### Aharonov-Bohm effect’s analogue in water waves

Rather indirectly through Claude (a not-so-short story there) I learned of a paper by Michael Berry and collaborators titled “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue” (a copy can be found on Berry’s website at number 96). It is among the most pleasurable papers I have read in physics to date.

To understand why this Chandrasekhar-esque paper tickles me so, I need to explain a little bit of the physics.

The Aharonov-Bohm effect
The Aharonov-Bohm effect is one of the classic illustrations of the difference between classical and quantum mechanics. More importantly though, it also demonstrated that the use of the vector magnetic potential is not a mere computational convenience, but that the potential also manifests itself in physical effects.

To begin with, consider the classical motion of a charged particle in an electro-magnetic field. The Lorentz-force formula gives that the force acting on a charged particle is

(Lorentz) $\displaystyle \vec{F} = q\vec{E} + q \vec{v}\times\vec{B}$

where $\vec{F},\vec{E},\vec{B}$ are respectively the force experienced by the charged particle, the ambient electric field, and the ambient magnetic field, $q, \vec{v}$ are the charge and velocity of the particle, and $\times$ as usual denotes the cross product. In a region of zero electro-magnetic field then, the charged particle should feel no force.

(As a side note, while the discussion below centers on the magnetic Aharonov-Bohm effect, which is the one most often discussed in textbooks and expository articles, there is also an electric variant. In both cases the trick is to create a region in which the electric/magnetic fields are zero but not the scalar/vector potentials.)

Now, consider an infinitely long magnetic solenoid. If we turn on the current, the electromagnetic field generated will be confined to inside the solenoid (in practice, we can make the magnetic field outside the solenoid very, very small just by making the solenoid really, really long). Now consider a classical beam of electrons shooting past the solenoid on the outside. Then regardless of whether the current is on or off inside the solenoid, since the region in which the electrons pass through contains no electromagnetic field, the motion of the electron is unchanged.

Now let us try to run the double-slit experiment with the electron, and forget for a moment that the interference patterns are caused by quantum effects. Forget that there is an interference pattern to even start with. Just consider running the double-slit experiment with an electron source capable of shooting one electron at a time. On the screen on the other side, we see the electrons arriving one by one. After a large number of electrons have been fired, a pattern (or a lack of one) should emerge on the other side. At the moment we are not particularly concerned with the exact pattern (purely classical physics predicts two concentrated dots corresponding to the two slits, quantum physics predicts an interference pattern). Suffice to know that some pattern will emerge.

Now suppose that in the area between the two slits, we placed our infinitely long solenoid. If electromagnetism were to interact with a charged particle purely classically, then our earlier analysis tells us the following: regardless of whether the current is switched on for the solenoid, the “pattern” that emerges from the double slit experiment will be the same. But that was not what was observed in experiments. Following the predictions of Aharonov and Bohm (and by Ehrenberg and Siday earlier), Osakabe et al (following Olariu and Popèscu) shown that, in fact, the quantum interference patterns observed in the double slit experiments “shift” depending on the current in the solenoid!

This observation confirms the Aharonov-Bohm intuition, that while the electromagnetic fields are, in some sense, the observables of the theory, the way they interact suggest that the potentials are in fact the more fundamental objects. To wit, while the magnetic field outside the solenoid remains the same when the current is turned from off to on, the magnetic vector potential becomes non-zero. And it is this change that effects the observed shift of interference patterns.

Here I am going to skip a bit of the physical motivation and the derivation of the equations, and just go straight to the mathematics. We are considering a charged particle minimally coupled to a background field (we assume the effect of the charged particle is small enough so it doesn’t give rise to back-reactions). Mathematically a particle (with no charge) is modeled as the section of some vector bundle over Minkowski space (for now ignore gravity). It obeys some equation of motion. In the non-relativistic case, a scalar/spin-zero particle is modeled by the Schroedinger equation. A particle with spin is modeled by the Pauli equation (but I am getting ahead of myself a bit here, since spin does not manifest unless the particle is coupled to a magnetic field; cf. the Stern-Gerlach effect). In the relativistic case, a scalar particle is modeled by the wave equation, while a spin-one-half particle is given by the Dirac equation. (In the relativistic case and assuming the particle has no mass, there is also a more general description using Penrose spinors.)

To minimally couple a background gauge field (say electro-magnetic, but we can also deal with Yang-Mills) to the model, we assume that the the background gauge group is $\mathfrak{G}$ with corresponding Lie algebra $\mathfrak{g}$. Whereas in the uncharged case the particle is taken to be a section of some vector bundle over $\mathbb{R}^{1,3}$, we now treat the particle as a section of some associated vector bundle to the principal $\mathfrak{G}$-bundle over $\mathbb{R}^{1,3}$. From the point of view of an analyst (such as myself), the more useful description is the local trivialization of the bundle. So if an uncharged particle is locally represented as a $V$-valued function on Minkowski space (where $V$ denote some finite-dimensional vector space [usually we also ask it to have an inner product]), the charged particle can be represented as a $V\otimes \mathfrak{g}$ valued function. Now, given a connection on the original vector bundle and a $\mathfrak{g}$-connection, the associated bundle naturally inherits a corresponding connection. In local coordinates, if $\nabla$ denotes the trivialization of the connection on the original vector bundle and $A$ denotes the trivialization of the $\mathfrak{g}$ connection one-form, we can write the charged connection as $\tilde{\nabla} = \nabla + A$. And the equation of motion for the charged particle can be obtained from the uncharged version by simply replacing all covariant derivatives relative to $\nabla$ by their charged counterparts.

As a simple example (and the one used for the Aharonov-Bohm effect), consider a charged particle with no-spin in non-relativistic quantum mechanics. The uncharged equation is (dropping physical constants)

$\displaystyle i \frac{\partial}{\partial t} u = - \triangle u + Vu$

where $V$ denotes some ambient potential and $\triangle = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} = \nabla\cdot\nabla$ is the Laplace operator. If we add a background electro-magnetic field and impose a charge on the particle, we modify the equation as follows. First notice that corresponding to the electric field $E$ and magnetic field $B$ is the four-potential $(\phi,A')$ where $\phi$ is a scalar and $A'$ is a vector. The four-potential is (non-uniquely) defined by $\nabla\phi = E, \nabla\times A' = B$, the non-uniqueness represents “gauge freedom”.

In the mathematical language given above, Maxwell theory corresponds to the gauge group $U(1)$ whose Lie algebra is isomorphic to $\mathbb{R}$. In a local trivialization, the connection one-form can be expressed as an imaginary-valued one-form $A$, which in the standard coordinates of Minkowski space can be written as $A = i(\phi,A')$. The physical fields $E,B$ are now just components of the curvature tensor associated to the connection $A$. So the transformation of the equation of motion by the minimal coupling is simply replacing $\frac{\partial}{\partial t} \to \frac{\partial}{\partial t} + i\phi$, and $\nabla \to \nabla + iA'$. Which implies that the equation of motion becomes

(charged Schrodinger) $\displaystyle i \frac{\partial}{\partial t} u = - \triangle u - i \nabla\cdot(A'u) - i A'\cdot\nabla u + A'\cdot A' u + \phi u + V u$

Now let us quickly illustrate (on a heuristic level) the Aharonov-Bohm effect. The important idea is that the coupling of a charged particle to the gauge field is not through the physically manifest curvature fields, but through the connection forms. Now assume $U$ is a simply connected domain (in space). Assume $E = B = \phi = 0$ in this domain. But assume $A' \neq 0$ is constant in time. The requirement that $\nabla\times A' = B = 0$ and simply-connectedness implies that there exists (well-defined up to a constant) a function $a$ on $U$ such that $\nabla a = A'$. Now, given a solution $u_0(x,t)$ to the uncharged Schrodinger equation in the domain

$\displaystyle i \partial_t u_0 + \nabla\cdot\nabla u_0 = V u_0$

Now consider the function $u(x,t) = u_0(x,t) e^{-i a(x)}$. A computation shows that

$\displaystyle (\nabla + i A') u(x,t) = [\nabla u_0(x,t) + i A' u_0(x,t) - u_0(x,t) i \nabla a(x) ]e^{-i a(x)} = (\nabla u_0(x,t))e^{-ia(x)}$

Basically, by inserting a the phase factor $e^{-ia}$ we are able to compensate (locally) for the gauge connection. This is because the region is locally gauge-flat ($B = 0$) so the fact that $A' \neq 0$ is due to a bad trivialization/gauge choice. Notice that the above identity implies that

$\displaystyle i\partial_t u + (\nabla + i A')\cdot(\nabla + i A') u = Vu$

i.e. $u$ is a solution in $U$ of the charged Schroedinger equation.

Observe that $a$ can be obtained by fixing it to be zero at some point in $U$ and then path-integrate $A'$. Simply connectedness implies that any two paths with shared endpoints are homologous, and $A'$ being curl free implies that path integrals over homologous paths take the same value.

Now let us consider the “particle” picture of the double-slit experiment. Assume (rather unreasonably) that there is a wave-packet solution (spatially localized at every time) to the uncharged Schroedinger equation that corresponding to “the electron passing through one, but not the other of the slits” for each of the slits. In particular, we ask that the wave-packets start “centered” around some point $x_0$ at time $t = 0$ and end up around some point $x_f$ at time $t = t_f$. (The strategy here is to study the interference pattern at point $x_f$ due to incidental waves originating at $x_0$.)

Just to repeat, we can then assume that we have two solutions $\tilde{u}_1, \tilde{u}_2$ of the uncharged Schroedinger equation where $\tilde{u}_1$ is spatially supported away from slit number 1, and similarly for $\tilde{u}_2$ and slit number 2. This means that, in particular, the supports of $u_i$ are each on a domain $U_i$ which is gotten by removing two non-intersecting rays from $\mathbb{R}^3$. And in particular they are simply connected.

Now, turning on the the magnetic potential, corresponding to each $\tilde{u}_i$ is a solution $u_i$ of the charged Schroedinger equation, given by $u_k = \tilde{u}_ke^{ia_k}$, using the construction outlined above. In particular,

$\displaystyle u_k(x_f,t_f) = \tilde{u}_k(x_f,t_f) e^{i \int_{\gamma_k} A'}$

where $\gamma_k$ is a path connecting $x_0$ and $x_f$ that avoids the slit $k$. This implies that the observed shift in interference pattern should be the same as the difference in phase-shifts, which is simply given by the circulation integral

$\displaystyle \mbox{phase shift } = \oint_\gamma A'$

where $\gamma$ is any path that winds around the solenoid (so necessarily passing each of the slits) once.

The topological interpretation of the classic Aharonov-Bohm effect is that by imposing the double-slit, the spatial topology becomes no longer simply connected. The de Rham theorem then tells us that there may exist closed but not exact one-forms.

A geometric version
(No, I am not going to talk about Foucault’s pendulum and Berry Phase.)

A simpler, geometric version can be had in the realm of Riemannian geometry. We use the intuition that the magnetic field $B$ in the Aharonov-Bohm effect, which vanishes outside a compact set, geometrically represents curvature. So let us construct a Riemannian manifold that has vanishing curvature outside of a compact set.

The easiest example can be found by the following physical construction: take a piece of paper and cut a slit on it. Pull the two sides of the slit together so they overlap. Now we have a cone. Except for the tip of the cone where the geometry becomes singular (it is a pointy end, so it is not smooth), each little patch of the cone is exactly a patch on our flat piece of paper, and thus is locally flat. If the pointy tip bother you, we can smooth it out by cutting off the tip and gluing on a spherical shell. (Whether there is a tip or not is irrelevant for this effect, so we’ll ignore the singularity there for the time being.) So now we have a manifold that is locally flat outside of a small set around the tip.

Now, recall from elementary non-Euclidean geometry that what characterizes Euclidean geometry in the large is the parallel line axiom, which asserts that “given a line $\ell$ and a point $p$ not on the line, there exists a unique line $\ell'$ through $p$ that does not intersect $\ell$; this new line is called the parallel line through the point $p$ to the given line $\ell$.” In the context of differential/Riemannian geometry, however, a manifold is locally Euclidean if on every small patch it has no curvature (i.e. in every small patch it looks exactly like a small patch in Euclidean space). The geometric version of Aharonov-Bohm effect now says that on a non-simply connected region, locally Euclidean is not enough to ensure globally Euclidean. And we can show this with a physical example.

Now start with a fresh sheet of paper. Draw a bunch of parallel lines on it. (Or you can start with a sheet of graph paper.) Again, cut a slit and fold the paper up into a cone. And then cut off the tip (to remove the non-flat points). the cone with the tip missing is now a locally flat manifold. But you see that “parallelism” becomes destroyed by the non-simply connected topology. If you trace out two line segments that starts parallel, if both of the line segments pass to the same side of the hole, then they stay parallel forever. On the other hand, if you trace out two line segments that starts parallel such that their extensions passes on opposite sides of the hole, they will eventually intersect! Furthermore, you will observe that on a given cone, any such two lines will always intersect at the same angle. The angle is given precisely by the angle of overlap in the construction of the cone. This “angle-defect” is the analogue of the “phase-shift” in the Aharonov-Bohm effect.

Water waves
What Berry et al did was to draw a physical analogue, and not a geometric analogue.

I am not going to do the mathematics here, but just appeal to common sense. If one imagines that the gauge connection $A'$ sort of represents the motion of “aether” (the medium in which the particle/wave propagates), then the following analogy is not too hard to see. Take a bathtub filled with water. Pull the stopper. The water forms a vortex/funnel at the drain and a “hole” forms in the bulk of the water, making the medium no longer simply connected. As the water goes down the drain it whirls around the drain hole. But inside the bulk of the water, the whirl is sort of uniform: there are no additional turbulence or eddies forming in the swirling water. This lack of additional eddies gives that locally the flow of water is irrotational, the natural analogue of geometrically flat.

Now envision a wave traveling across the bathtub. Since locally the water is irrotational, in so far as the wave is concerned, locally it propagates as if it is traveling on still water. As it crosses the drain hole however, on one side, it will be accelerated by the spin of the water, on the other, it will be slowed down. This means that as the wave crosses the drain hole, the crests and troughs from the two sides no longer match up. They interfere!

Now what really impressed me about Berry et al’s paper is two-fold. Firstly, they made the above Gedanken experiment precise in the theoretical way. They take the mathematical equations that models surface water waves on a irrotational fluid, and imposes that the fluid is swirling around like in a bathtub. They then solve the linearized version of the equations and showed that, indeed, a phase difference appears between the two paths around the drain hole. But what makes it even cooler is that secondly, they did the experiment. They built a bathtub with a drain, and a water inlet that minimizes effects on vorticity of the water, so that water level can be maintained without disturbing the experiment. They built a paddle on one end of the bathtub. And the filled the bathtub with water, unstopped the drain, let the water start swirling, and then used the paddle to create a surface wave. And they photographed the results. From the photographs they calculated the speed of the water swirl and the speed of the surface waves, and they were about to conclude that the physical experiment agrees with the theoretical prediction! Ain’t that something.

I highly recommend taking a look at the paper (I gave a link to it in the beginning of this post), even if just to see the wonderful photos.