## Rydberg Positronium Special Report, ICPEAC 2015

One of the conferences that we attended during the summer (ICPEAC 2015) had the necessary set-up to film one of our talks about our recent Rydberg paper, this was summarised on a published IOP abstract.

You can watch our talk along with the rest of the lectures on ICPEAC’s youtube channel: https://www.youtube.com/watch?v=Cytjc2Er2Co.

## ANTIMATTER: who ordered that?

The existence of antimatter became known following Dirac’s formulation of relativistic quantum mechanics, but this incredible development was not anticipated. These days conjuring up a new particle or field (or perhaps even new dimensions) to explain unknown observations is pretty much standard operating procedure, but it was not always so. The famous “who ordered that” statement of I. I. Rabi was made in reference to the discovery of the muon, a heavy electron whose existence seemed a bit unnecessary at the time; in fact it was the harbinger of a subatomic zoo.

The story of Dirac’s relativistic reformulation of the Schrödinger wave equation, and the subsequent prediction of antiparticles, is particularly appealing; the story is nicely explained in a recent biography of Dirac (Farmelo 2009). As with Einstein’s theory of relativity, Dirac’s relativistic quantum mechanics seemed to spring into existence without any experimental imperative. That is to say, nobody ordered it! The reality, of course, is a good deal more complicated and nuanced, but it would not be inaccurate to suggest that Dirac was driven more by mathematical aesthetics than experimental anomalies when he developed his theory.

The motivation for any modification of the Schrödinger equation is that it does not describe the energy of a free particle in a way that is consistent with the special theory of relativity. At first sight it might seem like a trivial matter to simply re-write the equation to include the energy in the necessary form, but things are not so simple. In order to illustrate why this is so it is instructive to briefly consider the Dirac equation, and how it was developed. For explicit mathematical details of the formulation and solution of the Dirac equation see, for example, Griffiths 2008.

The basic form of the Schrödinger wave equation (SWE) is

$(-\frac{\hbar^2}{2m}\nabla^2+V)\psi = i\hbar \frac{\partial}{\partial t}\psi.$                                                    (1)

The fundamental departure from classical physics embodied in eq (1) is the quantity $\psi$, which represents not a particle but a wavefunction. That is, the SWE describes how this wavefunction (whatever it may be) will behave. This is not the same thing at all as describing, for example, the trajectory of a particle. Exactly what a wavefunction is remains to this day rather mysterious. For many years it was thought that the wavefunction was simply a handy mathematical tool that could be used to describe atoms and molecules even in the absence of a fully complete theory (e.g., Bohm 1952). This idea, originally suggested by de Broglie in his “pilot wave” description, has been disproved by numerous ingenious experiments (e.g., Aspect et al., 1982). It now seems unavoidable to conclude that wavefunctions represent actual descriptions of reality, and that the “weirdness” of the quantum world is in fact an intrinsic part of that reality, with the concept of “particle” being only an approximation to that reality, only appropriate to a coarse-grained view of the world. Nevertheless, by following the rules that have been developed regarding the application of the SWE, and quantum physics in general, it is possible to describe experimental observations with great accuracy. This is the primary reason why many physicists have, for over 80 years, eschewed the philosophical difficulties associated with wavefunctions and the like, and embraced the sheer predictive power of the theory.

We will not discuss quantum mechanics in any detail here; there are many excellent books on the subject at all levels (e.g., Dirac 1934, Shankar 1994, Schiff 1968). In classical terms the total energy of a particle E can be described simply as the sum of the kinetic energy (KE) and the potential energy (PE) as

$KE+PE=\frac{p^2}{2m}+V=E$                                                 (2)

where p = mv represents the momentum of a particle of mass m and velocity v. In quantum theory such quantities are described not by simple formulae, but rather by operators that act on the wavefunction. We describe momentum via the operator $-i \hbar\nabla$ and energy by $i\hbar \partial / \partial t$ and so on. The first term of eq (1) represents the total energy of the system, and is also known as the Hamiltonian, H. Thus, the SWE may be written as

$H\psi=i\hbar\frac{\partial\psi}{\partial t}=E\psi$                                                              (3)

The reason why eq (3) is non-relativistic is that the energy-momentum relation in the Hamiltonian is described in the well-known non-relativistic form. As we know from Einstein, however, the total energy of a free particle does not reside only in its kinetic energy; there is also the rest mass energy, embodied in what may be the most famous equation in all of physics:

$E=mc^2.$                                                                    (4)

This equation tells us that a particle of mass m has an equivalent energy E, with c2 being a rather large number, illustrating that even a small amount of mass (m) can, in principle, be converted into a very large amount of energy (E). Despite being so famous as to qualify as a cultural icon, the equation E = mc2 is, at best, incomplete. In fact the total energy of a free particle (i.e., V = 0) as prescribed by the theory of relativity is given by

$E^2=m^2c^4 +p^2c^2.$                                                        (5)

Clearly this will reduce to E = mc2 for a particle at rest (i.e., p = 0): or will it? Actually, we shall have E = ± mc2, and in some sense one might say that the negative solutions to this energy equation represent antimatter, although, as we shall see, the situation is not so clear cut. In order to make the SWE relativistic then, one need only replace the classical kinetic energy E = p2/2m with the relativistic energy E = [m2c4+p2c2]1/2. This sounds simple enough, but the square root sign leads to quite a lot of trouble! This is largely because when we make the “quantum substitution” $p \rightarrow -i\hbar\nabla$  we find we have to deal with the square root of an operator, which, as it turns out, requires some mathematical sophistication. Moreover, in quantum physics we must deal with operators that act upon complex wavefunctions, so that negative square roots may in fact correspond to a physically meaningful aspect of the system, and cannot simply be discarded as might be the case in a classical system.

To avoid these problems we can instead start with eq (5) interpreted via the operators for momentum and energy so that eq (3) becomes

$(- \frac{1}{c^2}\frac{\partial^2}{\partial t^2} + \nabla^2)\psi=\frac{m^2 c^2}{\hbar^2}\psi.$                                                (6)

This equation is known as the Klein Gordon equation (KGE), although it was first obtained by Schrödinger in his original development of the SWE. He abandoned it, however, when he found that it did not properly describe the energy levels of the hydrogen atom. It subsequently became clear that when applied to electrons this equation also implied two things that were considered to be unacceptable; negative energy solutions, and, even worse, negative probabilities. We now know that the KGE is not appropriate for electrons, but does describe some massive particles with spin zero when interpreted in the framework of quantum field theory (QFT); neither mesons nor QFT were known when the KGE was formulated.

Some of the problems with the KGE arise from the second order time derivative, which is itself a direct result of squaring everything to avoid the intractable mathematical form of the square root of an operator. The fundamental connection between time and space at the heart of relativity leads to a similar connection between energy and momentum, a connection that is overlooked in the KGE. Dirac was thus motivated by the principles of relativity to keep a first order time derivative, which meant that he had to confront the difficulties associated with using the relativistic energy head on. We will not discuss the details of its derivation but will simply consider the form of the resulting Dirac equation:

$(c \alpha \cdot \mathrm{P}+\beta mc^2)\psi=i\hbar \frac{\partial\psi}{\partial t}.$                                                     (7)

This equation has the general form of the SWE, but with some significant differences. Perhaps the most important of these is that the Hamiltonian now includes both the kinetic energy and the electron rest mass, but the coefficients αi and $\beta$ have to be four-component matrices to satisfy the equation. That is, the Dirac equation is really a matrix equation, and the wavefunction it describes must be a four component wavefunction. Although there are no problems with negative probabilities, the negative energy solutions seen in the KGE remain. These initially seemed to be a fatal flaw in Dirac’s work, but were overlooked because in every other aspect the equation was spectacularly successful. It reproduced the hydrogen atomic spectra perfectly (at least, as perfectly as it was known at the time) and even included small relativistic effects, as a proper relativistic wave equation should. For example, when the electromagnetic interaction is included the Dirac equation predicts an electron magnetic moment:

$\mu_e = \frac{\hbar e}{2m} = \mu_B$                                                                   (8)

where $\mu_B$ is known as the Bohr magneton. This expression is also in agreement with experiment, almost: it was later discovered that the magnetic moment of the electron differs from the value predicted by eq (8) by about 0.1% (Kusch and Foley 1948).  The fact that Dirac’s theory was able to predict these quantities was considered to be a triumph, despite the troublesome negative energy solutions.

Another intriguing aspect of the Dirac equation was noticed by Schrödinger in 1930. He realised that interference between positive and negative energy terms would lead to oscillations of the wavepacket of an electron (or positron) about some central point at the speed of light. This fast motion was given the name zitterbewegung (which is German for “trembling motion”). The underlying physical mechanism that gives rise to the zitterbewegung effect may be interpreted in several different ways but one way to look at it is as an interaction of the electron with the zero-point energy of the (quantised) electromagnetic field. Such electronic oscillations have not been directly observed as they occur at a very high frequency (~ 1021 Hz), but since zitterbewegung also applies to electrons bound to atoms, this motion can affect atomic energy levels in an observable way. In a hydrogen atom the zitterbewegung acts to “smear out” the electron charge over a larger area, lowering the strength of its interaction with the proton charge. Since S states have a non-zero expectation value at the origin, the effect is larger for these than it is for P states. The splitting between the hydrogen 2S1/2 and 2P1/2 states, that are degenerate in the Dirac theory, is known as the Lamb Shift (Lamb, 1947). This shift, which amounts to ~1 GHz was observed in an experiment by Willis Lamb and his student Robert Retherford (not to be confused Ernest Rutherford!). The need to explain this shift, which requires a proper explanation of the electron interacting with the electromagnetic field, gave birth to the theory of quantum electrodynamics, pioneered by Bethe, Tomanoga, Schwinger and Feynman.

The solutions to the SWE for free particles (i.e., neglecting the potential V) are of the form

$\psi = A \mathrm{exp}(-iEt / \hbar).$                                                       (9)

Here A is some function that depends only on the spatial properties of the wavefunction (i.e., not on t). Note that this wavefunction represents two electron states, corresponding to the two separate spin states. The corresponding solutions to the Dirac equation may be represented as

$\psi_1 = A_1 \mathrm{exp}(-iEt / \hbar),$

$\psi_2 = A_2 \mathrm{exp}(+iEt / \hbar).$                                                   (10)

Here $\psi_2$ represents the negative energy solutions that have caused so much trouble. The existence of these states is central to the theory they cannot simply be labelled as “unphysical” and discarded. The complete set of solutions is required in quantum mechanics, in which everything is somewhat “unphysical”. More properly, since the wavefunction is essentially a complex probability density function that yields a real result when its absolute value is squared, the negative energy solutions are no less physical than the positive energy solutions; it is in fact simply a matter of convention as to which states are positive and which are negative. However you set things up, you will always have some “wrong” energy states that you can’t get rid of. Thus, Dirac was able to eliminate the negative probabilities and produce a wave equation that was consistent with special relativity, but the negative energy states turned out to be a fundamental part of the theory and could not be eliminated, despite many attempts to get rid of them.

After his first paper in 1928 (The quantum theory of the electron) Dirac had established that his equation was a viable relativistic wave equation, but the negative energy aspects remained controversial. He worried about this for some time, and tried to develop a “hole” theory to explain their seemingly undeniable existence. A serious problem with negative energy solutions is that one would expect all electrons to decay into the lowest energy state available, which would be the negative energy states. Since this would not be consistent with observations there must, so Dirac reasoned, be some mechanism to prevent it. He suggested that the states were already filled with an infinite “sea” of electrons, and therefore the Pauli Exclusion Principle would prevent such decay, just as it prevents more than two electrons from occupying the lowest energy level in an atom. (Note that this scheme does not work for Bosons, which do not obey the exclusion principle). Such an infinite electron sea would have no observable properties, as long as the underlying vacuum has a positive “bare” charge to cancel out the negative electron charge. Since only changes in the energy density of this sea would be apparent, we would not normally notice its presence. Moreover, Dirac suggested that if a particle were missing from the sea the resulting hole would be indistinguishable from a positively charged particle, which he speculated was a proton, protons being the only positively charged subatomic particles known at the time.

This idea was presented in a paper in 1930 (A Theory of Electrons and Protons, Dirac 1930). The theory was less than successful, however, and the deficiencies served only to undermine confidence in the entire Dirac theory. Attempts to identify holes as protons only made matters worse; it was shown independently by Heisenberg, Oppenheimer and Pauli that the holes must have the electron mass, but of course protons are almost 2000 times heavier. Moreover, the instability between electrons and holes completely ruled out stable atomic states made from these entities (bad news for hydrogen, and all other atoms). Eventually Dirac was forced to conclude that the negative energy solutions must correspond to real particles with the same mass as the electron and a positive charge. He called these anti-electrons (Quantised Singularities in the Electromagnetic Field, Dirac 1931).

This almost reluctant conclusion was not based on a full understanding of what the negative energy states were, but rather the fact that the entire theory, which was so beautiful in other ways that it was hard to resist, depended on them. It turns out that to properly understand the negative energy solutions requires the formalism of quantum field theory (QFT). In this description particles (and antiparticles) can be created or destroyed, so it is no longer necessarily appropriate to consider these particles to be the fundamental elements of the theory. If the total number of particles in a system is not conserved then one might prefer to describe that system in terms of the entities that give rise to the particles rather than the particles themselves. These are the quantum fields, and the standard model of particle physics is at its heart a QFT. By describing particles as oscillations in a quantum field not only do we have an immediate mechanism by which they may be created or destroyed, but the problem of negative energies is also removed, as this simply becomes a different kind of variation in the underlying quantum field. Dirac didn’t explicitly know this at the time, although it would be fair to say that he essentially invented QFT, when he produced a quantum theory that included quantized electromagnetic fields (Dirac, 1927, The Quantum Theory of the Emission and Absorption of Radiation). This led, eventually, to what would be known as quantum electrodynamics. Dirac would undoubtedly have been able to make much more use of his creation if he had not been so appalled by the notion of renormalization. Unfortunately this procedure, which in some ways can be thought of as subtracting infinite quantities from each other to leave a finite quantity, was incompatible with his sense of mathematical aesthetics.

So, despite initially struggling with the interpretation of his theory, there can be no question that Dirac did indeed explicitly predict the existence of the positron before it was experimentally observed. This observation came almost immediately in cloud chamber experiments conducted by Carl Anderson in California (C. D. Anderson: The apparent existence of easily deflectable positives, Science 76 238, 1932).  Curiously, however, Anderson was not aware of the prediction, and the proximity of the observation was apparently coincidental. We will discuss this remarkable observation in a later post.

*This post is adapted from an as-yet unpublished book chapter by D. B. Cassidy and A. P. Mills, Jr.

References:

Griffiths, D. (2008). Introduction to Elementary Particles Wiley-VCH; 2nd edition.

Farmelo, “The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom” Basic Books, New York, (2011).

Dirac, P.A.M. (1927). The Quantum Theory of the Emission and Absorption of Radiation, Proceedings of the Royal Society of London, Series A, Vol. 114, p. 243.

P. A. M. Dirac, Proc. Phys. Soc. London Sect. A 117, 610 (1928).

P. A. M. Dirac, Proc. Phys. Soc. London Sect. A 126, 360 (1930).

P. A. M. Dirac, Proc. Phys. Soc. London Sect. A 133, 60 (1931).

Anderson, C. D. (1932). The apparent existence of easily deflectable positives, Science 76, 238.

A.  Aspect, D. Jean, R. Gerard (1982). Experimental Test of Bell’s Inequalities Using Time- Varying Analyzers, Phys. Rev. Lett. 49 1804

P. Kusch and H. M. Foley “The Magnetic Moment of the Electron”, Phys. Rev. 74, 250 (1948).

## Controlling Positronium Annihilation with Electric Fields

To produce Rydberg (highly-excited) states of positronium we use a multi-photon $1 ^3S \rightarrow 2 ^3P \rightarrow nS/nD$ excitation scheme [1].  These high-$n$ Ps atoms are long-lived and could potentially be used for (anti)-gravity measurements, however, the intermediate state ($n=2$) has interesting properties of it’s own, as described in our latest article (Phys. Rev. Lett. 115, 183401).

Unlike regular atoms, Ps has the peculiar feature that it can self-annihilate into gamma-rays.  The amount of overlap between the positron and electron wave functions depends on the particular state the atom is in, and this determines how long before self-annihilation occurs (characterised by the average annihilation lifetime).  The quantum spin ($s=1/2$) of the electron and positron can combine in positronium to either cancel  ($S=0$) or sum ($S=1$), depending on the relative alignment between the two components.  In the former case (para-Ps) the atom has a very short ground-state lifetime of just 125 ps, whereas in the latter case (ortho-Ps) the atom lives in the $n=1$ state for an average of 142 ns (this may not sound very long but it’s actually plenty of time to do spectroscopy with pulsed lasers).

We produce $n=1$ ortho-Ps ($1^3S_1$) atoms then excite these using 243 nm laser light from our UV laser. The electronic dipole transition selection rules (principally, $\Delta S= 0$ and $\Delta \ell = \pm 1$) dictate that this single-photon transition drives the atoms to the $n = 2$, $\ell= 1$$S= 1$ state ($2 ^3P_J$).  For historical reasons the orbital angular momentum is written here as $S$ ($\ell= 0$) and $P$ ($\ell= 1$).

The fluorescence lifetime of an excited atom is the time it takes, on average, to spontaneously emit a photon and decay to a lower energy state. All of the $n=2, \ell = 1$ states have a fluorescence lifetime of 3.19 ns, and an annihilation lifetime of over 100 $\mu$s (practically infinite compared to the time-scale of our measurements, i.e., $2^3P$ states don’t annihilate directly, but can decay to a different state then annihilate). The $n=2, \ell = 0$ ortho and para states have annihilation lifetimes of 1136 ns and 1 ns, and they both fluoresce with a lifetime of $\simeq$ 0.24 s ($\approx \infty$).  The bottom line here is that there are a wide range of fluorescence and annihilation lifetimes for the various possible sub-states in the $n=1$ and $n=2$ manifolds.

In a magnetic field the short-lived $S=0$ and longer-lived $S=1$ states (with the same $\ell$) are mixed together (Zeeeman mixing).  Similarly, an electric field mixes states with different $\ell$ (but the same $S$) (Stark mixing).  By exciting Ps to $n = 2$ in a weak magnetic field then varying an electric field, we can tailor the extent of this mixing to increase or decrease the overall lifetime. This technique can be used to greatly increase the excitation efficiency to another state, since the losses due to annihilation can be reduced.  Conversely, increasing the annihilation rate can be used as an efficient way to detect excitation.

The polarization orientation of the UV  excitation laser gives us some control over which $M_J$ states are subsequently populated. More specifically, if the laser polarization is parallel to the applied magnetic field then only $\Delta M_J=0$ transitions are allowed, whereas if the polarization is perpendicular to it then  $\Delta M_J$ must change by $\pm 1$.

Below is a calculation of how the $n=2$ energy levels are shifted by an electric field, in zero magnetic field (red) and in a magnetic field of 13 mT (blue). Note the avoided crossing at 585 V/ cm in the 13 mT case.
So what can we actually measure? In most cases, laser excitation makes it more likely for ground state ortho-Ps to ultimately end up in the short-lived para-Ps state, thus applying the laser causes an increase in the annihilation gamma ray flux at early times. This change can be observed and quantified using the parameter $S$ (higher values means more gamma rays were detected compared to a measurement made without the laser). This is plotted below for various electric field strengths, and with the laser polarised either parallel (red) or perpendicular (green) to the magnetic field.  In both cases, the avoided crossing gives a sharp increase in annihilation rate (see the “ears” in both plots), whilst higher electric fields either reduce or increase the signal, depending on which $M_J$ states the laser initially populates.

Notice that when the laser polarisation is aligned parallel to the magnetic field (red), very high electric fields lead to negative $S$ values. This means that the lifetime of the Ps becomes longer than 142 ns (the ground-state ortho-Ps lifetime) if the laser is applied. This is due to the fact that in this field configuration there is significant mixing into the long lived $2^3S_1$ state.  This could be used to produce an ensemble of pure $2^3S_1$ states, by exciting Ps in this high field and then extracting the excited state into a region of zero field. These pure states could be exploited for $n=2$ microwave spectroscopy [3].

## High Temperature Coldhead for Positronium Photoemission

We have installed a new coldhead in our beamline with a high temperature heater, this will allow us to heat up samples up to 1000 K and cool them down to 10 K. The aim of this new addition to our system is to test Positronium emission from targets by implanting a positron pulse just after irradiating the sample with a high power laser. This has been done before with silicon samples [1]. In these previous studies samples of silicon were heated to similar temperatures and a green (532 nm) pulsed laser was fired at the samples before implanting bunched positron pulses, it was shown that heating the target increased the positronium yield, and likewise, increasing the laser power also enhanced the amount of positronium produced. It was also observed that the temperature of the target and the power of the laser pulse do not affect the energy of the produced positronium, only the yield. We plan to use this coldhead to repeat these measurements on germanium targets. Positronium produced out of these semiconductor targets has a very large emission energy, approximately 0.16 eV, compared to the 50 meV we can attain with porous silica. However, we can also cool these samples down to 10 K, potentially making efficient positronium emission at cryogenic temperatures more feasible if the same positronium photoemission mechanism is reproducible. This would be very useful for experiments in which antihydrogen will be produced at very low temperatures after interacting with positronium and trapped antiprotons [2].

## 3rd International Workshop on Antimatter and Gravity

The UCL Positronium Spectroscopy group recently hosted the 3rd International Workshop on Antimatter and Gravity (WAG). The aim of this meeting was to bring together those interested in all aspects of antimatter and gravity, from fundamental theory to direct tests of the weak equivalence principle of general relativity using antiparticles or elements that contain antimatter, e.g. positronium, muonium, or antihydrogen. In our case, we would aim to contribute towards this goal by ultimately making a gravity measurement with positronium.

The workshop was a great opportunity to hear new ideas and theories, to catch-up on the most recent successes and the milestones reached with on-going antimatter experiments, and to learn of the latest proposals for new ways to study antimatter and gravity.

We hope those who attended had as a good time here in London as we did hosting them, and we look forward to meeting again for the next WAG!

## Positronium Spectroscopy, Conference season!

The UCL Positronium Spectroscopy group has recently attended the final EU Marie Curie COHERENCE conference, iCoRD (International Conference on Rydbergs at Durham), where we presented our latest Rydberg Positronium results (PRL. 115, 173001). These results will also be presented at the 29th International Conference on Photonic, Electronic, and Atomic Collisions (ICPEAC) in Toledo, Spain.

We are now attending the 18th International Workshop on Low-Energy Positron and Positronium Physics & the 19th International Symposium on Electron-Molecule Collisions and Swarms (POSMOL) in Lisbon, Portugal, where we will also be presenting our Positronium time-of-flight spectroscopy (NJP. 17, 043059) results, as well as some of our latest results on Positronium at cryogenic temperatures.

The posters giving a brief overview of the contents in each 3 of these sets of results can be found in our Downloads page:

## New Nd:YAG laser

We have a “new” (used) Nd:YAG pulsed laser (labelled Nd:YAG B in the photo below) that can produce up to 600 mJ of 1064 nm light in 6 ns pulses which we frequency double to 532 nm to pump our Radiant Dyes Narrowscan laser.  The dye laser has a high-power, narrow bandwidth (5 GHz), near infra-red (730-750 nm) output.

As before, the 1064nm output of the “old” Nd:YAG (labelled as Nd:YAG A) is doubled to 532 nm and the sum-frequency of the first and second harmonic is used to generate 170 mJ of 355 nm light. This third harmonic pumps the Sirah Cobra-Stretch laser (another dye laser), which outputs broadband (85 GHz) 486 nm pulses that are then doubled to 243 nm (UV).

The 243 nm UV photons can resonantly excite ground-state Positronium into the 2P state; the excited atoms can then be driven to high-n Rydberg states with our infra-red laser (see Ref 1).

The laser systems (A and B) are completely independent, so we can easily fine-tune the timing of the two and optimise the two-step excitation process.

Refs:

## What happens to a silica film at cryogenic temperatures?

Since we are interested in making positronium atoms we are always looking to shoot positron beams at various materials, and under different conditions. In some cases we might need our Ps atoms to be made in a cold environment, so they can be excited to Rydberg states without being harassed by black body radiation. One of the best positronium formation targets we have used are porous silica films, which we get from collaborators in Paris (Laszlo Liskay and co-workers from CEA Saclay) [1]. Because of the way these materials make Ps they are not very sensitive to the temperature, so it should be possible to cool them down without changing the amount or character of Ps produced after a positron beam is implanted. This has already been seen at around 50 K [2] but we decided to have a look for ourselves at a slightly lower temperature (12K) to see if the impact of the positron beam might cause some damage at these temperatures (it can happen [3]).

With a cold head installed in our new positronium interaction chamber, we have cooled one of Lazslo’s silica films [1] to 12 Kelvin (~261˚C) which is about 100˚C colder than the dark side of the moon. It turns out that our positron beam didn’t do any damage at all and the sample was basically fine, so just for fun we decided to blast it with a laser beam (UV light, at 243 nm).

When you cool something down any gas in the region will tend to freeze on it. In ultra-high vacuum there isn’t that much gas around, but there is always a bit (known as residual gas, for obvious reasons) and after a while we do observe some fairly minor effects from all this freezing gas. Fortunately this takes a long time, and the sample is still useable for a week or so, and if you warm it up it will be restored to its original condition (since the frozen gas just evaporates away from the target). Once you start shooting the silica with a laser, however, things are not so stable, as shown in the figure. We observe a drastic reduction in the positronium formation efficiency after the silica is irradiated at low temperature (nothing happens at room temperature).

The delayed fraction f (black data points) measured for different sample temperatures (solid red lines), with the UV laser fired during the times indicated. Since f measures the amount of long-lived Ps present (it is more or less proportional to the fraction of incident positrons that form positronium) the sharp drop indicates that either less Ps is being created, or that it is being destroyed shortly after creation. The latter process is consistent with the experiment of Saito et al. Note that there is no effect from the laser at room temperature, and that the paramagnetic centers created at low temperature can be annealed out when the temperature is raised.

This is not very surprising, researchers in Japan already saw this many years ago [4]. Although they did not use lasers, and their experiments were done with slightly different samples (not thin films as we have been using) the physical mechanism is expected to be essentially the same. At low temperatures disturbed molecules are not able to repair themselves and so if they are distorted in some way by radiation they tend to remain in that configuration. This can create something called a paramagnetic center which is bad news for positronium atoms. Why? Well paramagnetic centers are essentially unpaired spins, and interactions with these makes it very easy for a long-lived (triplet) Ps state to be converted into a short-lived (singlet) state. In other words, paramagnetic centers kill positronium atoms. These killer centers are not stable at room temperature, and molecular thermal fluctuations can restore the system to its normal state (which generally does not contain any paramagnetic centers). This means that after we create these troublesome centers with a laser all we have to do to get rid of them is to warm the target up. When we do this (see figure above) we get an annealing/recovery process quite similar to the results of Saito et al [4].

Refs.

## System modification for Rydberg Ps imaging

A key milestone along the road to Ps gravity measurements is control of the motion of  long-lived states of positronium. Using methods previously developed for atoms and molecules we aim to manipulate low-field seeking Stark states within the Rydberg-Stark manifold (see below) using inhomogeneous electric fields [1, 2].

The force exerted on Rydberg atoms due to their electric dipole moment can be described as:

where n is the principal quantum number, k is the parabolic quantum number (ranging from –(n-1-|m|) to n-1-|m| in steps of 2), and F is the electric field strength [3, 4]. The figure above shows an example of Rydberg-Stark state manifold for n=11.

We have recently modified our experimental system to accommodate an MCP for imaging Ps atoms. This involved the extension of our beamline with another multi-port vacuum chamber, within which we should be able to reproduce laser excitation of Ps to Rydberg states.  These will be formed at the centre of the chamber and directed along a 45 degree path towards the MCP. If imaging Ps* proves successful we will then use electrodes to create the inhomogeneous electric fields needed to manipulate their flight path.

The addition of the new vacuum chamber to our beamline is shown below.

Refs.

## Positrons are cool (thanks to tetrafluoromethane)

The positronium spectroscopy experiments we perform are contingent on our ability to form and bunch dense clouds of positrons (anti-electrons).  These are implanted into porous silica where they pair up with electrons and form our favourite exotic element, Ps (which we then photo-ionise with lasers before the two components have chance to annihilate one another).

It’s been 24 years since the first buffer-gas trap [1] was used to collect positrons using a combination of electric and magnetic fields in a configuration known generically as a Penning trap.  The positron accumulation device is often termed a ‘Surko Trap’ after its inventor Cliff Surko, and there’s detailed information about how they work on his site.

The magnetic field lines of a solenoid guide a low-energy positron beam through a series of cylindrical electrodes that have been biased with voltages to create an electric potential minimum (along the axis of the magnetic field) –  see figure below.  These fields alone are not enough to capture positrons from the beam, as those particles with enough energy to enter the trap can also escape it.  Admitting a small amount of nitrogen into the vacuum chamber allows positrons to lose energy as they traverse the trap via inelastic collisions with the buffer-gas molecules, resulting in confinement.  Unfortunately positrons are also lost to annihilation with electrons in the gas.  Surko traps feature several sections with the pressure in each optimised to either capture (higher pressure: good chance of an inelastic collision) or to keep (lower pressure: less chance of annihilation) positrons, which gives the trap its characteristic asymmetric shape.  Thanks to years of optimisation and refinements [2] these devices can accumulate hundreds of millions of positrons from a small radioactive source in just a few minuets (our fairly small trap is capable of capturing roughly half a million e+ in 1 s).  Pulsing the electrode voltages ejects the positrons from the trap in a dense, time-focussed (<10 ns) cloud that’s ideal for creating Ps atoms.

The details of how the positrons interact with the nitrogen are critical for optimisation of the trap.  The likelihood that a positron will have a collision with a nitrogen molecule is related to the pressure of the gas and the scattering cross-section: a hypothetical area that describes the apparent “size” of the molecule.   The scattering cross-section has numerous components that correspond to different types of interaction (elastic collisions, various types of inelastic collisions, ionisation, direct annihilation, Ps formation, …. etc) with some interactions more likely than others.  However, these cross-sections can vary significantly depending on the energy of the collision.

Nitrogen is used in Surko traps because there is a small range of energies  (around 10 eV) where inelastic scattering by exciting an electronic transition in the molecule is reasonably probable, compared to annihilation.  After the positrons cool below the energy needed to excite the electronic transition, further cooling relies on exciting the vibrational states of the molecule, for which the cross-section is rather small.  To speed up cooling we use a second gas, tetrafluoromethane (CF4), which has a much larger cross-section for low-energy inelastic collisions with positrons [3].

Recent Monte Carlo simulations of Surko traps offer a way to further optimise and improve trap designs without the need to manufacture dozens of prototypes.

This past week Srdjan Marjanovic, a PhD student at the University of Belgrade, visited UCL to test some of the simulations he has been working on  (see above) using our positron trap.  One suggestion he made is to use CF4 for trapping, in lieu of nitrogen, the logic being that for the energies where inelastic collisions dominate many loss mechanisms are suppressed.  The main difficulty, however, is that many more collisions are needed to successfully capture the positrons, as the energy exchanged to excite vibrational transitions in CF4 is roughly 40 times less than for the electronic transition usually exploited with N2.  Unfortunately  we were unable to adapt our trap to work efficiently with CF4. Nonetheless, by trying we have learnt more about how our trap works and – most importantly –  Srdjan has new data he can use to refine his simulations.

Photo.  Srdjan hard at work on the positron beam.