Sougato Bose, Probing macroscopic quantum superpositions and the quantum nature of gravity through levitated objects

20170215_160711We often have internal speakers giving talks at the Atomic, Molecular Optical and Positron Physics (AMOPP) group about their cutting-edge research. Last wednesday (15th February 2017) Professor Sugato Bose presented some of his latest results and calculations based on experiments being performed by  Professor Peter Barker’s group on macroscopic quantum behaviour.

The talk had the broad interest of the department as you can see from the fully-occupied lecture hall above,  and Professor Sugato agreed to provide a copy of his slides, which you can download here, so you may too get an insight into this topic.

 

Abstract:

Probing macroscopic quantum superpositions and the quantum nature of gravity through levitated objects

Prof. Sougato Bose, Dept of Physics & Astronomy University College London

We will discuss theoretical proposals of how quantum superpositions of distinct centre of mass states of a nano-crystal may be created and probed purely by measuring a spin embedded in the object. The idea is to use a levitated diamond with an NV centre spin. Next we will also describe how to reach conditions whereby two such masses interacting purely through gravitational interaction can become entangled. Witnessing such an entanglement experimentally is equivalent to establishing the quantum nature of the gravitational field. Time permitting, we will discuss how the violation of macro-realism can be verified for a levitated nano-object in a loop-hole free manner simply by coarse grained position measurements.

Charles W. Clark, Twisting the neutron wavefunction

charles_clarkLast Wednesday (February 8th 2016) we were lucky enough to have Charles W. Clark from the Joint Quantum Institute at NIST as an invited speaker to give a talk about “Twisting the neutron wavefunction”. The talk focused on the importance of fundamental wave theory and how the wave-particle duality of neutrons studied with interferometers (particularly on a Mach-Zehnder configuration) can provide great insight about basic optical principles being applied to matter waves, such as the addition of angular momentum to neutron wavefunctions.

The abstract for his talk can be found below, and he kindly agreed to provide a copy of his presentation slides which you can download here.

We will be updating our blog with subsequent talks from visiting speakers that may visit UCL, so stay tuned!

Abstract:

Twisting the neutron wavefunction

Charles W. Clark, Joint Quantum Institute, University of Maryland , USA

Wave motions in nature were known to the ancients, and their mathematical expression in physics today is essentially the same as that first provided by d’Alembert and Euler in the mid-18th century. Yet it was only in the early 1990s that physicists managed to control a basic property of light waves: their capability of swirling around their own axis of propagation. During the past decade such techniques of control have also been developed for quantum particles: atoms, electrons and neutrons. I will present a simple description of these phenomena, emphasizing the most basic aspects of wave and quantum particle motion. Neutron interferometry offers a poignant perspective on wave- particle duality: at the time one neutron is detected, the next neutron has not yet even been born. Here, indeed, each neutron “then only interferes with itself.” Yet, using macroscopically-machined objects, we are able to twist neutron deBroglie waves with sub-nanometer wavelengths.

Rydberg Ps electrostatically guided in curved quadrupole

The latest efforts of our research at UCL have been focused on manipulating Positronium (Ps) atoms in highly-excited principal quantum number n states (Rydberg states) [PRL. 114, 173001]. In one of our latest works we showed how we can exploit the large electric dipole moment of low-field-seeking Rydberg states (those states which have positive Stark shift) to confine them in a quadrupole “guide” [PRL 117, 073202].

As a direct follow-up to that study, we devised a modified version of a quadrupole guide with a 45° bend that would allow us to perform velocity selection on the atoms being guided by tuning the efficiency with which the Rydberg Ps atoms are transmitted through the bend, in addition, in our previous set-up we experienced technical difficulties since the detection scheme was in-line with our positron beam, so having a curved guide would also be beneficial for that reason.

curvedguidechamber

The schematic figure above depicts our current experimental setup, which we have used to guide Rydber Ps atoms around a 45° bend into a region off-axis with our positron beam. We have not yet implemented velocity selection, but we have clear evidence that we can efficiently guide Ps atoms in this configuration.histtotal_lyso2nai

The left panel in the figure above shows the time of flight (TOF) distribution of n = 14 atoms excited to high-field seeking states (as measured by the detectors at the end of the curved guide, i.e. “LYSO C” and “NaI”), and a background wavelength with is off-resonant with any transition, essentially acting like a “laser on” and “laser off” measurement. The right panel shows the background-subtracted trigger rate for this measurement (“laser on” – “laser off”), which shows clear evidence of atoms with a TOF arrival time of ~8 \mu \mathrm{s}.

In addition to this being a stepping stone to demonstrate velocity selection due to the acceptance of the curved section of the guide, we may also improve this set-up into eventually developing a ring-like stark decelerator, and other Ps atom optics.

Production and time-of-flight measurements of high Rydberg states of Positronium

One of our recent studies focused on measuring the lifetimes of Rydberg states of Positronium (Ps) [PRA. 93, 062513]. However, some of the limitations that prevented us from measuring lifetimes of states with higher principal quantum number (n), is the fact that such states can be easily ionised by the electric fields generated by the electrodes in our laser-excitation region (these electrodes are normally required to achieve an excitation electric field of nominally ~ 0 V/cm).

We have recently implemented a simple scheme to overcome this complication, whereby we make use of a high-voltage switch to turn discharge the electrodes in the interaction region after the laser excitation has taken place.

n19ionThe figure shown above show the Background-subtracted spectra (the SSPALS detector trace is recorded with a background and resonant wavelength, they are then normalised and subtracted from each other) for n = 18 and n = 19. It is clear from the “Switch Off” that when the high voltage switched is not utilised (and the voltages to all electrodes are always on), that most of the annihilations happen at early times, especially around ~100ns, this is the time it takes for the atoms to travel out of the low-field region, and become field-ionised by the DC voltage on the electrodes.

On the other hand, the “Switch On” curves show that both n = 18 and 19 have many more delayed events (after ~ 400 ns) due to Rydberg Ps being able to travel for much longer distances before annihilating when the switch is used to discharge the electrode biases.

n19ion_tof

The figure above shows  data taken by a detector set up for single-gamma-ray detection, approximately 12 cm away from the Ps production target, on the same experiment as described for the previous figure. It is clear from this data that the time-of-flight (TOF) to this detector is ~2 \mu \mathrm{s} However, in this case it is clear that only the n = 19 state benefited from having the “switch on”, indicating that is the smallest-n state that this scheme is necessary for our current electric-field configuration.

Comparing the SSPALS and TOF figures it can be seen that even though the n = 18 SSPALS signal was changed drastically, the n = 18 TOF distribution remained the same, this is a clear example of how changes in the SSPALS spectrum discussed in the first figure are indicative of changes in atom distributions close to the Ps production region, but are not necessarily correlated to TOF distributions measured at different positions across the Ps flight paths

These methods will eventually lead to more accurate measurement of the lifetimes of higher n-states of Ps, and the possibility of using those states with higher electric dipole moments for future atom-optics experiments, such as Ps electrostatic lenses and Stark decelerators.

Efficient production of n = 2 Positronium in S states

We routinely excite Positronium (Ps) into its first excited state (n = 2) via 1-photon resonant excitation [NJP. 17 043059], and even though most of the time this is an intermediate step for subsequent excitation to Rydberg (high n) states [PRL. 114, 173001], there is plenty of interesting physics to be explored in n = 2 alone, as we discussed in one of our recent studies [PRL. 115, 183401 and  PRA. 93, 012506].

In this study we showed that the polarisation of the excitation laser, as well as the electric field that the atoms are subjected to, have a drastic effect on the effective lifetime of the excited states and when Ps annihilates.

qexp

Above you can see the data for two laser polarisations, showing the Signal parameter S(%) as a function of electric field, this is essentially a measure of how likely Ps is to annihilate compared to ground-state (n = 1) Ps, that is to say, if S(%) is positive then n = 2 Ps in such configuration annihilates with shorter lifetimes than n = 1 Ps (142 ns), whereas if S(%) is negative then n = 2 Ps will annihilate with longer lifetimes than 142 ns, These longer lifetimes are present in the parallel polarisation (pannel a).

Using this polarisation, and applying a large negative or positive electric field (around 3 kV/cm), provides such long lifetimes due to the excited state containing a significant amount of triplet S character (2S), a substate of = 2 with spin = 1 and \ell = 0. If the Ps atoms are then allowed to travel (adiabatically) to a region of zero nominal electric field (our experimental set-up [RSI. 86, 103101] guarantees such transport), then they will be made up almost entirely of this long-lived triplet S character, and will thus annihilate at much later times than the background n = 1 atoms. These delayed annihilations can be easily detected by simply looking at the gamma-ray spectrum recorded by our LYSO detectors [NIMA. 828, 163] when the laser is on resonance (“Signal”), and subtracting it from the spectrum when the laser is off resonance (“Background”).

The figure above shows such spectra taken with the parallel laser polarisation, at a field where there should be minimal 2S Production (a), and a field where triplet S character is maximised (b).   It is obvious that on the second case, there are far more annihilations at later times, indicated by the positive values of the data on times up to 800 ns. This is clear evidence that we have efficiently produced = 2 triplet S states of Ps using single-photon excitation. Previous studies of 2S Ps produced such states either by collisional methods [PRL34, 1541], which is much more inefficient than single-photon excitation,  or by two-photon excitation, which is also more inefficient, requires much more laser power and is limited by photo-ionisation [PRL. 52, 1689].

This observation is the initial step before we begin a new set of experiments where we  will attempt to measure the = 2 hyperfine structure of Ps using microwaves!

14th International Workshop on Slow Positron Beam Techniques & Applications

Members of the UCL positronium laser spectroscopy group recently attended the 14th International Workshop on Slow Positron Beam Techniques & Applications (SLOPOS14) in Matsue, Japan. The conference took place from the 22nd to the 27th of May 2016.  During this time we heard many great talks from groups working with positrons and positronium (Ps) from all over the world.

We also presented some of our work, including Rydberg-Stark states of Ps (PRL. 115, 173001), laser-enhanced time-of-flight spectroscopy (NJP. 17, 043059), Ps production in cryogenic environments (PRB 93, 125305), controlling annihilation of excited-state Ps (PRL115, 183401 & PRA93, 012506), and improved SSPALS measurements with LYSO scintillators (NIM. A,  828, 163). The talk “Controlling Annihilation Dynamics of n = 2
Positronium with Electric Fields”, given by Alberto. M. Alonso (PhD student), was awarded a prize for making an outstanding contribution to the conference!

SLOPOS14 was a great opportunity to meet fellow physicists working in our field, to learn of their progress and to share our own.  These meetings are important for discussing new results and new ideas, and for building collaborations for future work. We are extremely grateful to the organisers for their hard work in hosting the event.

slopos14photo

We look forward to the next SLOPOS, which will be held in Romania in 2019

 

Photoemission of Ps from single-crystal p-Ge semiconductors

The production of positronium in a low-temperature (cryogenic) environment is in general only possible using materials that operate via non-thermal processes. In previous experiments we showed that porous silica films can be used in this way at temperatures as low as 10 K, but that Ps formation at these temperatures can be inhibited by condensation of residual gas, or by laser irradiation.

It has been known for several years now that some semiconductors can produce Ps via an exciton-like surface state [12]. Si and Ge are the only semiconductors that have been studied so far, but it is likely that others will work in a similar way. The electronic surface state(s) underlying the Ps production can be populated thermally, resulting in temperature dependent Ps formation that is very similar to what is observed in metals (for which the Ps is actually generated via thermal desorption of positrons in surface states). Since laser irradiation can also populate electronic surface states, and is known to result in Ps emission from Si at room temperature, the possibility exists that this process can be used at cryogenic temperatures.

We have studied this possibility using p-type Ge(100) crystals. Initial sample preparation involves immersion in acid (HCl) and this process leaves the sample with Chlorine-terminated dangling bonds which can be thermally desorbed. We attached the samples to a cold head with a high temperature interface  that can be heated to 700 K and cooled to 12 K. The heating is necessary to remove Cl from the crystal surface, which otherwise inhibits Ps formation. Fig 1 shows the initial heating cycle that prepares the sample for use. The figure shows the delayed annihilation fraction (which is proportional to the amount of positronium) as a function of temperature.

photoweb

FIG. 1:  Delayed fraction as a function of sample temperature after initial installation into the vacuum system. After the surface Cl has been thermally desorbed the amount of Ps emitted at room temperature is substantially increased.

As has been previously observed [2] using visible laser light at 532 nm can increase the Ps yield. This occurs because the electrons necessary for Ps formation can be excited to surface states by the laser. However, these states have a finite lifetime, and as both the laser and positron pulses are typically around 5 ns wide these have to be synchronized in order to optimise the photoemission effect. This is shown in FIG 2.  These data indicate that the electronic surface states are fairly short lived, with lifetimes of less than 10 ns or so. Longer surface states were observed in similar measurements using Si.

phototime web

FIG 2: Delayed fraction as a function of the arrival time of the laser relative to the incident positron pulse. These data are recorded at room temperature.  The laser fluence was ~ 15 mJ/cm^2

When Ge is cooled the Ps fraction drops significantly. This is not related to surface contamination, but is due to the lack of thermally generated surface electrons. However, surface contamination does further reduce the Ps fraction (much more quickly than is the case for silica. This effect is shown in FIG 3. If a photoemission laser is applied to a cold contaminated Ge sample two things happen (1) the laser desorbs some of the surface material and (2) photoemission occurs .This means that Ge can be used to produce Ps with a high efficiency at any temperature, and we don’t even have to worry about the vacuum conditions (within some limits).

laser_powers

FIG 3: Delayed fraction as a function of time that the target was exposed to showing the effect that different laser fluences has on the photoemission process. During irradiation, the positronium fraction is noticeably increased.

There are many possible applications for cryogenic Ps production within the field of antimatter physics, including the formation of antihydrogen formation via Ps collision with antiprotons [3], Ps laser cooling and Bose Einstein Condensation [4], as well as precision spectroscopy.

[1] Positronium formation via excitonlike states on Si and Ge surfaces. D. B. Cassidy, T. H. Hisakado, H. W. K. Tom, and A. P. Mills, Jr. Phys. Rev. B, 84, 195312 (2011). DOI:10.1103/PhysRevB.84.195312.

[2] Photoemission of Positronium from Si. D. B. Cassidy, T. H. Hisakado, H. W. K. Tom, and A. P. Mills, Jr. Phys. Rev. Lett. 107, 033401 (2011). DOI:10.1103/PhysRevLett.107.033401.

[3] Antihydrogen Formation via Antiproton Scattering with Excited Positronium. A. S. Kadyrov, C. M. Rawlins, A. T. Stelbovics, I. Bray, and M. Charlton. Phys. Rev. Lett. 114, 183201 (2015). DOI:10.1103/PhysRevLett.114.183201.

[4] Possibilities for Bose condensation of positronium. P. M. Platzman and A. P. Mills, Jr. Phys. Rev. B 49, 454 (1994). DOI:10.1103/PhysRevB.49.454.

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= 1S= 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 \mus (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.
StarkmapFigSo 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.

Quenching

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].

[1] Selective Production of Rydberg-Stark States of Positronium. T. E. Wall, A. M. Alonso, B. S. Cooper, A. Deller, S. D. Hogan, and D. B. Cassidy, Phys. Rev. Lett. 114, 173001 (2015) DOI:10.1103/PhysRevLett.114.173001.

[2] Controlling Positronium Annihilation with Electric Fields.  A. M. Alonso, B. S. Cooper, A. Deller, S. D. Hogan, D. B. Cassidy, Phys. Rev. Lett. 115, 183401 (2015) DOI:10.1103/PhysRevLett.115.183401.

[3] Fine-Structure Measurement in the First Excited State of Positronium. A. P. Mills,  S. Berko, and  K. F. Canter, Phys. Rev. Lett. 34 1541 (1975) DOI:/10.1103/PhysRevLett.34.1541.