of Positronium (Ps). Measurements such as this allow us to test quantum mechanics, specifically the branch known as quantum electrodynamics (QED) which describes the electromagnetic force. If we perform a measurement that deviates from QED theory then the calculations, or the theory itself must be questioned. Deviations could come from so called ‘new physics’ such as new particles, new forces or even the source of dark matter [1,2]. However, more often than not it is an unknown feature of the experiment, as I will demonstrate here.

The previous measurement written about on this blog was of the 2³P₁ 2³P₀ transition, known as the interval, named after the subscript of the final state. These numbers and letters symbolise the values of different quantum numbers and therefore uniquely describe the energy state of the Ps atom. The image below explains what these numbers and letters represent.

In addition to this transition we also measured two others, and , see the published data here [3]. These showed something that the former did not. Figure 3 shows an example of the line shapes we measured, which are produced by creating Ps in the 2³S₁ energy state and then stimulating emission to the lower 2³P₁ or 2³P₂ energy state using microwave photons. The line shape represents how much of the initial state we transfer to the final state as a function of photon energy, given here in units of frequency in GHz. The closer the photons are to the exact transition frequency between the two states (i.e. their energy separation), the more atoms transfer from one state to another and the larger the signal we see. This central transition frequency is what QED predicts and what we want to measure. The key thing to note in these results is that these line shapes show an asymmetry, a bias to lower frequencies for the transition and higher frequencies for the transition, making the lineshapes look lopsided.

This is a problem. Line shapes of this kind are expected to conform to a specific shape described by an equation known as the Lorentzian function. This function is symmetric by definition, so if the line shape you are trying to extract data from is asymmetric then any transition frequency you obtain from a Lorentzian function fitted to the data is not trustworthy. The Lorentzian shape of this function is from QED theory. There is a possible source of asymmetry known as quantum interference (QI), whereby neighbouring states can cause a higher probability of excitation on the side of the measured line shape toward the neighbouring state. However, for these transitions the effect is much smaller than that observed and is beyond the current limit of our precision.

The question then is whether we can: (1) find a model (and corresponding equation) which correctly accounts for the asymmetric effects and get a meaningful value for the energy interval, or (2) identify the source of the asymmetry and try to remove it. During the experiments numerous tests were made which could not identify or remove the source of the asymmetry. Instead, we collaborated with Akopyan et al. who performed a large number of simulations, taking into account 28 states energy states with over 700 coupled differential equations, the results of which can be found here [4].

The conclusion was that the asymmetry was due to microwave radiation escaping from the ends of the waveguide used to confine it and being reflected back in from the walls of the vacuum chamber. This causes the strength of the microwaves to vary over the range of frequencies used to probe the Ps, causing some points to be artificially higher/lower and skewing our measurements.

This effect can produce asymmetric lineshapes like those measured, but also introduce shifts to the transition frequency without asymmetry, casting doubt on the systematic error of the first measurement. Indeed, the transitions measured here are highly susceptible to this effect as they are much broader than those found in most microwave spectroscopy studies meaning these effects are less noticeable. In conclusion, reflection-induced frequency dependent power is a large systematic error in the data described in this and the previous article and some method must be determined to remove this. The next step is to investigate these effects experimentally to confirm how they behave, which will be explained in Pt. II of the story…

[1] Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants. S. G. Karshenboim, Phys. Rep. 422, 1 (2005)

[2] Precision spectroscopy of positronium: Testing bound-state QED theory and the search for physics beyond the Standard Model. G. S. Adkins, D. B. Cassidy, J. Pérez-Ríos, Phys. Rep. 975, 1 (2022) DOI: 10.1016/j.physrep.2022.05.002

[3] Observation of asymmetric line shapes in precision microwave spectroscopy of the positronium 2³P₁ 2³P_J (J = 1, 2) fine-structure intervals. L. Gurung, T. J. Babij, J. Pérez-Ríos, S. D. Hogan and D. B. Cassidy, Phys. Rev. A.103, 042805 (2021) DOI:10.1103/PhysRevA.103.042805

[4] Line-shape modelling in microwave spectroscopy of the positronium n = 2 fine-structure intervals. L. A. Akopyan, T. J. Babij, K. Lakhmanskiy, D. B. Cassidy and A. Matveev, Phys. Rev. A. 104, 062810 (2021) DOI: 10.1103/PhysRevA.104.062810