# Aharonov–Bohm Effect, Experiments, Applications, etc

1. Apr 10, 2009

### swat4life

Hello!
I'm new to the forum and quite interested in the wealth of information and exchanges that seem to go on here.

I found my way here a couple of times before looking for some information on other experiments, but was too lazy to register, lol.

At any rate my query is related to the Aharonov–Bohm Effect. Now before I ask it specifically, know that my interest stems from a purely recreational/mental curiosity motive and nothing else - so thanks for your patience...

Basically, I've just discovered David Bohm and I can't get enough of him! So, as a result, I've been reading about his various experiments, contributions to physics, theories, worldview philosophies, etc.

What I'm curious about is what (if any) relationship there is between the results of said experiment/effect and a) the concept of nonlocality, b) the hidden variable concept (which I *somewhat* understand) and c)real world applications

I found a couple of links from some universities here:

http://msc.phys.rug.nl/quantummechanics/ab.htm [Broken]

http://www.physics.harvard.edu/~dtlarson/tutorial05/lecture6.pdf [Broken]

The former does a pretty good job of giving me some visual, rudimentary understanding of the principle concept(s) and experiment, the latter I find rather boring and tedious to understand.

But I am hungry to learn more. So if someone could shed some light in a paint-by-numbers kind of way, that would really rock...

After perusing the above, I happened upon a wiki on the material and I'd be delighted to have some clarification on the following:

"Nano rings were created by accident[16] during the manufacture of quantum dots 10-100nm in size. The process sometimes cause the material to splash when making deposits onto a surface leaving a defective dot that becomes a doughnut-shaped ring, an Aharonov–Bohm nano ring. These nano rings have been a source of study and are the right size for enclosing an exciton. The right size does not allow them to hold an exciton for long. But when a combination of magnetic and electric fields is applied, the electric field can tuned to freeze an exciton in place or let it collapse and re-emit a photon at a later time. This is the pairing of an electron that has been kicked into a higher state by a photon, with a hole it leaves within the shell around the nucleus. When an electron’s high energy state decays again, it is drawn back to the hole it is linked to and a photon is once again emitted. By holding an exciton in place one could delay the reemitting of a photon and effectively slow or even "freeze" light. While varying exotic states of matter have been used to slow the progress of light, the University of Warwick reported in March of 2009 that it was successful for the first time to completely freeze light by releasing individual photons at will[16] Application of these rings used as light capacitors or buffers includes photonic computing and communications technology."
http://en.wikipedia.org/wiki/Aharonov-Bohm_effect

So what's going on here? Are they saying that basically, during an experiment used to demonstrate the Aharonov-Bohm Effect, something's happening with the interaction between electric and magnetic fields which does *something* to excitons (need to look that up...) and photons whereby light is "frozen", held still, etc?

Then, there's some discussion about how the process of "capturing", slowing, etc the light, photons, etc could well pave the way for photonic computing.

So, are they basically saying that the nano-rings used/discovered as a result of the A-B Effect could be used as a central component for what - this "freezing" process?

I know this is a rather lengthy set of questions, but I'd be very appreciative for anyone's insight here.

Lastly, I've seen some great links on Youtube about David Bohm; I've also Wishlist-ed several of the books that seem to be popular about him.

Are there any other movies, documentaries, online videos, etc about him or with him being interviewed personally one could point me to?

Last edited by a moderator: May 4, 2017
2. Apr 10, 2009

### Peeter

You can't go wrong with Bohm's book 'Quantum Theory', where a great deal of effort to motivate and discuss the underlying ideas in addition to covering the core physics and mathematics.

The prerequisites for this book are fairly minimal. You do have to be comfortable with Fourier transforms, a bit of calculus, and some classical physics, but won't need to learn any of the Hilbert space algebraic methods, nor bras and kets, or other such methods that make the subject seem scary to a new learner.

3. Apr 10, 2009

### swat4life

Wishlist-ed it!

TX

PS
Is the Aharonov-Bohm Effect that hard to understand? Just curious about whether or not I'm on the "right track" with my current minimal understanding....

4. Apr 10, 2009

### sokrates

a) It is somewhat related to the concept of nonlocality, if you make a hypothetically huge ring (in reality coherence of electrons will be very hard to preserve in such a case because short-wave length phonons might affect electrons in different arms very differently). Others will argue that you can relate non-locality and entanglement whenever there's a quantum interference, so I don't think Aharonov-Bohm deserves a special status. Note that it's more related to the newly emerged field of Mesoscopic Physics and you'll notice that the orthodox QM community here will not be very knowledgeable about AB from a quantum standpoint.

It is basically a "which-way" experiment, and surely is a cousin of double-slit experiment.

b) So, yes, it might be related to the hidden variable concept just like the double-slit.

c) Real world applications... Yes, this is more relevant because you can make transistors out of an equivalent Aharanov-Bohm ring! You probably own about a billion transistors if you own a laptop; so if you don't know what that is, fly to wikipedia and at least read a few pages. You can google Moore's law and its colossal effect on current day technology.

If you are interested in probing more, I could refer you to the first spin based transistor, namely the Datta-Das SpinFET where an "equivalent" process ( just like double-slit and AB effect) is used to provide transistor operation.

5. Apr 10, 2009

### Peeter

I don't know to be honest. Am studying QM sporadically in my spare time, and haven't gotten to this effect. However, you were asking about Bohm material, and this book (Dover ~\$25) is an excellent QM resource which happens to also be written by him (of what I've read so far, chapters 3 & 9 have the meat, and are particularly well sequenced and easy to work through).

6. Apr 11, 2009

### Hans de Vries

I don't often criticize Wiki but this is an example of how not to do it.
http://en.wikipedia.org/wiki/Aharonov-Bohm_effect (at the date of this post)

All of Aharonov Bohm can be derived from the very elementary:

$$\Delta U=e\Phi~~~~~\mbox{and}~~~~~\omega=U/\hbar$$

That is: A charge in a potential field has a higher energy of $\Delta U=e\Phi$
and it has therefor a higher frequency. Special Relativity then gives you
the equivalent for the momentum.

$$\Delta \vec{p}=e\vec{A}~~~~~\mbox{and}~~~~~\vec{k}=\vec{p}/\hbar$$

So a particle on a trajectory which has a non-zero vector potential
acquires extra phase, and different vector potentials on two other-
wise equal paths leads to interference.

Alternatively, you can look at it this way: The particle moving along
the solenoid sees, in it's rest frame, a moving solenoid which is not
electrically neutral but has a dipole field instead due to non-simultaneity.

It has a positive potential $+\Phi$ at one side and a negative potential $-\Phi$
at the other side. The end result is the same: interference, but now
without the vector potential.

Regards, Hans

7. Apr 11, 2009

### swat4life

Insightful - Tx!
Answered my specific questions too ;)

Also gave me something interesting to to look up (i.e. Mesoscopic Physics)

8. Apr 11, 2009

### Creator

Very good Hans...
So a charged particle can acquired momentum without passing through the B or E field ?

....

Last edited: Apr 11, 2009
9. Apr 12, 2009

### Hans de Vries

Well, there are two types of momenta:

The particle can not change velocity: Its inertial momentum can not be changed.
The particle can accumulate more (or less) phase: Its canonical momentum can be changed.

The canonical momentum is defined by the phase change rate: $-i\hbar\partial_x[/tex]. The inertial momentum is defined by the velocity and the invariant mass: [itex]\gamma mv$

Their relation is: $p_c ~=~ p_i + e\vec{A}$

Regards, Hans

10. Apr 13, 2009

### Creator

Yes, Hans, sorry for not making the distinction.
Total momentum is inertial momentum plus electrodynamic momentum ($e\vec{A}$).
Now with regards to AB....Under what situations can electrodynamic momentum be transformed into inertial?

...

Last edited: Apr 13, 2009
11. Apr 13, 2009

### Hans de Vries

In other words: How do you derive the Lorentz Force
from the Aharonov Bohm effect .

If you first write the momenta in 4-vector form:

$$p^\mu_c ~=~ p^\mu_i + eA^\mu$$

recalling that the cannonical momentum is given by the
phase change rates

$$p^\mu_c\, \psi ~=~ i\hbar\partial^\mu\, \psi$$

Now we can write the wave function as an exponential.

$$\psi ~=~ \exp\Big(\, \frac{1}{i\hbar}\int dx^4\, (p^\mu_i + eA^\mu)\, \Big) ~=~ \exp\Big(\, \frac{1}{i\hbar}\mbox{\Huge\phi}\, \Big)$$

Where $\phi$ is the phase of wave function. This means that
the combination of two vectors must be the gradient
of the scalar function $\phi$:

$$p^\mu_i + eA^\mu ~~=~~ \partial^\mu\, \phi$$

It is impossible that a vector is the gradient of a scalar
whenever it has curl. This means that whenever $A^\mu$ has
curl then $p^\mu_i$ must compensate this with an opposite curl
in order to remove the overall curl in $p^\mu_i + eA^\mu$.

This change in the inertial momentum $p^\mu_i$ due to a curl
in $A^\mu$ is of course the Lorentz force.

The 4-curl of $A^\mu$ gives the EM field Tensor:

$$F^{\mu\nu} ~~=~~ \left(\,\partial^\mu A^\nu-\partial^\nu A^\mu\,\right)$$

There is much more detail in my book here:
http://www.physics-quest.org/Book_Lorentz_force_from_Klein_Gordon.pdf

Regards, Hans

12. Apr 14, 2009

### Hans de Vries

The simplest derivation possible of the Lorentz force:

$$p_c^\mu ~~=~~ p^\mu+eA^\mu ~~=~~ -\partial^\mu\,\phi$$

Where p is the inertial momentum depending only on the velocity
and $\phi$ is the phase of the field. The combination of p and eA can
not have any curl since $\phi$ is a scalar. So any curl in eA must
be compensated by an opposite curl in p.

$$-(\partial^\mu p^\nu-\partial^\nu p^\mu) ~~=~~ e(\partial^\mu A^\nu-\partial^\nu A^\mu)$$

One obtains the Lorentz force by using $U_\nu=\partial x_\nu/\partial \tau$ to turn all
the spatial derivatives into time derivatives.

$$-\frac{\partial p^\nu}{\partial x_\mu}\,\frac{\partial x_\nu}{\partial \tau} ~~+~~\frac{\partial p^\mu}{\partial x_\nu}\,\frac{\partial x_\nu}{\partial \tau}~~=~~ eF^{\mu\nu}\,U_\nu$$

The first term cancels, it represents the derivatives of the
invariant mass $p_\nu p^\nu$, so we obtain.

$$\frac{\partial p^\mu}{\partial \tau}~~=~~ eF^{\mu\nu}\,U_\nu$$

Regards, Hans

Last edited: Apr 14, 2009
13. Apr 14, 2009

### Creator

Hans thanks for the insight.

Good point.

Very clever. IOW, since the curl of the gradient of a scalar = 0 , that imposes the condition that the curl of $$(p^\mu_i + eA^\mu) = 0$$. So inertial 'force' arises (due to the curl in $eA^\mu$) in order to maintain the curl free condition. ?

I never would have recognized it.

So we must appeal to a curl free condition to extract 'force' from QFT. Very interesting.

For concreteness lets consider a 'multiply connected' superconducting ring whose center 'hole' is threaded by a time vaying magnetic flux, the flux of which is completely contained in the vacuum hole. IOW, all of the SC condensate is shielded from the B flux.

The Vector potential A from the external source current cannot be shielded and still interacts with the cooper pairs.
Accordingly, $$p^\mu_i + eA^\mu ~~=~~ \partial^\mu\, \phi$$ applies. So according to your above analysis why shouldn't a force arise which changes cooper pair inertial momenta ?

Creator

Last edited: Apr 14, 2009
14. Apr 15, 2009

### Hans de Vries

The vector potential is circular as the superconducting ring. A varying flux implies
an electric field $\textsf{E}=-\partial\vec{A}/\partial t$ around the ring which induces a current in the ring

Now what happens from an Aharonov Bohm point of view?

First of all there is the important fact that Special Relativity forbids spatial integration:

$$\Delta\phi ~~=~~ \int \vec{A}\cdot d\vec{r} ~~~~~\mbox{physically forbidden integration}$$

This would otherwise mean that phase phi instantaneously depends on the vector
potential field at an arbitrary distance. SR does allow time like integration like.

$$\Delta\phi ~~=~~ \int \Phi\cdot dt~~~~~\mbox{physically allowed integration}$$

or (more general) path integration which integrates over time along the trajectory
of the particle.

This means that $\vec{p}_i + e\vec{A} = \partial^i\, \phi$ may not change so that p has to change in
order to compensate for eA. This effect is at the origin of $\textsf{E}=-\partial\vec{A}/\partial t$ which is
part of the Lorentz force.

Regards, Hans

PS. This is all independent of the flux quantization effect which fixes the number
of phase changes of phi around the ring to an integer number.