Aharonov–Bohm effect

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The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon by which an electrically charged particle is affected by the electromagnetic potential A in regions in which both the magnetic field B and electric field E are zero. In general, the profound consequence of Aharonov–Bohm effect was the realisation that the electromagnetic 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 potential A is seen as being more fundamental or "real"; the E and B fields can be derived from the potential A, but the potential can not be derived from the E and B fields. Furthermore some physical effects are dependent only on A, which the E and B fields are simply unable to account for.

Werner Ehrenberg and Raymond E. Siday first predicted the effect in 1949,[1] and similar effects were later rediscovered by Aharonov and Bohm in 1959.[2] (After publication of the 1959 paper, Bohm was informed of Ehrenberg and Siday's work, which was acknowledged and credited[3] in Bohm and Aharanov's subsequent 1961 paper.[4])

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 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.[5] A general review can be found in Peshkin and Tonomura (1989).[6]

Contents

[edit] 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 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 \Delta\varphi determined by the magnetic flux Φ through the area between the paths (via Stokes' theorem and \nabla \times \mathbf{A} = \mathbf{B}), and given by:

\Delta\varphi = \frac{q\Phi}{\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 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 q=2e 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 London (1948)[7] using a phenomenological model.

The magnetic Aharonov–Bohm effect was experimentally confirmed by Osakabe et al. (1986),[8] following much earlier work summarized in Olariu and Popèscu (1984).[9] Its scope and application continues to expand. Webb et al. (1985)[10] demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986)[11] and Imry & Webb (1989).[12] Bachtold et al. (1999)[13] detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004).[14]

[edit] 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 an 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 qe and magnetic charge qm.

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

[edit] 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:

\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.[15]

[edit] Aharonov–Bohm nano rings

Nano rings were created by accident[16] while intending to make quantum dots. They have interesting optical properties associated with excitons and the Aharonov-Bohm effect.[16] 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.[17][18][19]

[edit] Mathematical interpretation

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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 + i A\,, 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 γ 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 eiα, 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 [20] .

[edit] References

  1. ^ Ehrenberg, W; Siday, RE (1949). "The Refractive Index in Electron Optics and the Principles of Dynamics"". Proceedings of the Physical Society B 62: 8–21. doi:10.1088/0370-1301/62/1/303. 
  2. ^ Aharonov, Y; Bohm, D (1959). "Significance of electromagnetic potentials in quantum theory". Physical Review 115: 485–491. doi:10.1103/PhysRev.115.485. 
  3. ^ Peat, FD (1997). Infinite Potential: The Life and Times of David Bohm. Addison-Wesley. ISBN 0-201-40635-7. http://www.fdavidpeat.com/bibliography/books/infinite.htm. 
  4. ^ Aharonov, Y; Bohm, D (1961). "Further Considerations on Electromagnetic Potentials in the Quantum Theory". Physical Review 123: 1511–1524. doi:10.1103/PhysRev.123.1511. 
  5. ^ Sjöqvist, E (2002). "Locality and topology in the molecular Aharonov-Bohm effect". Physical Review Letters 89 (21): 210401. doi:10.1103/PhysRevLett.89.210401. arΧiv:quant-ph/0112136. 
  6. ^ Peshkin, M; Tonomura, A (1989). The Aharonov-Bohm effect. Springer-Verlag. ISBN 3-540-51567-4. 
  7. ^ London, F (1948). "On the Problem of the Molecular Theory of Superconductivity". Physical Review 74: 562. doi:10.1103/PhysRev.74.562. 
  8. ^ Osakabe, N, et al. (1986). "Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor". Physical Review A 34: 815. doi:10.1103/PhysRevA.34.815. 
  9. ^ Olariu, S; Popescu, II (1985). "The quantum effects of electromagnetic fluxes". Reviews of Modern Physics 57: 339. doi:10.1103/RevModPhys.57.339. 
  10. ^ Webb, RA; Washburn, S; Umbach, CP; Laibowitz, RB (1985). "Observation of h/e Aharonov-Bohm Oscillations in Normal-Metal Rings". Physical Review Letters 54: 2696. doi:10.1103/PhysRevLett.54.2696. 
  11. ^ Schwarzschild, B (1986). "Currents in Normal-Metal Rings Exhibit Aharonov–Bohm Effect". Physics Today 39 (1): 17. doi:10.1063/1.2814843. 
  12. ^ Imry, Y; Webb, RA (1989). "Quantum Interference and the Aharonov-Bohm Effect". Scientific American 260 (4). 
  13. ^ Schönenberger, C. (1999). "Aharonov–Bohm oscillations in carbon nanotubes". Nature 397: 673. doi:10.1038/17755. 
  14. ^ Kong, J; Kouwenhoven, L; Dekker, C (2004). "Quantum change for nanotubes". Physics World. http://physicsworld.com/cws/article/print/19746. Retrieved 2009-08-17. 
  15. ^ van Oudenaarden, A (1998). "Magneto-electric Aharonov–Bohm effect in metal rings". Nature 391: 768. doi:10.1038/35808. 
  16. ^ a b Fischer, AM (2009). "Quantum doughnuts slow and freeze light at will". Innovation Reports. http://www.innovations-report.com/html/reports/physics_astronomy/quantum_doughnuts_slow_freeze_light_128981.html. Retrieved 2008-08-17. 
  17. ^ Borunda, MF et al. (2008). "Aharonov-Casher and spin Hall effects in two-dimensional mesoscopic ring structures with strong spin-orbit interaction". arΧiv:0809.0880 [cond-mat.mes-hall]. 
  18. ^ Grbic, B et al. (2008). "Aharonov-Bohm oscillations in p-type GaAs quantum rings". Physica E 40: 1273. doi:10.1016/j.physe.2007.08.129. arΧiv:0711.0489. 
  19. ^ Fischer, AM et al. (2009). "Exciton Storage in a Nanoscale Aharonov-Bohm Ring with Electric Field Tuning". Physical Review Letters 102: 096405. doi:10.1103/PhysRevLett.102.096405. arΧiv:0809.3863. 
  20. ^ V. Y. Chernyak and N. A. Sinitsyn (2009). "Robust quantization of a molecular motor motion in a stochastic environment". Preprint arXiv:0906.3032. http://arxiv.org/pdf/0906.3032v3. 

[edit] See also

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