Author: R E Sheldon

Microwaves & Positronium Pt. III: Positronium Spectroscopy Cubed

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A series of previous posts (found here, here and over here) described how our measurements of the positronium (Ps, an atom formed of an electron and its antimatter counterpart, the positron) 2 3S1 → 2 3PJ fine structure energy intervals were subject to significant shifts due to frequency dependent microwave power [1,2]. This variation in the power was due to reflections of the microwave radiation causing more power at some frequencies that others, skewing our measurements [3,4]. See the previous posts for a description of how we measure line shapes to determine the transition frequency. This post describes a new measurement of the 2 3S1 → 2 3P2 energy interval, known as the ν2 transition, performed using a waveguide with a new experimental design to eliminate reflection effects. The full published version of this work can be found in Reference [5].

The solution to the reflection problem, as determined from simulations of the microwave fields, was to use a vacuum chamber that minimised the possibility of microwave reflections going back into the waveguide and creating frequency dependent power variation (all our experiments are performed in a vacuum at <0.00000001% of atmospheric pressure to prevent the Ps scattering and annihilating). The vacuum chamber chosen was a cube, see Figure 1 for a diagram of the experimental layout. In this chamber the ends of the waveguide are just a few millimeters from the windows used to let in laser radiation, thus microwaves will pass out of the chamber as fused silica is transparent to microwaves, unlike metal which is highly reflective. This way we reduced the amount of reflected radiation and thus the frequency dependent power variation. Microwave absorbing foam with a reflectivity of <1% was placed on the windows, ensuring no microwaves were reflected back into the waveguide from outside the vacuum chamber.

Figure 1 A schematic diagram of the experimental setup showing the Ps (green) passing through the waveguide with the microwave absorbing foam (blue) at its exits. Adapted from Reference [5].

This experiment also had a few other improvements. Firstly, Doppler effects were minimised by retro-reflection of the UV laser beam used to make the 2 3S1 state Ps. If the laser Doppler selects atoms moving in one direction (away or towards the laser), then the retro-reflected beam, moving in an equal and opposite path, will select out atoms moving in the equal and opposite direction, resulting in a net zero velocity. Secondly, an extra wire mesh EG was included between the Ps production target ET, which has a large bias of -3.5 kV, and the waveguide to minimise electric field penetration into the microwave excitation region. Electric fields induce unwanted Stark shifts to ν2, but are attenuated strongly by wire meshes [6].

The quantum electrodynamic (QED) calculations we wish to test are for the zero-field case, where the atoms are in a region of no electric or magnetic field and do not interact with each other. The atoms in our experiment were a gas with less than 105 atoms per cubed centimeter and the electric fields present were very small. However, our experiment had to take place in a substantial magnetic field because this is how we guide our pulses of positrons to the location where we make Ps. A magnetic field will change the energy of quantum states (i.e. shifting where they lie in the electric potential well of the atom) and thus the energy intervals between states. This is called the Zeeman shift, and it is caused by an applied magnetic field distorting and polarising the probability distribution of the particles [7].

To get around this we measured the transition energy in multiple magnetic fields. The Zeeman shift is quadratic and could be extrapolated back to zero-field to obtain a value to compare with QED calculations. The measured transition energy νR (in MHz) is shown in Figure 2 as a function of magnetic field and the dashed lines are the extrapolations to zero-field. The figure shows data for microwave radiation moving toward the Ps atoms in two opposite directions, +x and –x, propagating in either direction along the waveguide. By taking the average of these two measurements we exactly cancel out any Doppler shifts between Ps and microwaves. However, the maximum Doppler shifts was expected to be ±0.26 MHz, based on the maximum possible misalignment of the Ps, laser and waveguide. This is much smaller than the 1.8 MHz difference observed, which is puzzling.

Figure 2 The measured Zeeman shifted transition frequency extrapolated back to zero field. The blue squares are the transition measured with microwave radiation travelling in the positive x direction, while the red circles are for the negative x direction. The green line and shaded band indicates the average measured zero-field transition frequency and its uncertainty. Hollow points are with microwave foam present and solid points are without microwave foam present. Adapted from Reference [5].

All other systematic effects are <0.1 MHz, yet the two directions disagree with each other. The prime suspect for this shift, reflection effects, was eliminated with the new chamber design and the foam. This was confirmed as shown in Figure 2 whereby data with foam (hollow points) and without foam (solid points) present does not display any change in the measured transition frequency. However, there was a small asymmetry to our line shape mesurements, which has previously indicated frequency dependent power [2]. This lead us to conclude that there were internal effects from defects in the waveguide construction or microwave circuit which caused frequency dependent power, distorting the line shapes. We treated this effect as a systematic error with a magnitude of 1.8/2 = 0.9 MHz, lowering the precision of our final result.

Our final value for the ν2 energy interval was 8627.94 ± 0.95 MHz, close enough to theory to be in broad agreement, as shown in Figure 3. This new value is the most precise measurement to date but fails to test the latest set of QED calculations, which can be described by a summation of smaller and smaller terms (i.e. ν2 = O4 + O5 + O6 + O7 + …). A measurement with just a ten times improvement in precision will be able to do this, and a 10000 times improvement would allow us to to test certain dark matter candidates [2]. We believe that this method has inherent vulnerabilities to frequency dependent microwave power and that other methods should be explored, a post on this will be coming soon. However, with improvements to the waveguide design and microwave circuit this methodology can be used for precision measurements nonetheless.

Figure 3 A comparison of our measurement of ν2 with historical ones, including the average of all the measurements so far. The vertical black line and shaded bar is the latest QED calculation and its associated error. Adapted from Reference [5].

[1] Precision Microwave Spectroscopy of the Positronium n = 2 Fine Structure. L. Gurung, T. J. Babij, S. D. Hogan and D. B. Cassidy; Phys. Rev. Lett.125, 073002 (2020)

[2] Observation of asymmetric line shapes in precision microwave spectroscopy of the positronium 23P1 →23PJ (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)

[3] 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)

[4] Microwave spectroscopy of positronium atoms in free space. R. E. Sheldon, T. J. Babij, S. H. Reeder, S. D. Hogan, and D. B. Cassidy; Phys. Rev. A 107, 042810 (2023)

[5] Precision microwave spectroscopy of the positronium 2 3S1 →2 3P2 interval. R. E. Sheldon, T. J. Babij, S. H. Reeder, S. D. Hogan, and D. B. Cassidy; Phys. Rev. Lett. 131, 043001 (2023)

[6] Penetration of electrostatic fields and potentials through meshes, grids, or gauzes. F. H. Read, N. J. Bowring, P. D. Bullivant, and R. R. A. Ward; Rev. Sci. Instruments, 69 (5), 2000–2006 (1998)

[7] Atomic physics. C. J. Foot; Oxford master series in physics, Oxford University Press (2005)

Microwaves & Positronium Pt. II: Giving Positronium the Horn (Antenna)

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A previous post of ours (found here) described how our measurements of the energy intervals of the n = 2 positronium (Ps) fine structure produced asymmetric line shapes [1], see Figure 1(a). A line shape represents the probability of transferring an atom from one quantum energy state to another as a function of the applied photon energy. In this case the photons are in the microwave regime at a frequency of ~8.6 GHz as we are looking at the 2 3S1 to 2 3P2 state transition, known as ν2. For details on how we make and measure these lineshapes, and what 2 3S1 means see the previous post linked above.

The reason we do these measurements is that they are tests of quantum electrodynamics, and can reveal new physics beyond the standard model [2]. The energy interval ν2, given in MHz, can be retrieved as the central frequency of the line shape and compred to theoretical calculations. But if there is something distorting the measurement that we cannot account for then the precision we obtain will be limited and the test not as effective. Thus the asymmetry in the previous measurement meant that a value of ν2 could not be extracted from the line shape due to the lack of a suitable theoretical model that would account for the asymmetry.

Figure 1 A comparison of line shapes generated using a waveguide (a) and a horn antenna (b). Adapted from References [1, 4].

The previous measurement used a waveguide, which is a structure designed to allow certain microwave frequencies to propagate with high intensity and uniformity. The cause of the line shape asymmetry was reflection effects, whereby microwaves escaped the open ends of the waveguide and were reflected back in [3]. The power reflected back into the waveguide varied as a function of microwave frequency because the changing wavelength altered the reflections in the chamber. This frequency dependent power distorted the line shapes, introducing asymmetry and apparent shifts to the energy interval we want to measure.

So having performed a measurement suffering from reflection effects we decided to make them much much worse. For the peer reviewed work summarised in this article click here [4]. Instead of a waveguide we used a horn antenna which allows microwave radiation to be coupled from a source into free space, removing the spatial restrictions of the waveguide (which was 12.6 mm x 25.8 mm in size, a WR-112 as it is known). In this experiment we placed the horn antenna outside the vacuum chamber where the Ps is made at a distance of up to 34 cm from the Ps, see Figure 2. The microwaves entered the chamber through a fused silica window where they could address the Ps atoms.

Figure 2 A schematic diagram of the experimental setup. The Ps atoms (green shading) are emitted toward the right hand side of the vacuum chamber (dark grey). Microwave radiation (yellow shading) is incident from the horn antenna (purple) into the vacuum where it intersects the Ps. Adapted from Reference [4].

The distance between the horn antenna and the chamber should allow the radiation to propagate and become plane waves (radiation that has each peak of the wave travelling parallel to one another), in what is known as the far-field regime [5]. Plane waves should have uniform polarisation and power distribution. However, due to the metal chamber and electrodes there was a significant amount of reflection inside the chamber, much more than in the waveguide. This can be seen in Figure 3 (a & c) which shows a 2-D map of the simulated microwave field strength inside the vacuum chamber, and 3 (b & d) which shows the polarisation of the same data. In an ideal world these would show a uniform block of colour indicating uniform field strength and polarisation, but it really does not. This effectively randomised microwave field increases the frequency dependent power variation felt by the Ps atoms, amplifying distortions to the line shape.

Figure 3 A simulated map of the electric field strength (b & d) and polarisation (a & c) of the microwave radiation ued in this work. The horn antenna and vacuum chamber were simulated, with the outline of the latter shown in grey. The upper and lower set of plots are for different microwave frequencies and you can see the difference in field pattern between the two for both field strength and polsarisation. From Reference [6].

The lineshape you see in Figure 1(b) is symmetric, unlike the previous measurement of ν2. However, this measurement does show a shift from the theoretical prediction of the energy interval. There were two causes we thought most likely to explain this shift: (a) the atoms had a preferential motion towards/away from the horn creating a Doppler shift (the change in frequency of light due to the motion of the target, i.e. when moving towards something you percieve the peaks of the wave being closer together and so see a higher frequency of light), or (b) the microwave reflections were causing a shift even without an asymmetry. The way we tested this was by varying the horn antenna angle θH with respect to the Ps. By rotating the horn we can select the subset of Ps atoms moving towards or away from the microwave radiation and change the Doppler shift with a certain angular dependency.

What we found was a ~300 kHz/degree shift, see Figure 4, which is much larger and of opposite magnitude compared to the expected Doppler shift of -40 kHz/degree. We therefore ruled out Doppler shifts as a possible explanation. In fact the shift we saw would have to come from all atoms moving directly towards the antenna at -10o and away from the antenna at +10o which, given our experiment, was implausible. Other possibilities such as ac Stark shift, spatial selection effects, polarisation effects, and stray electric fields were excluded for being too small in magnitude or not dependent on the horn orientation.

Figure 4 The angle dependent shift of the transition frequency as measured during our experiment. The horizontal dashed line is the expected theoretical transition frequency. The angled dashed lines are linear fits to the data to quantify the observed shift. Adapted from Reference [4].

It would appear that this shift is therefore due to reflection effects, which is not surprising given the distorted line shapes observed in waveguide measurements where reflections were much less dominant. We therefore concluded that a horn antenna as a source of free space microwaves is not an ideal way to perform precision measurements of Ps. But the technique further demonstrates the nature of reflection effects in line shape measurements over a broad frequency range. An interesting feature is that the line shapes show no asymmetry but display large shifts, confirming the previously simulated data that showed a shift can manifest without an associated asymmetry [3] and therefore removal of reflections must be demonstrated beyond verification of symmetric line shapes.

If we wish to test QED in Ps using these microwave regime transitions we must remove reflections as a source of error. We believe that by using a different chamber with less reflective surfaces at the ends of the waveguide we can remove these reflections. Simulations with this new chamber have demonstrated this is indeed the case. Figure 5 shows the change in strength of the microwave field in a waveguide inside a new, modified vacuum chamber (called the Cube) versus the one used to make the previous asymmetric measurements (called the Cross) [1]. The field is over three times more uniform in the newly designed chamber than the old. Thus we intend to replicate the waveguide measurement with the new chamber (and other improvements) to probe bound state QED.

Figure 5 The variation in the electric field strength of the microwave radiation in a waveguide as a function of microwave frequency for three different cases. Adapted from Reference [7].

[1] Observation of asymmetric line shapes in precision microwave spectroscopy of the positronium 2 3P1 → 2 3PJ (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)

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

[3] 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)

[4] Microwave spectroscopy of positronium atoms in free space. R. E. Sheldon, T. J. Babij, S. H. Reeder, S. D. Hogan, and D. B. Cassidy; Phys. Rev. A 107, 042810 (2023)

[5] Microwave Engineering. David M. Pozar; 4th Edition (2012)

[6] Tests of Quantum Electrodynamics Using n = 2 Positronium. R. E. Sheldon; PhD Thesis, University College London (2024)

[7] Precision microwave spectroscopy of the positronium 2 3S1 →2 3P2 interval. R. E. Sheldon, T. J. Babij, S. H. Reeder, S. D. Hogan, and D. B. Cassidy; Phys. Rev. Lett.131, 043001 (2023)

CATMIN III

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In July this year UCL hosted the CATMIN III: Frontiers in Rydberg Physics conference, co-organised by groups from Innsbruck, Harvard and UCL. Topics included Rydberg atom qubits for quantum computers [1], precision measurements of QED and fundamental constants [2], Rydberg molecule studies [3] and field sensing [4]. Our whole group attended, including our undergraduate summer students Yizhen and Carolina.

The group photo of CATMIN III in front of the UCL portico

David gave a broad presentation on Positronium Rydberg physics, including studies performed at UCL, most of which can be found on our publications page and a full video of the talk can be found on YouTube. Sam and I presented posters on ongoing work, both of which can be found in the downloads section. Sam is using Rydberg helium (He) to probe fields inside a microwave waveguide recently used to perform precision studies on the Ps n = 2 fine structure in which significant energy level shifts have been observed [5]. Given the sensitivity of Rydberg He to radio frequency, electric and magnetic fields and much better statistics compared to Ps we can investigate imperfections in the waveguide, a possible source of the shift found in Ps. A full blog post on this will be published soon.

My poster was on experiments done to observe THz (mm-wave) transitions in Rydberg Ps. For years the THz regime of the electromagnetic spectra (0.3 – 30 THz) was known as ‘the THz gap’, a band of EM radiation that is very hard to produce, lying between electronic methods at low photon energy and optical methods at high photon energy. However, much progress has been made in the last decade and there now exists commercially available technology (albeit for an extortionate price, a total of £90,000 for everything required) to generate radiation up to 1.7 THz. We are trying to measure the transition frequency between high n Rydberg states, in particular n = 21 – 24 which has a frequency of 0.874 THz. My poster described apparatus for this measurement but unfortunately, we have not yet seen any THz transition. We think the reason for this is that we have too little power (£90k only gets you 16 uW of THz power) or our statistics are too poor. Once this is working we can obtain a value for the Rydberg constant in a purely leptonic system which has advantages over the hadronic systems typically used (i.e. conventional atoms) [6]. This THz measurement can be done to examine systematics before an n = 2 – 21 measurement is performed for high precision result.

As well as the talks, we engaged in a different form of collaboration on Wednesday afternoon as there was a football match and a rounders game in Regents Park (we, of course, did very well… ish). After a somewhat achy Thursday there was a spectacular conference dinner at the Ambassador Hotel in Bloomsbury where the organiser of CATMIN IV gave a speech. We’d like to thank the organisers for hosting the event and allowing us to speak and look forward to CATMIN IV in Grenoble in the near future.

Sport!!
Rounders in Regents Park at CATMIN III

[1] Collectively Encoded Rydberg Qubit. N. L. R. Spong, Y. Jiao, O. D. W. Hughes, K. J. Weatherill, I. Lesanovsky, and C. S. Adams, Phys. Rev. Lett, 127, 063604 (2021)

[2] Precision millimetre-wave spectroscopy and calculation of the Stark manifolds in high Rydberg states of para-H2. N. Holsch, I. Doran, M. Beyer and F. Merkt, J. Mol. Spectroscopy, 387, 111648 (2022)

[3] Quantum-state-dependent decay rates of electrostatically trapped Rydberg NO molecules. M. H. Rayment and S. D. Hogan. Phys. Chem. Chem. Phys. 23 (34), 18806-18822 (2021)

[4] Rydberg-atom based radio-frequency electrometry using frequency modulation spectroscopy in room temperature vapor cells. S. Kumar, H. Fan, H. Kübler, A. J. Jahangiri, and J. P. Shaffer, Optics Express, 25(8), 8625-8637 (2017)

[5] Precision microwave spectroscopy of the positronium interval. R. E. Sheldon, T. J. Babij, S. H. Reeder, S. D. Hogan, and D. B. Cassidy, Phys. Rev. Lett. 131, 043001 (2023)

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

Microwaves & Positronium Pt. I: Lopsided Lorentzian Line shapes

Published on by R E Sheldon3 Comments

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.

Figure 1. The triplet energy levels of the Ps n = 2 state and the associated fine structure transitions.

The previous measurement written about on this blog was of the 23P1 \rightarrow 23P0 transition, known as the \nu_0 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.

Figure 2. An example of spectroscopic notation for Ps.

In addition to this transition we also measured two others, \nu_1 and \nu_2, 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 \nu_1 transition and higher frequencies for the \nu_2 transition, making the lineshapes look lopsided.

Figure 3. The asymmetric lineshapes measured during this work shown as the detuning from the theoretical resonant frequency, from Reference [3]. (a) The \nu_0 transition, (b) the \nu_1 transition and (c) the \nu_2 transition.

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.

Figure 4. The simulated variation in microwave field strength as a function of frequency and lineshapes calculated from the associated field strengths showing asymmetry and shifts from the expected transition frequency. (a) The \nu_0 transition, (b) the \nu_1 transition and (c) the \nu_2 transition. From Reference [4].

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 \nu_0 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 \rightarrow 23PJ (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