Aharonov-Bohm effect

The Aharonov-Bohm effect, sometimes called the Ehrenberg-Siday-Aharonov-Bohm effect, is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded. Werner Ehrenberg and R.E. Siday first predicted the effect in 1949 [cite journal |first=W. |last=Ehrenberg |coauthors=Siday, R. E.|title=The Refractive Index in Electron Optics and the Principles of Dynamics" |journal=Proc. Phys. Soc. |volume=B62 |pages=8–21 |year=1949 |doi=10.1088/0370-1301/62/1/303] , and similar effects were later rediscovered by Aharonov and Bohm in 1959 [cite journal |first=Y. |last=Aharonov |coauthors=Bohm D.|title=Significance of electromagnetic potentials in quantum theory
journal=Phys. Rev. |volume=115 |pages=485–491 |year=1959 |doi=10.1103/PhysRev.115.485] . (After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited [ [http://www.fdavidpeat.com/bibliography/books/infinite.htm Peat, F. David] , "Infinite Potential: The Life and Times of David Bohm" (Addison-Wesley: Reading, MA, 1997). ISBN 0-201-40635-7.] in Bohm and Aharanov's subsequent 1961 paper [cite journal |first=Y. |last=Aharonov |coauthors=Bohm D. |title=Further Considerations on Electromagnetic Potentials in the Quantum Theory
journal=Phys. Rev. |volume=123 |pages=1511–1524 |year=1961 |doi=10.1103/PhysRev.123.1511] .) Such effects are predicted to arise from both magnetic fields and electric fields, but the magnetic version has been easier to observe. In general, the profound consequence of Aharonov-Bohm effects is that knowledge of the classical electromagnetic field acting "locally" on a particle is not sufficient to predict its quantum-mechanical behavior.

The most commonly described case, sometimes called the Aharonov-Bohm solenoid effect, is 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 zero in the region through which the particle passes. This phase shift has been observed experimentally by its effect on interference fringes. (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, and this has also seen experimental confirmation. A separate "molecular" Aharonov-Bohm effect was proposed for nuclear motion in multiply-connected regions, but this has been argued to be essentially different, depending only on local quantities along the nuclear path (Sjöqvist, 2002 [Sjöqvist, E. "Locality and topology in the molecular Aharonov-Bohm effect," "Phys. Rev. Lett." 89 (21), 210401/1–3 (2002)] ).

A general review can be found in Peshkin and Tonomura (1989) [ [http://www.phy.anl.gov/theory/staff/mp.html Peshkin, M.] and Tonomura, A., "The Aharonov-Bohm effect" (Springer-Verlag: Berlin, 1989). ISBN 3-540-51567-4] .

Magnetic Aharonov-Bohm 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 vector potential A. This implies that a particle with electric charge "q" travelling along some path P in a region with zero magnetic field ($mathbf\{B\}\; =\; 0\; =\; abla\; imes\; mathbf\{A\}$) must acquire a phase $varphi$, given in SI units by

:$varphi\; =\; frac\{q\}\{hbar\}\; int\_P\; mathbf\{A\}\; cdot\; dmathbf\{x\},$

with a phase difference $Deltavarphi$ between any two paths with the same endpoints therefore determined by the magnetic flux Φ through the area between the paths (via Stokes' theorem and $abla\; imes\; mathbf\{A\}\; =\; mathbf\{B\}$), and given by:

:$Deltavarphi\; =\; frac\{qPhi\}\{hbar\}.$

This phase difference can be observed by placing a solenoid between the slits of a double-slit experiment (or equivalent). An ideal solenoid 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 classical effect. However, there is a (curl-free) 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. 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 is 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 "q"=2"e" for the electron Cooper pairs), and thus the flux Φ must be a multiple of "h"/2"e". The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by London (1948) [London, F. "On the problem of the molecular theory of superconductivity," "Phys. Rev." 74, 562–573 (1948)] using a phenomenological model.

The magnetic Aharonov-Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole necessarily implies that both electric and magnetic charges are quantized. A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as an infinitely long Dirac string of infinitesimal diameter that contains the equivalent of all of the 4π"g" flux from a monopole "charge" "g". 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: $2qg/chbar$ must be an integer (in cgs units) for any electric charge "q" and magnetic charge "g".

The magnetic Aharonov-Bohm effect was experimentally confirmed by Osakabe et al. (1986) [Osakabe, N., T. Matsuda, T. Kawasaki, J. Endo, A. Tonomura, S. Yano, and H. Yamada, "Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor." "Phys Rev A." 34(2): 815-822 (1986).] , following much earlier work summarized in Olariu and Popèscu (1984) [Olariu, S. and I. Iovitzu Popèscu, "The quantum effects of electromagnetic fluxes," "Rev. Mod. Phys." 57, 339–436 (1985)] . Its scope and application continues to expand. Webb et al. (1985) [Webb, R., S. Washburn, C. Umbach, and R. Laibowitz. "Phys. Rev. Lett." 54, 2696 (1985)] demonstrated Aharonov-Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) [Schwarzschild, B. "Currents in Normal-Metal Rings Exhibit Aharonov-Bohm Effect." "Phys. Today" 39, 17–20, Jan. 1986] and Imry & Webb (1989) [Imry, Y. and R. A. Webb, "Quantum Interference and the Aharonov-Bohm Effect," "Scientific American", 260(4), April 1989] . Bachtold et al. (1999) [Bachtold, A., C. Strunk, J. P. Salvetat, J. M. Bonard, L. Forro, T. Nussbaumer and [http://pages.unibas.ch/phys-meso/ C. Schonenberger] , “Aharonov-Bohm oscillations in carbon nanotubes”, "Nature" 397, 673 (1999)] detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004) [Kong, J., L. Kouwenhoven, and C. Dekker, "Quantum change for nanotubes", " [http://physicsweb.org/articles/world/17/7/3/1 Physics Web] " (July 2004)] .

Electric Aharonov-Bohm 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:

:$Deltaphi\; =\; -frac\{qVt\}\{hbar\}\; ,$

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 [van Oudenaarden, A., M. H. Devoret, Yu. V. Nazarov, and J. E. Mooij, "Magneto-electric Aharonov-Bohm effect in metal rings," "Nature" 391, 768–770 (1998)] .

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$abla\; =\; d\; +\; i\; A,,$ where "d" means partial derivation in the Minkowski space $mathbb\; M^4$. The curvature form of the connection, $mathbf\; F=dmathbf\; A$, 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 $abla$ 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 $e^\{ialpha\}$, which results in a shift in the interference pattern.

See also a related effect, the Berry phase.

References