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 field (E, B), despite being confined to a region in which both the magnetic field B and electric field E are zero. The underlying mechanism is the coupling of the electromagnetic potential with the complex phase of a charged particle's wavefunction, 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, and similar effects were later published by Yakir Aharonov and David Bohm 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 Aharanov's subsequent 1961 paper. A general review can be found in Peshkin and Tonomura (1989).

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 (Coulomb's electrostatic potential, entirely analogous, mathematically, to the classical gravitational potential) and a stationary magnetic field as the curl of a vector potential (then a new concept - the idea of a scalar potential was already well accepted by analogy with the gravitational 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 (unlike fields) 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 (Maxwell's) 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:

1. whether potentials are "physical" or a just convenient tool for calculating force fields;
2. whether action principles are fundamental;
3. the principle of locality.

Because of reasons like these, the Aharonov–Bohm effect was chosen by the New Scientist magazine as one of the "seven wonders of the quantum world".

Potentials vs. fields

The Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, Φ and A, whereas previously it was possible to argue that only the electromagnetic fields, E and B, were physical and that the electromagnetic potentials, Φ and A, were purely mathematical constructs (Φ and A being non-unique, in addition to not appearing in the Lorentz Force formula). The non-uniqueness of the electromagnetic potentials is a manifestation of electromagnetic gauge freedom, with the electric and magnetic fields and forces being gauge invariant and therefore directly observable (the E and B fields do appear in the Lorentz force formula).

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 E and B, and he wished later in life he had been taught to think in terms of the A field instead, as this would be more fundamental. In Feynman's path-integral view of dynamics, the A 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, A, must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the E and B fields alone, or using the A 4-potential alone. But when using just the E and B fields, the effect depends on the field values in a region from which the test particle is excluded, not only classically but also quantum mechanically. In contrast, when using just the electromagnetic four-potential A only the effect depends on the A potential in the region where the test particle is allowed. Therefore we can either abandon the principle of locality (which most physicists are reluctant to do) or we are forced to accept that the electromagnetic 4-potential - composed of Φ and A - offers a more complete description of electromagnetism than the electric and magnetic fields can. 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.  The E and B fields can be derived from the 4-potential, but the 4-potential, by gauge freedom, cannot be derived from the E and B fields.

Magnetic solenoid effect

The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential A forms part.

Electromagnetic theory implies that a particle with electric charge q travelling along some path P in a region with zero magnetic field B, but non-zero A (by $\mathbf {B} =0=\nabla \times \mathbf {A}$), acquires a phase shift $\varphi$, given in SI units by

$\varphi ={\frac {q}{\hbar }}\int _{P}\mathbf {A} \cdot d\mathbf {x} ,$

Therefore particles, with the same start and end points, but travelling along two different routes will acquire a phase difference Δφ determined by the magnetic flux Φ_B through the area between the paths (via Stokes' theorem and $\nabla \times \mathbf {A} =\mathbf {B}$), and given by:

$\Delta \varphi ={\frac {q\Phi _{B}}{\hbar }}.$

Schematic of double-slit experiment in which 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 turned on in the cylindrical solenoid.

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 (or equivalent). An ideal solenoid (i.e. infinitely long and with a perfectly uniform current distribution) encloses a magnetic field B, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron) passing outside experiences no magnetic field B. However, there is a (curl-free) vector potential A 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 Δφ around a closed loop must be an integer multiple of 2π (with the charge for the electron Cooper pairs), and thus the flux Φ must be a multiple of h/2e. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenological model.

The magnetic Aharonov–Bohm effect was experimentally confirmed by Osakabe et al. (1986), following much earlier work summarized in Olariu and Popèscu (1984). Its scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004).

Monopoles and Dirac strings

The magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole can be accommodated by the existing magnetic source-free Maxwell's equations if both electric and magnetic charges are quantized.

A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. The Dirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization. That is, $2{\frac {q_{\text{e}}q_{\text{m}}}{\hbar c}}$ must be an integer (in cgs units) for any electric charge q_e and magnetic charge q_m.

Like the electromagnetic potential A the Dirac string is not gauge invariant (it moves around with fixed endpoints under a gauge transformation) and so is also not directly measurable.

Electric effect

Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov–Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the Schrödinger equation, the phase of an eigenfunction with energy E goes as $\exp(-iEt/\hbar )$. The energy, however, will depend upon the electrostatic potential V for a particle with charge q. In particular, for a region with constant potential V (zero field), the electric potential energy qV is simply added to E, resulting in a phase shift:

$\Delta \phi =-{\frac {qVt}{\hbar }},$

where t is the time spent in the potential.

The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a varying potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a bias voltage V relating the potentials of the two halves of the ring. This situation results in an Aharonov–Bohm phase shift as above, and was observed experimentally in 1998.

Aharonov–Bohm nano rings

Nano rings were created by accident while intending to make quantum dots. They have interesting optical properties associated with excitons and the Aharonov–Bohm effect. Application of these rings used as light capacitors or buffers includes photonic computing and communications technology. Analysis and measurement of geometric phases in mesoscopic rings is ongoing.

Mathematical interpretation

In the terms of modern differential geometry, the Aharonov–Bohm effect can be understood to be the monodromy of a flat complex line bundle. The U(1)-connection on this line bundle is given by the electromagnetic four-potential A as $\nabla =d+iA\,,$ where d means partial derivation in the Minkowski space $\mathbb {M} ^{4}$. The curvature form of the connection, $\mathbf {F} =d\mathbf {A}$, is the electromagnetic field strength, where $\mathbf {A}$ is the 1-form corresponding to the four-potential. The holonomy of the connection, $e^{i\int _{\gamma }\mathbf {A} }$ around a closed loop $\gamma$ is, as a consequence of Stokes' theorem, determined by the magnetic flux through a surface bounded by the loop. This description is general and works inside as well as outside the conductor. Outside of the conducting tube, which is for example a longitudinally magnetized infinite metallic thread, the field strength is $\mathbf {F} =0$ ; in other words outside the thread the connection is flat, and the holonomy of a loop contained in the field-free region depends only on the winding number around the tube and is, by definition, the monodromy of the flat connection.

In any simply connected region outside of the tube we can find a gauge transformation (acting on wave functions and connections) that gauges away the vector potential. However, if the monodromy is non trivial, there is no such gauge transformation for the whole outside region. If we want to ignore the physics inside the conductor and only describe the physics in the outside region, it becomes natural to mathematically describe the quantum electron by a section in a complex line bundle with an "external" connection $\nabla$ rather than an external EM field $\mathbf {F}$ (by incorporating local gauge transformations we have already acknowledged that quantum mechanics defines the notion of a (locally) flat wavefunction (zero momentum density) but not that of unit wavefunction). The Schrödinger equation readily generalizes to this situation. In fact for the Aharonov–Bohm effect we can work in two simply connected regions with cuts that pass from the tube towards or away from the detection screen. In each of these regions we have to solve the ordinary free Schrödinger equations but in passing from one region to the other, in only one of the two connected components of the intersection (effectively in only one of the slits) we pick up a monodromy factor $e^{i\alpha }$, which results in a shift in the interference pattern.

Effects with similar mathematical interpretation can be found in other fields. For example, in classical statistical physics, quantization of a molecular motor motion in a stochastic environment can be interpreted as an Aharonov–Bohm effect induced by a gauge field acting in the space of control parameters.

See also

• Geometric phase
• Hannay angle
• Wannier function
• Berry phase
• Wilson loop
• Winding number
• Aharonov-Casher effect

External links

• A video explaining the use of the Aharonov-Bohm effect in nano-rings.