Aharonov–Bohm effect
The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum-mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential, despite being confined to a region in which both the magnetic field and electric field are zero. The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wave function, and the Aharonov–Bohm effect is accordingly illustrated by interference experiments.
The most commonly described case, sometimes called the Aharonov–Bohm solenoid effect, takes place when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being negligible in the region through which the particle passes and the particle's wavefunction being negligible inside the solenoid. This phase shift has been observed experimentally. There are also magnetic Aharonov–Bohm effects on bound energies and scattering cross sections, but these cases have not been experimentally tested. An electric Aharonov–Bohm phenomenon was also predicted, in which a charged particle is affected by regions with different electrical potentials but zero electric field, but this has no experimental confirmation yet. A separate "molecular" Aharonov–Bohm effect was proposed for nuclear motion in multiply connected regions, but this has been argued to be a different kind of geometric phase as it is "neither nonlocal nor topological", depending only on local quantities along the nuclear path.
Werner Ehrenberg and Raymond E. Siday first predicted the effect in 1949. Yakir Aharonov and David Bohm published their analysis in 1959. After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited in Bohm and Aharonov's subsequent 1961 paper. The effect was confirmed experimentally, with a very large error, while Bohm was still alive. By the time the error was down to a respectable value, Bohm had died.
Significance
In the 18th and 19th centuries, physics was dominated by Newtonian dynamics, with its emphasis on forces. Electromagnetic phenomena were elucidated by a series of experiments involving the measurement of forces between charges, currents and magnets in various configurations. Eventually, a description arose according to which charges, currents and magnets acted as local sources of propagating force fields, which then acted on other charges and currents locally through the Lorentz force law. In this framework, because one of the observed properties of the electric field was that it was irrotational, and one of the observed properties of the magnetic field was that it was divergenceless, it was possible to express an electrostatic field as the gradient of a scalar potential and a stationary magnetic field as the curl of a vector potential. The language of potentials generalised seamlessly to the fully dynamic case but, since all physical effects were describable in terms of the fields which were the derivatives of the potentials, potentials were not uniquely determined by physical effects: potentials were only defined up to an arbitrary additive constant electrostatic potential and an irrotational stationary magnetic vector potential.The Aharonov–Bohm effect is important conceptually because it bears on three issues apparent in the recasting of classical electromagnetic theory as a gauge theory, which before the advent of quantum mechanics could be argued to be a mathematical reformulation with no physical consequences. The Aharonov–Bohm thought experiments and their experimental realization imply that the issues were not just philosophical.
The three issues are:
- whether potentials are "physical" or just a convenient tool for calculating force fields;
- whether action principles are fundamental;
- the principle of locality.
Chen-Ning Yang considered the Aharonov–Bohm effect to be the only direct experimental proof of the gauge principle. The philosophical importance is that the magnetic four-potential over describes the physics, as all observable phenomena remain unchanged after a gauge transformation. Conversely, the Maxwell fields under describe the physics, as they do not predict the Aharonov-Bohm effect. Moreover, as predicted by the gauge principle, the quantities that remain invariant under gauge transforms are precisely the physically observable phenomena.
Potentials vs. fields
It is generally argued that the Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, in quantum mechanics. Classically it was possible to argue that only the electromagnetic fields are physical, while the electromagnetic potentials are purely mathematical constructs, that due to gauge freedom are not even unique for a given electromagnetic field.However, Vaidman has challenged this interpretation by showing that the Aharonov–Bohm effect can be explained without the use of potentials so long as one gives a full quantum mechanical treatment to the source charges that produce the electromagnetic field. According to this view, the potential in quantum mechanics is just as physical as it was classically. Aharonov, Cohen, and Rohrlich responded that the effect may be due to a local gauge potential or due to non-local gauge-invariant fields.
Two papers published in the journal Physical Review A in 2017 have demonstrated a quantum mechanical solution for the system. Their analysis shows that the phase shift can be viewed as generated by a solenoid's vector potential acting on the electron or the electron's vector potential acting on the solenoid or the electron and solenoid currents acting on the quantized vector potential.
Global action vs. local forces
Similarly, the Aharonov–Bohm effect illustrates that the Lagrangian approach to dynamics, based on energies, is not just a computational aid to the Newtonian approach, based on forces. Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, and potential energies must be used instead. In fact Richard Feynman complained that he had been taught electromagnetism from the perspective of electromagnetic fields, and he wished later in life he had been taught to think in terms of the electromagnetic potential instead, as this would be more fundamental. In Feynman's path-integral view of dynamics, the potential field directly changes the phase of an electron wave function, and it is these changes in phase that lead to measurable quantities.Locality of electromagnetic effects
The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential,, must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, or using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded. In contrast, when using just the four-potential, the effect only depends on the potential in the region where the test particle is allowed. Therefore, one must either abandon the principle of locality, which most physicists are reluctant to do, or accept that the electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields can. On the other hand, the Aharonov–Bohm effect is crucially quantum mechanical; quantum mechanics is well known to feature non-local effects, and Vaidman has argued that this is just a non-local quantum effect in a different form.In classical electromagnetism the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials Φ and A are seen as being more fundamental. Despite this, all observable effects end up being expressible in terms of the electromagnetic fields, E and B. This is interesting because, while you can calculate the electromagnetic field from the four-potential, due to gauge freedom the reverse is not true.
Magnetic solenoid effect
The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics must be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential forms part.Electromagnetic theory implies that a particle with electric charge traveling along some path in a region with zero magnetic field, but non-zero , acquires a phase shift, given in SI units by
Therefore, particles, with the same start and end points, but traveling along two different routes will acquire a phase difference determined by the magnetic flux through the area between the paths, and given by:
Image:Aharonov-Bohm effect.svg|thumbnail|right|250px|Schematic of double-slit experiment in which the Aharonov–Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field B is changed in the whisker. The direction of the B field is outward from the figure; the inward returning flux is not shown, but is outside the electron paths. The arrow shows the direction of the A field which extends outside the boxed region even though the B field does not.
In quantum mechanics the same particle can travel between two points by a variety of paths. Therefore, this phase difference can be observed by placing a solenoid between the slits of a double-slit experiment. An ideal solenoid encloses a magnetic field, but does not produce any magnetic field outside of its cylinder, and thus the charged particle passing outside experiences no magnetic field. However, there is a vector potential outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on or off. This corresponds to an observable shift of the interference fringes on the observation plane.
The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wave function must be single valued: its phase difference around a closed loop must be an integer multiple of , and thus the flux must be a multiple of. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenological model.
The first claimed experimental confirmation was by Robert G. Chambers in 1960, in an electron interferometer with a magnetic field produced by a thin iron whisker, and other early work is summarized in Olariu and Popèscu. However, subsequent authors questioned the validity of several of these early results because the electrons may not have been completely shielded from the magnetic fields. An early experiment in which an unambiguous Aharonov–Bohm effect was observed by completely excluding the magnetic field from the electron path was performed by Tonomura et al. in 1986. The effect's scope and application continues to expand. Webb et al. demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild and Imry & Webb. Bachtold et al. detected the effect in carbon nanotubes; for a discussion, see Kong et al..