As far as we know, Birkbeck College in London was the first public university in the world to open its doors to women, in 1830. Its physics Department rose to fame in the 1930s with the brief passage of Patrick Blackett at the Chair of physics, which he then handed over to J.D. Bernal in 1937. Bernal practically invented the field of protein crystallography, and after 1945 he established the Birkbeck’s Biomolecular Research Laboratory where also Rosalind Franklin did pioneering works on x-ray structures. However, Birkbeck College deserves a little extra fame also in the 1960s, for welcoming David Bohm as Chair of theoretical physics, and Werner Ehrenberg, who returned after a long leave, as Chair of experimental physics and head of the Department. Bohm and Ehrenberg are the key actors (plus one) in today’s story.

Bohm had been fired from Princeton after refusing to testify against Oppenheimer. The “black list” of the McCarthy committee had him banned for more than ten years, because of which he fled first to Brazil and then to Israel, before finally landing in England. Two years before joining Birkbeck College he was in Bristol, where he carried out a seminal work with his young student Yakir Aharonov. Their two papers in the Physical Review of 1959 and 1961 are today considered the first and cleanest example of quantum gauge phenomena. Their discovery, named Aharonov-Bohm effect (or AB), is about quantum mechanical effects that arise when charged particles pass through regions where the electromagnetic potentials 𝜙 and are nonzero, whereas the physical fields and are zero. OK, maybe this explanation does not sound quite like one… Let’s try better.

Going back to the already touched subject of “what do we really teach to our students”, I am not quite sure that in our teachings of classical physics we always make the following point sufficiently clear: the electromagnetic potentials 𝜙 and are introduced just as a practical means of obtaining and B, but they have no physical significance of their own. Indeed, they serve only as convenient tricks to obtain the physical fields and B, which are the only measurable quantities responsible for the electrodynamic forces. A battery in a device gives you a difference of potential over its finite size, the absolute value of its potential is nonsense. In fact, the potentials are not even unique: electric and magnetic fields are obtained as derivatives of the potential, so that adding to the potential a term that disappears after derivation, will not affect and B. For example, with = 𝛻 x , adding to A the gradient 𝛻𝜒 of a scalar 𝜒, leaves untouched, since 𝛻 x 𝛻 = 0. Such modifications are the gauge transformations, a tool of paramount importance in modern physics. Differently from global symmetries, such as conservation of energy, charge, angular momentum, a gauge symmetry is local in space and/or time, and essentially tells you that there is some redundancy in the way you are writing your theory. That should be obvious, by observing that and B have 6 components, while 𝜙 and A have only 4, therefore something remains undetermined and free to choose. (Notably, both Maxwell in regard to the electromagnetism, and Hilbert in regard to general relativity, had noticed such property, but failed to recognize its importance; it was Hermann Weyl who first saw the gauge transformation as a local symmetry of the physical theories.)

When we move to quantum mechanics, gauge transformations reappear as a phase 𝜙 that multiplies the wavefunction by exp(i𝜙) (that is a U(1) symmetry, or a rotation on the unit circle). Changing the phase of the wave function has no measurable consequences, since in either case we get the same expectation value for any observable operator. So, after an arbitrary phase change the wavefunction represents always the same physical state. It’s no chance that I used here the same symbol for the scalar potential and the phase: in fact, if we consider even the most elementary case, a free-electron wave packet traveling in a region of constant potential energy, = – e𝜙, it acquires a phase e𝜙t/ℏ. This follows simply from the hamiltonian H= p2/2m– e𝜙, in which case the free-electron eigenfunctions are each multiplied by exp(ie𝜙t/ℏ). But, as said, this will have no measurable consequences, the squared modulus of the wavefunction will take care of that.

Then, Bohm and Aharonov asked the following question: what happens to the wave packet if we split it in two, let each half pass in two regions of different, constant voltage  𝜙1 and 𝜙2 for some flight time T, then move the two halves again in a region of zero voltage and let them recombine? The inescapable conclusion was that the two half-packets acquire a phase difference of (𝜙1𝜙2)T, and a completely similar result would be obtained with the vector potential for the magnetic field. This means that, at odds with classical mechanics, electromagnetic potentials do have observable consequences in quantum mechanics: electrons move in a region where and B are zero, so the Lorentz force is zero, yet they will show quantum interference because of an effect coming from the potential… a purely mathematical trick that has physical, measurable effects! (I note that something similar seems to be happening recently with complex numbers, maybe a good subject for a future letter.)

Well, “measurable” may not be the easiest definition. Experiments of the kind ideally suggested by Aharonov and Bohm, playing with the vector potential, were performed since the 1960s, but were subject to hard criticism. The first unambiguous results were obtained only around 1986, and since then just a handful of other proofs have been demonstrated. Before being hallowed as one truly defining property of the quantum world, the AB-effect spurred controversies, because it appears to be at the same time gauge invariant and non-gauge invariant, or at the same time local and non-local (the interference pattern appears to connect two states physically separated and, like in the EPR paradox, it seems to require instantaneous exchange of information). The great Victor Weisskopf wrote, back in 1961: “The first reaction to this work is that it is wrong; the second is that it is obvious.” Most of these debates seem resolved today, and examples of AB-like phase are described in various fields of physics, from cosmology to lasers and molecules, although experiments remain difficult. On the other hand, as of today, no experiment could ever prove the time-dependent electric-AB effect with the scalar potential 𝜙. In 1995 at Argonne, I followed a lecture by Murray Peshkin (already retired but still very active) in which he described how the latter implies time-reversed causation: it can be proved from the Schrödinger equation that the electric-AB effect should arise on charges traversing a region where an electric field existed previously.

However, these are not the only puzzles that still surround the AB effect. Basil Hiley was a student at Birkbeck in 1961, when he showed professor Ehrenberg the first paper by Aharonov and Bohm. He reports that Ehrenberg looked at the article, and replied in his strong German accent: “Ach Hiley, zis AB effect that you are discussing, izit the one that Siday and I discovered?” As it turns out, Ray Siday had joined Ehrenberg’s laboratory in 1946 to work on beta-ray spectrometry, and while designing electromagnetic focusing he realized that, since the electron path depends on the applied potential A, and not on the magnetic field B, the refractive index is not gauge invariant. He started discussing with Ehrenberg the specific case when B would be confined in a region such that the electrons do not pass through it, and they eventually published an obscure paper titled “The refractive index in electron optics and the principles of dynamics” (Proc. Phys. Soc. B 62 (1949) 8), which went unnoticed. Ten years before AB, they had clearly identified that wave-optical effects can arise from an isolated magnetic field, even though the rays travel in a field-free region. And in fact, Aharonov and Bohm recognized Ehrenberg and Sidai’s priority in their second paper of 1961, however misleadingly calling it a ‘semi-classical’ treatment.

But… Ehrenberg and Sidai could have not been the first ones either! Gottfried Möllenstedt, who performed some of the earliest experiments on the AB effect, once wrote a chapter on the electron biprism in a book of Introduction to Electron Holography (E. Völk et al. eds., Kluwer, New York 1999). There he quotes a work by Walter Franz, that he had heard as a student at a conference in Danzig, 1939, just a few weeks before the beginning of WWII. According to his summary of the lecture Elektroneninterferenzen im Magnetfeld, Franz had considered a magnetic field limited to some region, and a charge going from a to b along two different paths, defining a closed loop around the magnetic field; and showed that the phase difference 𝛥𝜙 between the two paths equals the loop integral of the charge momentum p=mv+eA. Using Stokes’ theorem, the loop integral of A can be turned into the surface integral of 𝛻 x A that is, actually, the magnetic flux. Hence, Möllenstedt concluded that Franz had already shown in 1939 how the phase difference between electron rays depends on the magnetic flux included between them, even if the rays do not run through the magnetic field.

So, was the AB effect discovered in 1959, or 1949, or 1939…? Anyway, I wanted to propose you this brief recollection of history and gossips about the AB effect, because I stumbled in a paper that just appeared in Science 375, p.226 “Observation of a gravitational Aharonov-Bohm effect”. The authors are Mark Kasevich and his group at Stanford, who had already published a prequel of their work in Nature (“Quantum superposition at the half-metre scale”, 528 (2015) 530). They use matter-wave interferometers, in which a single atom interferes with itself, to probe quantum superposition over macroscopic time and length scales. To reach this difficult objective, hampered by dephasing and decoherence over the many centimeters and seconds of the atom path, they constructed an equivalent of the classical Mach-Zender interferometer, in which a photon beam is split in two, and rejoined after being reflected onto a second beam splitter. The apparatus is a vertical, 10-m long vacuum tube, with laser beams at each end that provide the equivalent of optical beam splitters and mirrors. Cold atoms from a Bose-Einstein condensate enter from the lower end and are “launched” up in a parabolic, free fall trajectory. This is called an “atomic fountain”. Along their path, atoms are hit by laser pulses at times 0, T, and 2T: the first shot splits each atom in two quantum states, that now move up in the tube along different trajectories like in the two “arms” of the interferometer, at slightly different speed and thus reaching different heights; the second shot reflects the trajectories at the top; the last shot recombines the two states at the bottom, which now can be “read” in the interference pattern.

In their most recent experiments published in Science, Kasevich placed a large mass of 1.25 kg of pure tungsten at the top end, to create a non-uniform gravity field. The key point is that both atom paths reach the uppermost point of the trajectory, at which they stall and change direction, like a ball thrown in the air reaching its highest point. At the stall point the kinetic energy equals the gravity, therefore the atom experiences zero total force. However, the higher path (originated by the larger momentum transfer of the first laser pulse) gets closer to the large tungsten mass, therefore it “sees” a different value of the gravitational potential compared to the other one. This is just the gravitational equivalent of the AB-effect: a mass (the atom) moving in a region of zero force and non-zero potential displays quantum interference, that is a phase difference induced only by the gravitational potential. But, while the electromagnetic AB phase is measured at the scale of electron clouds, here the quantum superposition is observed over lengths of meters and times of seconds, achieved with an astonishing sensitivity of one part in 1015.

This is actually a superfine measure of the relativistic curvature of space-time, which is linked to the gradient of gravity around the tungsten mass. It is like placing two clocks attached to each of the split atom paths, and measuring the difference between their two proper times: an atomic analog of the thought-experiment of the two twins, one travelling around a star and the other remaining on the Earth. Such an experiment marks a major step in the emerging field of gravitational quantum mechanics.

Gravitational quantum paths

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