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| The '''Aharonov–Bohm effect''', sometimes called the '''Ehrenberg–Siday–Aharonov–Bohm''' effect, is a [[quantum mechanics|quantum mechanical]] phenomenon in which an [[electric charge|electrically charged]] [[point particle|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 (physics)|coupling]] of the [[electromagnetic potential]] with the [[Argument (complex analysis)|complex phase]] of a charged particle's [[wavefunction]], and the Aharonov–Bohm effect is accordingly illustrated by [[double-slit experiment|interference experiments]].
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| 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 (waves)|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.<ref name="The Aharonov–Bohm effects: Variations on a Subtle Theme.">{{cite journal |first=A. |last=Batelaan |lastauthoramp=yes |first2=A. |last2=Tonomura |title=The Aharonov–Bohm effects: Variations on a Subtle Theme |journal=Physics Today |date=Sept 2009|volume=62|issue=9|pages=38–43 }}</ref> 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 potential]]s but zero electric field, but this has no experimental confirmation yet.<ref name="The Aharonov–Bohm effects: Variations on a Subtle Theme." /> 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.<ref>
| | Also visit my homepage :: cheap luke bryan tickets - [http://www.netpaw.org http://www.netpaw.org] - |
| {{cite journal
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| |author=Sjöqvist, E
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| |year=2002
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| |title=Locality and topology in the molecular Aharonov–Bohm effect
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| |journal=[[Physical Review Letters]]
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| |volume=89 |issue=21 |pages=210401
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| |doi=10.1103/PhysRevLett.89.210401
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| |id=
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| |pmid=12443394
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| |arxiv=quant-ph/0112136
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| |bibcode=2002PhRvL..89u0401S
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| }}</ref>
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| [[Werner Ehrenberg]] and [[R.E. Siday|Raymond E. Siday]] first predicted the effect in 1949,<ref>
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| {{cite journal
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| |author=Ehrenberg, W; Siday, RE
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| |year=1949
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| |title=The Refractive Index in Electron Optics and the Principles of Dynamics
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| |journal=[[Proceedings of the Physical Society]] | series = Series B
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| |volume=62 |pages=8–21
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| |doi=10.1088/0370-1301/62/1/303
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| |bibcode = 1949PPSB...62....8E }}</ref> and similar effects were later published by [[Yakir Aharonov]] and [[David Bohm]] in 1959.<ref>
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| {{cite journal
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| |author=Aharonov, Y; Bohm, D
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| |year=1959
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| |title=Significance of electromagnetic potentials in quantum theory
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| |journal=[[Physical Review]]
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| |volume=115 |pages=485–491
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| |doi=10.1103/PhysRev.115.485
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| |bibcode = 1959PhRv..115..485A }}</ref> 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.<ref>
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| {{cite book
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| |author=Peat, FD
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| |year=1997
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| |title=Infinite Potential: The Life and Times of David Bohm
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| |url=http://www.fdavidpeat.com/bibliography/books/infinite.htm
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| |page=
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| |publisher=[[Addison-Wesley]]
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| |isbn=0-201-40635-7
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| }}</ref><ref>
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| {{cite journal
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| |author=Aharonov, Y; Bohm, D
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| |year=1961
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| |title=Further Considerations on Electromagnetic Potentials in the Quantum Theory
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| |journal=[[Physical Review]]
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| |volume=123 |pages=1511–1524
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| |doi=10.1103/PhysRev.123.1511
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| |bibcode = 1961PhRv..123.1511A }}</ref>
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| Subsequently, the effect was confirmed experimentally by several authors; a general review can be found in Peshkin and Tonomura (1989).<ref>
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| {{cite book
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| |author=Peshkin, M; Tonomura, A
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| |year=1989
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| |title=The Aharonov–Bohm effect
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| |publisher=[[Springer-Verlag]]
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| |isbn=3-540-51567-4
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| }}</ref>
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| | |
| == Significance ==
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| In the 18th and 19th centuries, physics was dominated by Newtonian dynamics, with its emphasis on [[force (physics)|forces]]. Electromagnetic phenomena were elucidated by a series of experiments involving the measurement of forces between charges, [[electrical current|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 [[curl (mathematics)|irrotational]], and one of the observed properties of the magnetic field was that it was [[divergence]]less, 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 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).
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| The Aharonov–Bohm effect is important conceptually because it bears on three issues apparent in the recasting of ([[James Clerk Maxwell|Maxwell]]'s) [[classical electromagnetism|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 experiment]]s and their experimental realization imply that the issues were not just philosophical.
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| The three issues are:
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| #whether potentials are "physical" or just a convenient tool for calculating force fields;
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| #whether [[action (physics)|action]] [[principle of stationary action|principle]]s are fundamental;
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| #the [[principle of locality]].
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| 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".<ref>[http://www.newscientist.com/special/seven-wonders-of-the-quantum-world "Seven wonders of the quantum world"], newscientist.com</ref>
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| === Potentials vs. fields ===
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| The Aharonov–Bohm effect illustrates the physicality of electromagnetic potentials, ''Φ'' and '''A'''. Previously it was possible to argue that only the [[electromagnetic field]]s are physical, while the electromagnetic potentials are purely mathematical constructs, that due to [[gauge freedom]] aren't even unique for a given electromagnetic field. Electric and magnetic fields and forces are, however, [[gauge invariant]] and therefore directly observable, and unlike potentials, appear in the [[Lorentz force]] formula.
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| === Global action vs. local forces ===
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| Similarly, the Aharonov–Bohm effect illustrates that the [[Lagrangian dynamics|Lagrangian approach to dynamics]], based on [[action (physics)|energies]], is not just a computational aid to the [[Newtonian dynamics|Newtonian approach]], based on [[force (physics)|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{{Citation needed|date=January 2011}} 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 formulation|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.
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| === Locality of electromagnetic effects ===
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| 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]], (''Φ'','''A'''), 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 electromagnetic four-potential, the effect only depends on the 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 four-potential 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. <ref>
| |
| {{cite book
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| |author=Feynman, R
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| |year=
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| |title=[[The Feynman Lectures on Physics]]
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| |volume=2 |pages=15–5
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| |publisher=
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| |isbn=
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| |quote=knowledge of the classical electromagnetic field acting locally on a particle is not sufficient to predict its quantum-mechanical behavior.'' and ''...is the vector potential a "real" field? ... a real field is a mathematical device for avoiding the idea of action at a distance. .... for a long time it was believed that '''A''' was not a "real" field. .... there are phenomena involving quantum mechanics which show that in fact '''A''' is a "real" field in the sense that we have defined it..... '''E''' and '''B''' are slowly disappearing from the modern expression of physical laws; they are being replaced by '''A''' [the vector potential] and <math>\varphi</math>[the scalar potential]}}</ref> 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.
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| ==Magnetic solenoid effect==
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| The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the [[gauge fixing|gauge choice]] for the [[electromagnetic potential]], of which the [[magnetic vector potential]] '''A''' forms part.
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| 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 <math>\mathbf{B} = 0 = \nabla \times \mathbf{A}</math>), acquires a phase shift <math>\varphi</math>, given in [[SI]] units by
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| :<math>\varphi = \frac{q}{\hbar} \int_P \mathbf{A} \cdot d\mathbf{x},</math>
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| Therefore particles, with the same start and end points, but travelling along two different routes will acquire a phase difference <math>\Delta \varphi</math> determined by the [[magnetic flux]] <math>\Phi_B</math> through the area between the paths (via [[Stokes' theorem]] and <math>\nabla \times \mathbf{A} = \mathbf{B}</math>), and given by:
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| :<math>\Delta\varphi = \frac{q\Phi_B}{\hbar}.</math>
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| [[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 turned on in the cylindrical solenoid.]]
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| In [[quantum mechanics]] the same particle can travel between two points by a [[sum over histories|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 (mathematics)|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.
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| The same phase effect is responsible for the [[magnetic flux quantum|quantized-flux]] requirement in [[superconductivity|superconducting]] loops. This quantization occurs because the superconducting wave function must be single valued: its phase difference <math>\Delta \varphi</math> around a closed loop must be an integer multiple of 2π (with the charge {{val|''q'' = 2''e''}} for the electron [[Cooper pair]]s), and thus the flux must be a multiple of ''h''/2''e''. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenological model.<ref>
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| {{cite journal
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| |author=London, F
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| |year=1948
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| |title=On the Problem of the Molecular Theory of Superconductivity
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| |journal=[[Physical Review]]
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| |volume=74 |issue= |pages=562
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| |doi=10.1103/PhysRev.74.562
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| |bibcode = 1948PhRv...74..562L }}</ref>
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| The first claimed experimental confirmation was by [[Robert G. Chambers]] in 1960,<ref>{{cite doi|10.1103/PhysRevLett.5.3}}</ref><ref>{{cite doi|10.1038/nphys1619}}</ref> in an electron interferometer with a magnetic field produced by a thin iron whisker, and other early work is summarized in Olariu and Popèscu (1984).<ref>
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| {{cite journal
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| |author=Olariu, S; Popescu, II
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| |year=1985
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| |title=The quantum effects of electromagnetic fluxes
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| |journal=[[Reviews of Modern Physics]]
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| |volume=57 |issue= |pages=339
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| |doi=10.1103/RevModPhys.57.339
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| |bibcode=1985RvMP...57..339O
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| }}</ref> 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.<ref>P. Bocchieri and A. Loinger, Nuovo Cimento Soc. Ital. Fis. 47A, 475 (1978); P. Bocchieri, A. Loinger, and G. Siragusa, Nuovo Cimento Soc. Ital. Fis. 51A, 1 (1979); P. Bocchieri and A. Loinger, Lett. Nuovo Cimento Soc. Ital. Fis. 30, 449 (1981). P. Bocchieri, A. Loinger, and G. Siragusa, Lett. Nuovo Cimento Soc. Ital. Fis. 35, 370 (1982).</ref><ref>S. M. Roy, Phys. Rev. Lett. 44, 111 (1980)</ref> An early experiment in which an unambiguous Aharonov–Bohm effect was observed by completely excluding the magnetic field from the electron path (with the help of a [[superconductor|superconducting]] film) was performed by Tonomura et al. in 1986.<ref>Akira Tonomura, Nobuyuki Osakabe, Tsuyoshi Matsuda, Takeshi Kawasaki, and Junji Endo, "Evidence for Aharonov-Bohm Effect with Magnetic Field Completely Shielded from Electron wave", ''Phys. Rev. Lett.'' vol. 56, pp. 792–795 (1986).</ref><ref>
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| {{cite journal
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| |author=Osakabe, N; ''et al''.
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| |year=1986
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| |title=Experimental confirmation of Aharonov–Bohm effect using a toroidal magnetic field confined by a superconductor
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| |journal=[[Physical Review A]]
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| |volume=34 |issue=2 |pages=815–822
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| |doi=10.1103/PhysRevA.34.815
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| |pmid=9897338
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| |bibcode = 1986PhRvA..34..815O }}</ref> The effect's scope and application continues to expand. Webb ''et al.'' (1985)<ref name=WebbRA>
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| {{cite journal
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| |author=Webb, RA; Washburn, S; Umbach, CP; Laibowitz, RB
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| |year=1985
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| |title=Observation of h/e Aharonov–Bohm Oscillations in Normal-Metal Rings
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| |journal=[[Physical Review Letters]]
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| |volume=54 |issue= 25|pages=2696–2699
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| |doi=10.1103/PhysRevLett.54.2696
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| |pmid=10031414
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| |bibcode=1985PhRvL..54.2696W
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| }}</ref> demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986)<ref>
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| {{cite journal
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| |author=Schwarzschild, B
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| |year=1986
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| |title=Currents in Normal-Metal Rings Exhibit Aharonov–Bohm Effect
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| |journal=[[Physics Today]]
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| |volume=39 |issue=1 |pages=17
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| |doi=10.1063/1.2814843
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| |bibcode = 1986PhT....39a..17S }}</ref> and Imry & Webb (1989).<ref>
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| {{cite journal
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| |author=Imry, Y; Webb, RA
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| |year=1989
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| |title=Quantum Interference and the Aharonov–Bohm Effect
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| |journal=[[Scientific American]]
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| |volume=260 |issue=4 |pages=
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| |doi=
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| }}</ref> Bachtold ''et al.'' (1999)<ref>
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| {{cite journal
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| |author=Schönenberger, C
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| |year=1999
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| |last2=Bachtold
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| |first2=Adrian
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| |last3=Strunk
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| |first3=Christoph
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| |last4=Salvetat
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| |first4=Jean-Paul
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| |last5=Bonard
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| |first5=Jean-Marc
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| |last6=Forró
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| |first6=Laszló
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| |last7=Nussbaumer
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| |first7=Thomas
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| |title=Aharonov–Bohm oscillations in carbon nanotubes
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| |journal=[[Nature (journal)|Nature]]
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| |volume=397 |issue= 6721|pages=673
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| |doi=10.1038/17755
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| |bibcode = 1999Natur.397..673B }}</ref> detected the effect in carbon nanotubes; for a discussion, see Kong ''et al.'' (2004).<ref>
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| {{cite web
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| |author=Kong, J; Kouwenhoven, L; Dekker, C
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| |year=2004
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| |title=Quantum change for nanotubes
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| |url=http://physicsworld.com/cws/article/print/19746
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| |work=[[Physics World]]
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| |accessdate=2009-08-17
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| }}</ref>
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| ===Monopoles and Dirac strings===
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| The magnetic Aharonov–Bohm effect is also closely related to [[Paul Dirac|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.
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| 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 4π''g'' 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, <math>2\frac{q_\text{e}q_\text{m}}{\hbar c}</math> must be an integer (in [[Gaussian units|cgs]] units) for any electric charge ''q''<sub>e</sub> and magnetic charge ''q''<sub>m</sub>.
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| 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.
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| ==Electric effect==
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| 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.
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| From the [[Schrödinger equation]], the phase of an eigenfunction with energy ''E'' goes as <math>\exp(-iEt/\hbar)</math>. 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:
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| :<math>\Delta\phi = -\frac{qVt}{\hbar} ,</math>
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| where ''t'' is the time spent in the potential.
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| 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.<ref>
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| {{cite journal
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| |author=van Oudenaarden, A
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| |year=1998
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| |last2=Devoret
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| |first2=Michel H.
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| |last3=Nazarov
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| |first3=Yu. V.
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| |last4=Mooij
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| |first4=J. E.
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| |title=Magneto-electric Aharonov–Bohm effect in metal rings
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| |journal=[[Nature (journal)|Nature]]
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| |volume=391 |issue= 6669|pages=768
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| |doi=10.1038/35808
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| |bibcode = 1998Natur.391..768V }}</ref>
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| ==Aharonov–Bohm nano rings==
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| Nano rings were created by accident<ref name=Warwick>
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| {{cite web
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| |author=Fischer, AM
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| |year=2009
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| |title=Quantum doughnuts slow and freeze light at will
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| |url=http://www.innovations-report.com/html/reports/physics_astronomy/quantum_doughnuts_slow_freeze_light_128981.html
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| |publisher=Innovation Reports
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| |accessdate=2008-08-17
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| }}</ref> while intending to make [[quantum dots]]. They have interesting optical properties associated with [[exciton]]s and the Aharonov–Bohm effect.<ref name=Warwick/> 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.<ref name=Borunda>
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| {{cite arXiv
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| |author=Borunda, MF; ''et al''.
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| |year=2008
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| |title=Aharonov–Casher and spin Hall effects in two-dimensional mesoscopic ring structures with strong spin-orbit interaction
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| |class=cond-mat.mes-hall
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| |eprint=0809.0880
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| }}</ref><ref name=Grbic>
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| {{cite journal
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| |author=Grbic, B; ''et al.''
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| |year=2008
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| |title=Aharonov–Bohm oscillations in p-type GaAs quantum rings
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| |journal=[[Physica E]]
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| |volume=40 |page=1273
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| |doi=10.1016/j.physe.2007.08.129
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| |id=
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| |arxiv=0711.0489
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| |bibcode = 2008PhyE...40.1273G }}</ref><ref name=Fischer>
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| {{cite journal
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| |author=Fischer, AM; ''et al''.
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| |year=2009
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| |title=Exciton Storage in a Nanoscale Aharonov–Bohm Ring with Electric Field Tuning
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| |journal=[[Physical Review Letters]]
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| |volume=102 |issue= |pages=096405
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| |doi=10.1103/PhysRevLett.102.096405
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| |id=
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| |arxiv=0809.3863
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| |bibcode=2009PhRvL.102i6405F
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| }}</ref>
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| Several experiments, including some reported in 2012,<ref>{{cite journal|author= M. Tsubota, K. Inagaki, T. Matsuura and S. Tanda|title= Aharonov-Bohm effect in charge-density wave loops with inherent temporal current switching|journal=EPL (Europhysics Letters)|year=2012|volume=97|issue=5|pages=57011|doi= 10.1209/0295-5075/97/57011|url= http://iopscience.iop.org/0295-5075/97/5/57011/ |arxiv = 0906.5206 |bibcode = 2012EL.....9757011T }}</ref> show Aharonov-Bohm oscillations in [[charge density wave]] (CDW) current versus magnetic flux, of dominant period ''h''/2''e'' through CDW rings up to 85 [[µm]] in circumference above 77 K. This behavior is similar to that of the superconducting quantum interference devices (see [[SQUID]]).
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| ==Mathematical interpretation==
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| {{Refimprove section|date=October 2012}}
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| The Aharonov–Bohm effect can be understood from the fact that we can only measure absolute values of the wave function. While this allows us to measure phase differences through quantum interference experiments, we have no way to specify a wavefunction with constant absolute phase. In the absence of an electromagnetic field we can come close by declaring the eigenfunction of the momentum operator with zero momentum to be the function "1" (ignoring normalization problems) and specifying wave functions relative to this eigenfunction "1". In this representation the i-momentum operator is (up to a factor <math>\hbar/i</math>) the differential operator <math> \partial_i = \frac{\partial}{\partial x^i}</math>. However, by gauge invariance, it is equally valid to declare the zero momentum eigenfunction to be <math> e^{-i\phi(x)} </math> at the cost of representing the i-momentum operator (up to a factor) as <math>\nabla_i = \partial_i + i(\partial_i \phi)</math> i.e. with a pure gauge vector potential <math>A = d\phi</math>. There is no real asymmetry because representing the former in terms of the latter is just as messy as representing the latter in terms of the former. This means that it is physically more natural to describe wave "functions", in the language of [[differential geometry]], as sections in a complex line bundle with a hermitian metric and a U(1)-[[connection (mathematics)|connection]] <math>\nabla</math>. The [[curvature form]] of the connection, <math>i F = \nabla\wedge\nabla </math>, is, up to the factor i, the [[Faraday tensor]] of the electromagnetic [[field strength]]. The Aharanov–Bohm effect is then a manifestation of the fact that a connection with zero curvature (i.e. [[flat connection|flat]]), need not be trivial since it can have [[monodromy]] along a topologically nontrivial path fully contained in the zero curvature (i.e. field free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field free region.
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| Given a trivialization of the line-bundle, a non-vanishing section, the U(1)-connection is given by the 1-[[Differential form|form]] corresponding to the [[electromagnetic four-potential]] ''A'' as <math> \nabla = d + i A\,</math> where ''d'' means [[exterior derivative|exterior derivation]] on the [[Minkowski space]]. The monodromy is the [[holonomy]] of the flat connection. The holonomy of a connection, flat or non flat, around a closed loop <math>\gamma</math> is <math>e^{i \int_\gamma A}</math> (one can show this does not depend on the trivialization but only on the connection). For a flat connection we can find a [[Gauge theory|gauge transformation]] in any [[simply connected]] field free region(acting on wave functions and connections) that gauges away the vector potential. However, if the monodromy is nontrivial, there is no such gauge transformation for the whole outside region. In fact as a consequence of [[Stokes' theorem]], the holonomy is determined by the magnetic flux through a surface <math>\sigma</math> bounding the loop <math>\gamma</math>, but such a surface may exist only if <math>\sigma</math> passes through a region of non trivial field:
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| :<math> e^{i \int_{\partial \sigma} A} = e^{i \int_{\sigma} dA} = e^{i\int_\sigma F}</math>
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| The monodromy of the flat connection only depends on the topological type of the loop in the field free region (in fact on the loops [[Homology (mathematics)|homology]] class). The holonomy description is general, however, and works inside as well as outside the superconductor. Outside of the conducting tube containing the magnetic field, the field strength <math>F = 0</math>. In other words outside the tube the connection is flat, and the monodromy of the loop contained in the field-free region depends only on the [[winding number]] around the tube. The monodromy of the connection for a loop going round once (winding number 1) is the phase difference of a particle interfering by propagating left and right of the superconducting tube containing the magnetic field.
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| If we want to ignore the physics inside the superconductor and only describe the physics in the outside region, it becomes natural and mathematically convenient to describe the quantum electron by a section in a complex line bundle with an "external" flat connection <math>\nabla</math> with monodromy
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| :<math>\alpha = </math> magnetic flux through the tube / <math>(\hbar/e)</math>
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| rather than an external EM field <math>F</math>. The Schrödinger equation readily generalizes to this situation by using the [[Laplacian]] of the connection for the (free) Hamiltonian
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| :<math> H = \frac{1}{2m}\nabla^*\nabla </math>.
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| Equivalently, 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 <math> e^{i\alpha}</math>, which results in the shift in the interference pattern as we change the flux.
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| 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.<ref name='chernyak-09prl'>
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| {{cite journal
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| |author=Chernyak, VY; Sinitsyn, NA
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| |year=2009
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| |title=Robust quantization of a molecular motor motion in a stochastic environment
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| |journal=[[Journal of Chemical Physics]]
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| |volume=131
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| |issue=18 |page=181101
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| |doi=10.1063/1.3263821
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| |bibcode=2009JChPh.131r1101C
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| |id=
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| |pmid=19916586
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| |arxiv=0906.3032
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| }}</ref>
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| ==See also==
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| *[[Geometric phase]]
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| *[[Hannay angle]]
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| *[[Wannier function]]
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| *[[Berry phase]]
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| *[[Wilson loop]]
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| *[[Winding number]]
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| *[[Aharonov–Casher effect]]
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| ==References==
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| {{Reflist|colwidth=30em}}
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| ==General references==
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| *{{cite book |title=Topological quantum numbers in nonrelativistic physics |author=D. J. Thouless |isbn=981-02-3025-7 |publisher=World Scientific |year=1998 |chapter=§2.2 Gauge invariance and the Aharonov–Bohm effect |pages=18''ff''|url=http://books.google.com/books?id=6XC_PBEXnAEC&pg=PA18}}
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| ==External links==
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| *[http://www2.warwick.ac.uk/newsandevents/news/quantumdoughnuts A video explaining the use of the Aharonov–Bohm effect in nano-rings.]
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| {{DEFAULTSORT:Aharonov-Bohm Effect}}
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| [[Category:Quantum mechanics]]
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| [[Category:Physical phenomena]]
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| [[Category:Mesoscopic physics]]
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