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9-1-2009

The Aharonov-Bohm Effects: Variations on aSubtle ThemeHerman BatelaanUniversity of Nebraska - Lincoln, hbatelaan2@unl.edu

Akira TonomuraHitachi, akira.tonomura.zk@hitachi.com

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The Aharonov-Bohmeffects: Variations ona subtle themeHerman Batelaan and Akira Tonomura

The notion, introduced 50 years ago, that electrons could be affected by electromagnetic potentialswithout coming in contact with actual force fields was received with a skepticism that has spawneda flourishing of experimental tests and expansions of the original idea.

Herman Batelaan is a professor of physics at the University of Nebraska-Lincoln. Akira Tonomura is a feilow at Hitachi Ltd and a groupdirector at the RIKEN Frontier Research System in Saitama, Japan. He is also a principal investigator at the Okinawa Institute of Science andTechnology.

Quantum phenomena that were once thought to limitmeasurement capabilities are now being harnessed to en-hance them and to improve the sensitivity of nanometer-sizedevices. Persistent questioning and probing of quantum phe-nomena has yielded many such advances. However, theAharonov-Bohm effect, a modern cornerstone of quantummechanics, is not yet a good example of that kind of progress.The AB effect was already implicit in the 1926 Schrodingerequation, but it would be another three decades before theo-rists Yakir Aharonov and David Bhm pointed it out.' Andto this day, the investigation and exploitation of the AB effectremain far from finished.

Our discussion of the AB ef-fect begins with a seemingly in-nocent question: What happensto an electron as it passes by aninfinitely long ideal solenoid?One might expect that the elec-

Figure 1. Two electrons passingby a long current-carrying sole-noid on opposite sides (along theblue trajectories) illustrate themagnetic Aharonov-Bohm effect.Although the solenoid's magneticfield (purple flux lines) is almostentirely restricted to the coil's in-terior, the vector potential A out-side (tangential to the greencoaxial circles) fallsoff only as 1/rwith distance r from the axis. Oneof the two trajectories is parallelto A (at the point of closest ap-proach), while the other is anti-parallel. Even in the absence ofany Lorentz force on the elec-trons, that difference produces aquantum mechanical phase shiftbetween their wavefunctions.

tron is unaffected. Outside the solenoid, the magnetic andelectric fields B and E, and thus the Lorentz force, are all zero.But in quantum mechanics one describes an interaction notby the forces involved but rather by the Hamiltonian.

The canonical momentum in the electromagneticHamiitonian has a term proportional to the magnetic vectorpotential A, which is defined (with some gauge freedom) byB = V X A. Before the AB paper, the vector potential was gen-erally thought of as a useful mathematical artifice withoutindependent physical reality. Aharonov and Bhm changedall that.

38 September 2009 Physics Today 2009 American Institute ol Physics. S-0031 -922B-0909-020-6

Batelaan & Tonomura in Physics Today (September 2009) 62. Copyright 2009, American Institute of Physics. Used by permission.

The first interesting fact is that A is not zero outside thesolenoid, even though the force fields E and B vanish there(see figure 1). The vector potential is everywhere azimuthalto the solenoid's axis, and Stokes's theorem says that its lineintegral around any loop enclosing the solenoid equals themagnetic flux O inside. So outside the solenoid, A falls offlike 1/r with distance r from the axis.

In figure 1, two electrons traveling together pass by thesolenoid on opposite sides. At the points of closest approachto the solenoid, one is traveling parallel to A on its side, theother antiparallel. Quantum mechanically, one says that theelectrons interact with the solenoid because the wavefunctionof each accumulates a phase shift

- -e/hJA-dl (1)as it traverses the vector potential.

The dot product in equation 1 implies that the phasechanges are of opposite sign for electrons passing the sole-noid on opposite sides. So the two electrons on their differentpaths past a long solenoid enclosing flux

Figure 3. Electron interference pattern demonstrating themagnetic Aharonov-Bohm effect in an experiment thatstrictiy excludes all stray fields."" A coherent electron beamtraveling normal to the page is made to pass around atoroidal magnet (seen as a shadow) or through its 4-Mm-diameter hole. The magnet's superconducting cladding pre-vents all stray fields. Having threaded or passed around themagnet, the beam is made to interfere with a referenceplane wave. The resulting pattern, with the interferencefringe inside the hole offset by half a cycle from those out-side whenever the magnet flux is an odd multiple of h/2e, in-dicates an AB phase shift of TT (modulo 2n) between thethreading and bypassing electrons.

has sometimes been dismissed as an extreme limiting casebased on an unphysical geometry. A 1986 experiment by agroup at Hitachi led by one of us (Tonomura), done with atiny toroidal magnet, is illuminating in that regard.^ The mag-net's superconducting niobium cladding excluded any strayfields in that experiment. Its excellent agreement with the ABprediction answers ail objections about stray fields and un-physical geometries.

The electron interferogram in figure 3 manifests the ABeffect demonstrated in the Hitachi experiment. A coherentelectron beam is directed at a micro fabrica ted toroidal mag-net. Most of the beam passes by the outside of the magnet,but some of it threads through the magnet's 4-fam-diametercentral hole. A gold outer cladding prevents any electronsfrom penetrating the magnet itself, and the superconductinginner cladding keeps the hole and the entire vicinity of themagnet free of any stray B fields. But within the magnet'snonsuperconducting core, where no beam electrons venture,the circling 4 can be varied over several multiples of h/e.Equation 2 says that when O equals an odd multiple of theflux quantum li/2e. there should be an AB phase shift of TT(modulo 2n) between the electrons that threaded the hole andthose that bypassed the magnet. And that's precisely what isseen in the interferogram that results from bringing the beamtogether with a coherent reference beam that avoided themagnet altogether.

The electric AB effectGiven the many and varied confirmations of the magnetic ABeffect, it is no surprise that textbooks often present the AB ef-fect as a beautiful and closed subject. Nevertheless, an in-creasing citation rate has pushed the total number of citationsof the original AB paper beyond 2500, hardly what one ex-pects for a closed subject with no applications in prospect.The continuing interest is generated, in part, by the fact thattwo versions of the AB effect were already described in theoriginal 1959 paper: the magnetic version discussed aboveand the electric version.

If an enclosed magnetic flux can cause phase shifts, one

might expect that an enclosed electric flux can do the same.And that is indeed what Aharonov and Bhm predicted. Tiieelectric analogue of equation 1 is a time integral of the electricscalar potential V:

(p = e/hSVt. (3)That's plausible if one considers that A and V make up a rel-afivistic four-vector just like x and /.

The figure in box 1 compares the two original AB effects(as well as two others discussed below). Just as a split electronbeam run past a solenoid magnet manifests the magnetic ABeffect (panel a), the two beams passed through separatecharged metallic tubes demonstrate its electric analogue(panel b). There is essentially no electric field inside the tubes,and the electrons in the beam never experience the field be-tween the tubes. Nonetheless, by maintaining a voltage dif-ference between the tubes, one sees a measurable phase shiftwhen the two beams subsequently interfere.

There are two versions of the electric AB experiment. Inthe steady-state version, the electrons do experience fringeelectric fields as they enter and exit tlie tubes. But in thepulsed version, rapid charging and discharging of the tubesis done while the electron wavepackets are fully shielded in-side them.

The steady-state version of the experiment was per-formed in 1985 by Giorgio Matteucci and Giulio Pozzi at theUniversity of Bologna.^ The small size of electron interferom-eters prevented the use of actual metallic tubes. Instead, theexperimenters ran the electrons past a bimetallic wirecharged in such a way as to form a linear dipole field. In thatconfigurafion, the nonvanishing electric field produced nodefiecting forces. The expected electric AB phase shift frompassage on one side of the dipole field had the opposite signof that on the other side. And that's what Matteucci and Pozzifound. At the time, their experiment was considered to be ademonstrafion of what is called a type II Aharonov-Bohm ef-fect. The type I designation is reserved for experiments inwhich in principle the beam particles never traverse an Eor B field (see box 1).Insidious mimicsIt is now understood, however, that something insidious oc-curs in the nonpulsed version of the electric AB experiment.Although the electrons are not deflected, they are delayed orsped up by Coulomb forces. When the electrons enter andexit the potentials, those Coulomb forces act in such a way asto mimic the expected phase shift,'' So the steady-state exper-iments do not really demonstrate the electric AB effect.

But can the pulsed versions enfirely avoid those insidi-ous forces? Perhaps there is also a force descripfion that ex-plains away the magnetic AB effect too. Maybe, f>ace Einstein,God is malicious as well as subtle. We could ask: Might theelectron that passes by the solenoid be inducing changes inthe solenoid that act back and impose on the electron a forcethat mimics the AB prediction?'' (See box 2.)

To mimic the magnefic AB phase shift, the solenoid's re-action force would have to shift the electron wavepacket bya translation Ax =/\(/J^[,/2TI, where A is the electron'sde Broglie wavelength. For a wavepacket moving at velocityz>, the resulting Hme delay would be Ax/v. Led by one of us(Batelaan), a group at the University of Nebraska has recentlyruled out that reaction-force explanation with an experimentin which electron pulses were shot past solenoids and theirarrival times were measured for different solenoid currents(see figure 4).^ The experiment found no time delays thatcould mimic the magnetic AB effect.

40 September 2009 Physics Today www. physicstoday. org

Box 1. Types and dualsThe original magnetic and electric Aharonov-Bohm effects (pan-els a and b) are type 1 effects in the sense that in an ideal experi-ment, the electron sees no B or E fields, though it does traversedifferent potentials A and V. In their respective dual effectstheAharonov-Casher effect (panel c) and the so-called neutron-scalar AB effect (panel d}polarized neutrons (neutral particleswith magnetic dipoie moments) replace unpolarizedelectrons, and electrostatic configurations changeplaces v^ iith solenoids.'^ In panel c, a neutron interfe-rometer encloses a line of charge, and in panel d,neutrons pass through pulsed solenoids.These dualsare classified as type II effects because the neutronmust traverse a nonvanishing E or B field.

In either case, to acquire an AB phase shift, theelectron or neutron must pass through a region ofnonzero electromagnetic potential. That quantummechanical result seems to elevate the status of thepotentials to a physical reality absent from classicuelectromagnetism, Yakir Aharonov has pointed outthat the potentials do overdetermine the experi-mental outcome; the phase shift need only beknown modulo In. An alternative view is that theorigina! magnetic AB effect shows electromagneticfields acting nonlocally.'

For type II effects, the wavepackets can plowstraight through force fields, and forces are allovwedin the interaction. But the AB interpretation requires

that the emerging wavepackets not be deflected or delayed inany way. Quantum mechanical descriptions generally circum-vent the notion of forces. But one can use here an operationaldefinition of forces that might be mimicking an AB effect: If theinteraction has produced no deflection or delay, there were noforces.

A loophole in the interpretation of the Nebraska exper-iment might be that its solenoids were much bigger thanthose that demonstrated the magnetic AB effect." Small sole-noids might respond differently. It all depends on the inter-action time versus the solenoid's response time. For electronvelocities of about 10" m/s and a closest approach of about10 j.mi between the electron and a small solenoid, the shortestinteraction time would be about 10"'' seconds. For the samebeam velocity, a larger solenoid like those in the Nebraska ex-periment, with diameters on the order of millimeters, has in-teraction times a hundred times longer.

For comparison, the response time of electrons in a metalis typically 10"'''or 10'''seconds. TTius the Nebraska delay ex-periment and the demonstrations of the magnetic AB effectdo aliow the solenoids enough time to respond, albeit justbarely for the small solenoids used in the demonstrations.

Perhaps the ultimate experiment is the one proposed in1985 by Anton Zeilinger." So far in this discussion, we haveonly considered what a force does in the classical regime.Zcilinger considered what a force would do in the quantumregime. A large enough solenoid reaction force, he pointedout, would shift wavepackets so much in opposite directionsthat they would not overlap when recombined. That is to say,if solenoid reaction forces really were mimicking the pre-dicted AB effect, all interference contrast would disappearwhen the flux in the solenoid was big enough. But if the phaseshift really is due to the AB effect, the interference contrastwould remain unaffected beyond the length of thewavepacketor, in a continous electron beam, the beam's co-herence length L^.. The coherence length of a continuous beamwith a wavelength spread AA is given by AVAA. It's the sep-aration along the beam beyond which any two points havelost all phase coherence.

DispersionTo visualize the shift of a wavepacket, as distinguished froma quantum phase shift, it is useful to describe what happensto the packet's frequency components. If each frequency com-ponent accumulates the same phase shift after passage by thesolenoidwhich is what the AB effect predicts thewavepacket's envelope function does not shift. Tliat generallyaccepted view is captured by the concise statement that theAB effect is dispersionless. But if, on the other hand, someback-reaction force were giving different Fourier compo-nents different phase shifts that depended linearly on fre-quency, the wavepacket's envelope fimction would indeedshift. {See the interactive Flash movie accompanying the on-line version of this article.) Experimentally, the observationof electron fringes beyond the beam's coherence lengthwould verify the AB effect's dispersioniess character.

The extraordinary difficulty of such an experiment be-comes clear when one compares the electrons' de Brogliewavelength to the coherence length, which is also the small-est possible wavepacket length for a given AA. In the relevantexperimental regime, the typical L^ . is lO"^ times A. A longer Ameans unacceptably low electron momentum, and a shorterL^ means an unacceptably large momentum spread. So onewould have to observe 10^ interference fringes to test for dis-persion. Thus far such experiments, crucial as they are to thecharacterization of the AB effects, have remained out ofreach.

Nor has the pulsed version of the original {type I) electricAB effect ever been performed. The development of newpulsed electron sources and detection schemes may changethat situation. In 1999 Ahmed Zewail received the NobelPrize in Chemistry for developing ultrafast sensing tech-niques (see PHYSICS TODAY, December 1999, page 19). Those

www.physicstoday.org September 2009 Physics Today 41

Box 2. A paradoxYakir Aharonov and Daniel Rohrlich have posed a fully classicalparadox that they regard as "crucial for clarifying the entirelyquantum interactions of fluxons [flux quanta] and chargesthegeneralized Aharonov-Bohm effect.""The paradox involves theinteraction of a charge with a magnetic flux tube, just as in theoriginal magnetic AB effect. The magnetic flux tube is realizedwith symmetric charge and current distributions (see the figurebelow). Two coaxial, counterrotating, and oppositely chargedtubes create the flux. The difference between the two tubediameters is negligible; they rotate in opposite directions withthe same rim speed v.

Outside the tubes there Is no electric or magnetic field (thelatter assuming the tubes are very long). So a charged particleat rest outside the tubes should stay where it is. But if frictionbetween the tubes causes theirspins to decelerate, somethingcurious occurs. Faraday's law '.t ., , 'dictates that the diminishingmagnetic flux inside the tubesinduces an electric field outside.The electron experiences a forceF and accelerates, but the tubesdo not. What happened tomomentum conservation?

Aharonov and Rohrlich replythat one must take account ofmomentum in the electromag-netic field and "hidden" relativistic mechanical momentum inthe tubes. Many paradoxes are based on field momentum, andhidden momentum is a difficult concept that has been debatedfor decades. With regard to the AB effect, identifying all the{relativistic) classical momentum terms and all the forces associ-ated with the interaction between flux quanta and charges isnot easy.'^ So experiments have an important role to play.

techniques have until now been focused on probing femto-second molecular dynamics. But one techniqueultrafastelectron diffractionalso appears to be well suited for study-ing foundational issues of quantum mechanics such as theAB effect. That is especially true in view of the newest pulsed-electron sources such as those developed by Mark Kasevichand coworkers at Stanford Umversity.''

DualsAre the original magnetic and electric AB effects isolated phe-nomena? If they really exist, should they not be mirrored inother physical systems? Aharonov's answer is affirmative. In

with the Mott-Schwinger effect, in which neutron scatteringoff nuclei is attributed to the force between the nuclear chargeand the incident neutron's magnetic dipole." On the otherhand, the interactions of neutrons with line charges is con-sidered to be a true AB effect. As far as we know, there hasn'tbeen a unified theoretical or experimental treatment of thetwo effects. With increasingly refined neutron sources be-coming available, it may soon be possible to distinguish ex-perimentally between the force-driven scattering of neutronsoff a charged sphere and the presumed absence of such aforce in their interaction with a charged line.

Still more AB phenomenaAlthough solid-state manifestations of the AB effect are notthe main topic of this article, we would hardly do justice tothe breadth of the effect without mentioning them and theexperimental breakthroughs that have made them possible.The first step was the 1985 observation, by Richard Webb andcoworkers at IBM, of the AB effect for electrons circulating ina mesoscopic ring of nonsuperconducting metal'"' (seePHYSICS TODAY, January 1986, page 17). Although the elec-trons were propagating within the metal, their interactionwith the lattice did not cause enough decoherence to disruptAB interference, even in the absence of superconductivity.

Not having to rely on superconductivity in the creationof a solid-state electron interferometer lets the experimenteravoid the complication of flux quantization, and it permitsthe investigation of new quantum effects in disordered ma-terials. More recently, the observation of electron interferencein carbon nanotubes suggests that the microscopic regime isalso accessible for AB investigations.'^

In all the solid-state experiments, mesoscopic and micro-scopic, the AB phase shift was observed even though mag-netic fields were present throughout the interferometer. Thatmakes them a bit different from the enclosed-flux experi-ments with electron beams in vacuum. An important promisethat those small solid-state electronic devices offer is theirpossible application in such areas as quantum computing.

Our discussion thus far has been based on interferencebetween quantum mechanical amplitudes for alternativepaths of a single particle. The physics of two-electron inter-ference phenomena has to deal with the additional issue ofFermi-Dirac quantum statistics. A number of two-electron in-terference experiments in recent years have reported observ-ing the predicted statistical phenomena of antibunching andHanbury Brown-Twiss interference.'" (For background onHBT interference, see PHYSICS TODAY, August 208, page 8.)The most recent of those experiments exhibited HBT interfer-ence that created orbital quantum entanglement betweenelectron beams from two distinct sources.

Nowadays electrons can be used to demonstrate in thelaboratory all the elements at the heart of quantum mechan-ics: wave-particle duality, quantum statistics, nonlocaiity,and entanglement. Because those elements are important tothe prospects For quantum cryptography, quantum telepor-tation, and quantum computation, a new era of quantumelectronics may be at hand.

Aharonov stresses that the arguments that led to the pre-diction of the various electromagnetic AB effects applyequally well to any other gauge-invariant quantum theory. Inthe standard model of particle physics, the strong and weaknuclear interactions are also described by gauge-invarianttheories. So one may expect that particle-physics experi-menters will be looking for new AB effects in new domains.

We thank Adnm Caprez for the artwork. This article is based on worksupported by the NSF under grant no. O653'IS2.

September 2009 Physics Today 43

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University of Nebraska - LincolnDigitalCommons@University of Nebraska - Lincoln9-1-2009

The Aharonov-Bohm Effects: Variations on a Subtle ThemeHerman BatelaanAkira Tonomura