Aharonov-Bohm phase

A term that is nearly synonymous with complex numbers or functions is their phase. The rising preoccupation with the wave function phase in the last few decades is beyond doubt, to the extent that the importance of phases has of late become comparable to that of the moduli. (We use Dirac s terminology [7], which writes a wave function by a set of coefficients, the amplitudes, each expressible in terms of its absolute value, its modulus, and its phase. ) There is a related growth of literatm e on interference effects, associated with Aharonov-Bohm and Berry phases [8-14], In parallel, one has witnessed in recent years a trend to construct selectively and to manipulate wave functions. The necessary techifiques to achieve these are also anchored in the phases of the wave function components. This bend is manifest in such diverse areas as coherent or squeezed states [15,16], elecbon bansport in mesoscopic systems [17], sculpting of Rydberg-atom wavepackets [18,19], repeated and nondemolition quantum measurements [20], wavepacket collapse [21], and quantum computations [22,23], Experimentally, the determination of phases frequently utilizes measurement of Ramsey fringes [24] or similar" methods [25]. 

The phase-change nale, also known as the Ben phase [101], the geometric phase effect [102,103] or the molecular Aharonov-Bohm effect [104-106], was used by several authors to verify that two near-by surfaces actually cross, and are not repelled apart. This point is of particular relevance for states of the same symmetry. The total electronic wave function and the total nuclear wave function of both the upper and the lower states change their phases upon being bansported in a closed loop around a point of conical intersection. Any one of them may be used in the search for degeneracies. 

The full quantum mechanical study of nuclear dynamics in molecules has received considerable attention in recent years. An important example of such developments is the work carried out on the prototypical systems H3 [1-5] and its isotopic variant HD2 [5-8], Li3 [9-12], Na3 [13,14], and HO2 [15-18], In particular, for the alkali metal trimers, the possibility of a conical intersection between the two lowest doublet potential energy surfaces introduces a complication that makes their theoretical study fairly challenging. Thus, alkali metal trimers have recently emerged as ideal systems to study molecular vibronic dynamics, especially the so-called geometric phase (GP) effect [13,19,20] (often referred to as the molecular Aharonov-Bohm effect [19] or Berry s phase effect [21]) for further discussion on this topic see [22-25], and references cited therein. The same features also turn out to be present in the case of HO2, and their exact treatment assumes even further complexity [18],... 

C2H-molecule (1,2) and (2,3) conical intersections, 111-112 H3 molecule, 104-109 Wigner rotation matrix and, 89-92 Yang-Mills field, 203-205 Aharonov-Anandan phase, properties, 209 Aharonov-Bohm effect. See Geometric phase effect... 

The holonomy represents a parallel-transport operator around C assuming values in a non-Abelian Lie group G. (Interestingly, in the Abelian case, the holonomy has a physical role it is an object playing the role of the phase that can be observed in the Aharonov-Bohm experiment, whereas ,- itself does not have such an interpretation.)... 

A second interpretation of the Aharonov-Bohm effect was devised by Boyer [65,66], who used matter waves associated to moving electrons. Waves coming from each slit interfere with a phase shift = 2jidistance between two slits. If P is the impulse of an electron in the beam, the de Broglie relation gives us P 2nh/X. This results in the fact that the phase... 

If the velocity U of an electron within the beam is constant outside the solenoid, the variation of the vector potential A as a function of time in the medium, and thus also in the solenoid, will induce a modification of the phase, as indicated by the equations written above. This will produce a modification of the boundary conditions on the boundary of the solenoid for the quantities a and b. We must also stress that the modification of the vector potential outside the solenoid is generated by either an external or an internal source feeding the solenoid. This can explain the existence of the Aharonov-Bohm effect for toroidal, permanent magnets. The interpretation of the Aharonov-Bohm effect is therefore classic, but the observation of this effect requires the principle of interference of quantum mechanics, which enables a phase effect to be measured. 

Therefore, the distinction between the topological and dynamical phase has vanished, and the realization has been reached that the phase in optics and electrodynamics is a line integral, related to an area integral over Bt3> by a non-Abelian Stokes theorem, Eq. (553), applied with 0(3) symmetry-covariant derivatives. It is essential to understand that a non-Abelian Stokes theorem must be applied, as in Eq. (553), and not the ordinary Stokes theorem. We have also argued, earlier, how the non-Abelian Stokes explains the Aharonov-Bohm effect without difficulty. 

It is well known that the change in phase difference of two electron beams in the Aharonov-Bohm effect is described in the conventional U(l) invariant theory by... 

The typical explanation of the Aharonov-Bohm effect continues with the observation that a phase difference, 5, between the two test electrons is caused by the presence of the solenoid... 

We next observe that cpM is in units of volt-seconds (V s) or kg m 2/ (A s-2) = J/A. From Eq. (12) it can be seen that A8 and the phase factor, , are dimensionless. Therefore we can make the prediction that if the magnetic flux, (pM, is known and the phase factor, magnetic charge density, gm, can be found by the following relation ... 

We note that the phase effect is dependent on B2 and B, but not on B alone. Previous treatments found no convincing argument around the fact that whereas the Aharonov-Bohm effect depends on an interaction with the A field outside the solenoid, B, defined in U(l) electromagnetism as B = V x A, is zero at that point of interaction. However, when A is defined in terms associated with an SU(2) situation, that is not the case as we have seen. 

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