Antiproton Interferometry and Aharonov–Bohm Effect (AIABE)

Abstract

The Aharonov–Bohm (AB) effect is one of the most striking features of quantum physics, and it is of critical importance to elucidate the role of the electromagnetic potential and its physical meaning. In spite of the fact that it was first proposed in 1949, up to now it has only been demonstrated for electrons and neutrons. We propose the realization of an experiment to study the AB effect for the antiproton, which will have the double feature of being a non-elementary charged particle and at the same time an antiparticle. The experiment will be conducted in collaboration with the ASACUSA (Atomic Spectroscopy And Collisions Using Slow Antiprotons) group at the CERN AD (Antiproton Decelerator), making use of a slow antiproton beam.

1. Introduction

The Aharonov–Bohm (AB) effect is a quantum interference phenomenon showing that electromagnetic potentials can have observable consequences through the phase of a charged-particle wave function, even when the particle propagates in regions where the corresponding classical fields vanish and no Lorentz force acts on it. The idea was first formulated, in the form of a gedanken experimental scheme, by Ehrenberg and Siday in 1949 [1], and was later developed by Aharonov and Bohm in 1959, who emphasized its conceptual significance and its role in revealing the fundamental status of electromagnetic potentials in quantum physics [2].

In the standard magnetic AB configuration, typically realized in a two-slit interferometric geometry, a coherent charged-particle beam is split into two components that (using a classical language) propagate along different paths and are then recombined, while a confined magnetic flux is placed between the two paths (Figure 1). In the ideal case, the magnetic field is confined within the solenoid and therefore vanishes in the region accessible to the particles, whereas the vector potential A is non-zero. As a result, the relative phase of the two wave-function components is shifted by an amount determined by the magnetic flux $\mathsf{\Phi }$ enclosed by the interferometer, even though no classical force acts along either path.

Even before its experimental demonstration, the AB effect generated a lively debate between theoreticians and experimentalists, thoroughly described in ref. [3], and prompted several experimental investigations, culminating with the 1986 experiment of Tonomura and colleagues [4]. This clear demonstration of the effect, obtained by excluding the magnetic field from the electron classical path with the help of a superconducting film, was then supplemented by additional experiments devoted to the verification of the “zero force” condition. This verification was made through a careful study of the arrival times of the electrons in the Caprez experiment [5], with the direct demonstration that no electromagnetic forces were acting on the propagation of the particle wave-packet.

In parallel with these developments, theoretical studies highlighted the connection between the AB effect and the broader concept of geometric (Berry’s) phases in quantum mechanics [6]. The AB effect belongs to that subset of geometrical effects that are called topological, since they depend only the general topology of the system (and not on the details of the geometrical path classically followed by the particles). Moreover, in 2016 the effect was demonstrated in the original two-slit Ehrenberg–Siday configuration, as in Figure 1, by G. Pozzi and collaborators [7]; they made use of a uniformly magnetized bar with uniform cross-section, whose effect is the same as that of a straight, ideally infinite solenoid.

From an experimental point of view, in addition to the demonstration of the electric AB effect with electrons (see Appendix A), related interferometric studies have also been performed with neutrons. Since the neutron is electrically neutral, these experiments do not constitute direct realizations of the standard charge-based magnetic AB effect considered here. Rather, they involve AB-type phase phenomena associated with the neutron magnetic dipole moment, such as the Aharonov–Casher effect [8], where a magnetic moment encircles an electric charge distribution and the scalar AB analogue is produced by a pulsed magnetic field. For completeness, a brief discussion of these neutron experiments and of their conceptual distinction from the charged-particle AB effect is given in Appendix B.

Despite this extensive experimental and theoretical background, the magnetic AB effect for charged particles has so far been demonstrated for a single elementary particle of the Standard Model: the electron. It has never been studied with another charged particle, and in particular never with a composite charged hadron, such as the antiproton.

First of all, this would constitute a test of the wave-like nature of the antiproton, in addition to the well-known case of hadron scattering, low energy antiproton scattering [9] and protonium lifetime studies [10].

Secondly, such an experiment would confirm that the sign of a particle’s electromagnetic coupling does not alter the topological, field-free nature of quantum phase shifts, therefore studying in a novel way the gauge invariance for composite particles. This would be relevant for the CVC (Conserved Vector Current) concept and the Ward–Takahashi identity [11,12] describing how gauge symmetry restricts the quantum interaction of particles. The interest in interferometric studies with antiprotons is well recognized in the literature (see, e.g., Ref. [13]).

A further objective of the experiment is to investigate the dependence of the interference pattern on the direction of magnetization, and hence on the sign of the enclosed magnetic flux. A left-right asymmetry was recently observed in a high-statistics AB experiment with electrons [14], indicating a dependence on the sign of the magnetization. This effect has been suggested to originate from a breakdown of the paraxial approximation normally used in scattering theory, as proposed by Shelankov [15] and further discussed by Berry [16].

The detection of the AB effect relies on the observation of a phase-induced change in the interference pattern produced by a grating configuration. For this reason, the first step of the present study is the direct demonstration of quantum interference with antiprotons in the proposed grating configuration. This would represent the second observation of matter-wave interference with an antiparticle, after the positron interferometry experiment reported in 2019 [17], and would provide the necessary experimental basis for the subsequent measurement of the antiproton AB phase.

2. Experimental Strategy

In order to study the AB effect with the antiproton, the realization of quantum interference is a necessary precondition. Given the mass of the particle, it is important to use a low-energy beam, such as the one that is available at the CERN ASACUSA experiment. In ASACUSA, antiprotons coming from the CERN accelerator system are trapped and cooled in the MUSASHI Penning–Malmberg trap [18]. This allows the subsequent formation of a slow antiproton beam with a kinetic energy of about 250 eV.

This slow antiproton beam has already been used in another experimental study [19] and, given its low energy, has a de Broglie wavelength of

$${\lambda }_{dB}=h/mv\simeq 1.8pm$$

(1)

in the non-relativistic approximation. The beam has a velocity of 220,000 m/s and an intensity in the range of $2\times {10}^3$ particles/s, for an overall spill duration of 10 s. In these conditions the total statistical accumulation will be of $2\times {10}^4$ incoming antiprotons every 130 s, which is the repetition rate of the CERN Antiproton Decelerator, which supplies the ASACUSA setup [20].

The incoming antiprotons will be studied in a 1 m long interferometer, schematically shown in Figure 2. Given the beam velocity, the transit time will be about 5 $\mathsf{\mu }$s and the mean particle arrival rate is around 100 $\mathsf{\mu }$s⁻¹; therefore, one can safely conclude that the experiment will work in single-particle mode. This is another novelty of this approach with respect to the work done with electrons.

Detecting with good resolution an antiproton interference pattern is a prerequisite for the demonstration of the AB effect. For this reason, as a first step, we propose a Fraunhofer diffraction experiment according to the conceptual scheme presented in Figure 2.

Given the de Broglie wavelength under consideration, a 100 nm grating structure would diffract at 18 $\mathsf{\mu }$m after a 1 m flight length. The detection therefore appears feasible with a very-high-resolution detector. For this reason, we propose the use of emulsion detectors, to be used as integrators of the diffraction signal. Emulsion films have a superior space resolution, in our case better than 1 $\mathsf{\mu }$m [21,22], and they can detect the impact point of antiprotons by reconstructing the relevant annihilation pions.

While traveling through the interferometer, the antiproton will be subjected to the Earth’s magnetic field, which requires the presence of a mu-metal shielding. The deflection of a 250 eV antiproton traveling 1 m in the Earth magnetic field is of the order of 1 cm, which is reduced to only 10 $\mathsf{\mu }$m by the kind of mu-metal used in [17]. In principle, a static magnetic field does not compromise the experiment, since it only produces an overall shift of the interference pattern. However, its space variations and fluctuations in time can be problematic. For this reason we assume a shielding factor of ${10}^3$, as achieved with the shielding employed in [17].

Following the scheme of Figure 2, a first direct demonstration of antiproton interferometry can be obtained by a Fraunhofer interference–diffraction experiment with material grating. In order to substantiate this idea, a simulation was made assuming a material grating with $N=1000$ slits, each having an aperture of 200 nm and an inter-slit distance of 10 $\mathsf{\mu }$m. A 250 eV antiproton beam impinging on the grating would produce an interference pattern on the emulsion (after 1 m flight path), such as the one shown in Figure 3 (left). The simulation assumes 10⁶ detected particles and includes effects of energy and space resolution of the emulsion.

Given the geometric configuration, and the open fraction of the grating (about 2%), an amount of $5\times {10}^7$ impinging antiprotons is necessary to produce the pattern in the figure, which is achievable with 3–4 days of data taking at the Antiproton Decelerator.

Once antiproton interferometry has been demonstrated, the next logical step is the demonstration of the AB effect. This will be achieved following the conceptual scheme presented in Figure 4, where the classical double slit configuration as well as an anti-configuration (small obstacle instead of a grating with slits) are shown.

To properly simulate the AB effect in our proposed configuration, we have modeled the wave mechanical free-space propagation of the antiproton wavefront from the (anti)slit(s) by solving analytically the Fresnel–Kirchhoff integral in the paraxial approximation all the way from Fresnel to Fraunhofer, following the standard image formation scheme universally adopted in optics [23].

Under these conditions, the interference–diffraction of the 250 eV antiprotons would produce a theoretical pattern on the detector, as shown in Figure 5, where the difference between the two cases consists in a single unit of magnetic flux. The pattern of the minima and maxima are within the range of sensitivity of the emulsion detector even after 50 cm of propagation.

In a configuration like this, let us consider the following (realistic) parameters:

• Slit opening $a=250$ nm;
• Inter-slit distance $d=1$$\mathsf{\mu }$m for the double slit case;
• Grating-detector distance of 1 m.

The working principle of the emulsion detector is based on the measurement of the antiproton impact point (see Figure 5) through the reconstruction of the annihilation tracks produced by the outgoing pions. This technique relies on tracing, within the emulsion, the tracks produced by the annihilation of the antiproton. The 2–3 pions typically produced leave distinct tracks that can be reconstructed as discussed in [21,22]. According to this reconstruction technique, the expected uncertainty in the reconstructed antiproton impact position is about one micron.

However, the use of a classical two-slit configuration with material gratings has the significant disadvantage of reducing the incoming antiproton flux. In addition, antiprotons stopped by materials generate pions, and that background can be important.

In order to overcome these problems, we will use a configuration like the one shown on the right-hand side of Figure 4. The interferometric result of a configuration of this type is shown in the lower part of Figure 6. By contrast, the upper part of Figure 6 illustrates the case with no magnetic flux present.

As the magnetic element, we will use the bars developed for the AB experiment with electrons, as in [7,24]. Therefore, we propose a configuration like the one outlined in the right part of Figure 4, producing the interference pattern displayed in Figure 6.

The study of the magnetic AB effect will be conducted by varying the angle of inclination of the magnetized bar, so that the amount of magnetic flux enclosed by the path of the particles can be varied continuously.

The specific signature of the AB effect will be the asymmetry between the two cases of currents described in [14] and shown in Figure 7, corresponding to a rotation of the magnetic bar by 180 degrees. In addition, all the intermediate rotation angles will be available, including 90 degrees, which produces the case of zero magnetic flux and provides a normalization of the effect.

3. Conclusions

We propose the exploration of antiproton interferometry in conjunction with the AB effect, by making use of the slow antiproton beam available at the ASACUSA setup in the CERN Antiproton Decelerator.

The experiment, divided into a first demonstration of antiproton interferometry and a following demonstration of the AB effect, is intended to study this quantum–mechanical phenomenon with a non-elementary antiparticle.