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The Story of the Electron That Was Affected by Fields That Weren’t There

The Story of the Electron That Was Affected by Fields That Weren’t There

anomalous quantum Hall effect, non-Abelian fractional quantum Hall effect, Abelian group, non-Abelian group, symmetry, magnetic field, electromagnetic field, electromagnetic potential, gauge potential, gauge theory, Lorentz invariance, Maxwell's equations, Aharonov-Bohm effect, David Bohm, Yakir Aharonov, commutativity, noncommutativity, solenoid, doublt-slit experiment, Faraday rotator, interference pattern,

In 1959, two physicists, Yakir Aharonov and David Bohm, that were working with quantum mechanics stumbled upon a strange problem. They found that the math of quantum mechanics allowed an electrically charged particle – such as an electron – to ‘feel’ the effects of an electromagnetic field in a region where there were no electric or magnetic fields.

This bizarre phenomenon is called the Aharonov-Bohm effect. It illustrates how charged particles can experience a force even when there seems to be nothing around them to deliver that force. More specifically, it shows how simply working out where the electric and magnetic fields are present in space isn’t enough to figure out how charged particles will behave in that space. You also need to figure out, with help from quantum mechanics, how the particle is affected by fields outside that space.

This is why the Aharonov-Bohm effect has been called one of the “seven wonders of the quantum world” by New Scientist magazine.

Last month, physicists reported that they’d performed an experiment in which they observed the Aharonov-Bohm effect in a kind of system in which it hadn’t been seen before.

It’s an exciting development in the field because it paves the way for future work in other fields, including a special class of powerful quantum computers and other quantum mechanical effects that will help scientists better understand the world of quantum particles.

To begin understanding it, let’s take a trip to the 19th century.

Force fields

Electromagnetism is one of the most impressive edifices of the physical sciences. It’s a set of laws governing the dynamics of electricity, magnetism and the behaviour of charged particles.

In the 18th and 19th centuries, scientists, engineers and mathematicians pieced together the classical theory of electrodynamics. These efforts culminated in an elegant synthesis by James Clerk Maxwell in 1861.

Michael Faraday provided the physical picture underlying this theory: he imagined that electric and magnetic fields permeated all parts of the universe, and guided the motion of charged particles along ‘lines of force’. Maxwell built on this idea and created a set of mathematical equations that describe how electric and magnetic fields are related to each other, how their presence varies through space and time, and how they are produced and affected by charged particles.

When iron filings are placed near a magnet, they experience a force that aligns them along the magnetic field. Photo: Wikimedia Commons
When iron filings are placed near a magnet, they experience a force that aligns them along the magnetic field. Photo: Wikimedia Commons

Faraday’s idea of a ‘force field’ occupies a central role in fundamental physics. For example, physicists know today that the strong nuclear, weak nuclear and electromagnetic interactions – three of the four types of interactions observed in nature – are the result of their corresponding force fields exchanging little packets of energy in the form of particles.

This background is essential to understand the new experiment, setup by a team of physicists from the US, China and Croatia. They successfully synthesised the Aharonov-Bohm effect in a non-Abelian system, a phenomenon first predicted in 1975.

* * *

About fifty years ago, physicists and mathematicians made sense of how electric and magnetic fields interact to explain the things classical electrodynamics was created to explain – but they noticed something curious.

Consider an analogy: Selvi and Shankar have been asked to compete to see how high they can throw a tennis ball. Selvi picks up the tennis ball and flings it straight up towards the sky, and an observer records the height it reaches. But Shankar is somewhat of an ass and decides to take the elevator to the top floor of their office building, and throws the ball from there. Later, he claims that since his ball reached further with respect to sea level, he ought to win the contest.

It’s clear that Shankar is being unfair. The question ‘Who can throw the ball higher?’ is a physical question that attests to the player’s strength. It shouldn’t depend on which high-rise they decide to climb before playing. So what is relevant here is the change in the height of the ball the player is able to effect.

Similarly, physicists noticed that it was more productive to instead track a third quantity, whose changes contained information about electric and magnetic fields. This auxiliary quantity is called the gauge potential.

Roughly speaking, how a gauge potential changes in space and time tells physicists something about the electric and magnetic fields. The potential is not uniquely determined by physical effects. There is a gauge freedom in this description that allows physicists to change the gauge potentials without affecting the results of experiments.

Aharonov-Bohm phases

Now, while physicists could formulate classical electromagnetism in terms of fields or in terms of potentials, they saw potentials as a mathematical curiosity: they’re fun but they’re not essential.

Let’s recall the Aharonov-Bohm effect here: where a charged particle is affected by electromagnetic fields even when it’s not supposed to because there’s no electromagnetic field around it.

The electromagnetic potential here is a gauge potential.

Also read: Catching a Glimpse of S. Pancharatnam in History’s Precisest Chemical Reaction

The simplest way to see this effect is with a solenoid: an electrical conductor coiled like a helix (remember the telephone cord of the 1990s?). When you pass a current through the helix, it creates a magnetic field passing through the axis of the helix; this field is confined to the axis and isn’t manifested anywhere beyond the solenoid.

A solenoid. Image: Wikimedia Commons
A solenoid. Image: Wikimedia Commons

Normally, you’d conclude that since there’s no magnetic field outside the helix, an electron moving outside the helix shouldn’t be affected by such a field. But Aharonov and Bohm 1 showed that a charged particle, like an electron, could in fact keep track of the absent field’s gauge potential, and even be affected by an external field after it has been removed.

* * *

We’ve been taught in school that quantum particles like protons and electrons are the stuff that matter is made of – but the actual picture isn’t so simple. What you see as a particle moving in a fixed pattern, physicists see as a diffuse cloud of probability. The denser the cloud in a particular part of space, the more likely the particle will be found there.

Mathematically, physicists describe this cloud (in part) as a number called the phase; its value can’t be determined by experiments. Aharonov and Bohm found that when a charged particle moves through a region that has some electromagnetic potential – even though there’s no electromagnetic field there – the particle will acquire a phase. This phase depends on the electromagnetic potential and the path the particle takes through this region.

How do we test this idea?

Yakir Aharonov (left) and David Bohm. Photos: פרס א.מ.ת and Karol Langner (resp.), Wikimedia Commons
Yakir Aharonov (left) and David Bohm. Photos: פרס א.מ.ת and Karol Langner (resp.), Wikimedia Commons

Aharonov and Bohm figured a double-slit experiment with a solenoid placed between the two slits would do the trick. When electrons are beamed at the two slits and no current is passed through the solenoid (‘off’), the electrons will pass through the slits like waves and produce the famous interference pattern. But when the electromagnet is turned ‘on’, Aharonov and Bohm concluded that the electrons will now acquire a phase related to the gauge potential – even though there’s no electromagnetic field outside the helix – and thus the interference pattern would shift.

An illustration showing the double-slit experiment (i.e. without the solenoid). Credit: Wikimedia Commons

This phase shift is known as the Aharonov-Bohm phase. It’s related to the flux of the magnetic field along the axis of the helix. Physicists first observed it in an experiment in the 1980s.

What’s non-Abelian, though?

According to the popular theory physicists use to understand how particles interact 2, all interactions are mediated by fields, each with its corresponding gauge potential. These potentials are associated with two broadly defined symmetries. They’re called Abelian and non-Abelian.

Imagine a circle on a piece of paper with a point marked in red. Now imagine rotating this circle about its centre by an angle of xº. The red point will have moved by a certain distance. Next, rotate the circle by yº. The red point will have moved again.

Now imagine performing the same rotations, but rotate by yº first and then by xº. The result of both these processes – xº followed by yº and yº followed by xº – looks the same, i.e. the red point sits at the same place in the end. Mathematicians call this property, of the order of processes being irrelevant to the final state, commutativity. Symmetries that have this property are called Abelian.

Now imagine a book, held up with its front cover facing you. Let’s call the axis along its spine the z-axis; let’s call the line pointing towards you and perpendicular to the front cover the x-axis. Say you rotate the book 90º about the z-axis and then 90º degrees about the x-axis. Then imagine starting with the same configuration, with the front cover facing you, and performing the same rotations but in the opposite order. The book would be oriented differently this time! Physicists call this sensitivity to the order in which the rotations are performed non-commutativity, and the symmetries that have this property are called non-Abelian.

Most of us know commutativity in the multiplication of real numbers: 5 × 3 = 3 × 5 = 15. Most of us also know about matrices, and probably still remember matrix multiplication. But unlike with real numbers, matrix multiplication is generally non-commutative.

* * *

The non-Abelian Aharonov-Bohm phase is a non-commutative – i.e. non-Abelian – phase. So instead of being just a number, it’s a matrix.

In their newly published experiment, the scientists from the US, China and Croatia used fibre optic cables to observe the Aharonov-Bohm effect in a non-Abelian system for the first time.

To have non-commutativity, we need to have multiple axes along which to rotate – or reorient – the system, like with the book.

The scientists did this by passing light through a special crystal placed in a magnetic field, and then modulating the phase of light using electrical signals. They inspected the light’s interference pattern for signs that it had undergone both these rotations (but in opposite sequences). They found their results of their experiment agreed with the theory.

At long last, physicists have been able to build a setup that mimics the behaviour of non-Abelian gauge fields, and have confirmed that they supply the expected non-commutative phases when subjected to an Aharonov-Bohm test.

“The finding allows us to do many things, opening the door to a wide variety of potential experiments, including classical and quantum physical regimes, to explore variations of the effect,” Yi Yang, a graduate student at the Massachusetts Institute of Technology and one of the members of the team, told Sci-News.

Madhusudhan Raman is a postdoctoral fellow at the Tata Institute of Fundamental Research in Mumbai.

Vasudevan Mukunth is science editor, The Wire.

  1. And Werner Ehrenberg and Raymond Siday a decade before them

  2. Through the strong nuclear, weak nuclear and electromagnetic interactions

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