Measuring time in a quantum tunnel

In the quantum mechanical tunnelling effect, particles take a few attoseconds to overcome an energy barrier

July 26, 2017

Harry Potter can do a lot of things that are impossible for mere mortals - he can even walk through walls: To reach platform 9 3/4, at which the train to Hogwarts School of Witchcraft and Wizardry stops, he and his classmates slip through a wall between platforms nine and ten. This impossible feat in real life is normal in the weird world of quantum physics. Particles such as electrons are actually able to cross insuperable energy barriers. Physicists refer to this effect as quantum-mechanical tunnelling. Now, researchers at the Max Planck Institute for Nuclear Physics in Heidelberg have been able to show, for the first time, that it takes electrons a finite amount of time to tunnel. Although the phenomenon has been known for nearly a century, it had been unclear whether the process of electron tunnelling takes time or is instantaneous.

Over or through the energy wall? Classical particles can only overcome energy barriers if energy is expended to lift them above it. Quantum particles, on the other hand, have a probability of crossing the wall even if they do not have enough energy to do so by tunnelling through the obstacle.

The tunnelling effect plays a role not only in sophisticated experiments by quantum physicists but also in well-known processes such as radioactive decay. Inside an atomic nucleus, protons and neutrons are bound together by a strong field that confines them within an energy barrier. The effect is analogous to peas contained in a bowl. According to the laws of classical physics, the particles are unable to overcome the energy wall and escape from the nucleus. However, in the world of quantum physics, there is a certain probability that one of the particles will be able to move through the wall. Having crossed the energy barrier, it is outside the nucleus, which will then have undergone radioactive decay.

In this randomly occurring nuclear process, it cannot be determined whether the particle takes time to tunnel through the wall set up by the nucleus. Even textbooks are silent on this point. Now the Heidelberg-based group has deliberately produced a tunnelling effect and measured its duration.

Tunnelling may be instantaneous or take a finite amount of time

The researchers used atoms to study the tunnelling effect. Within atoms, the nucleus generates an electric field, which confines the electrons surrounding it. The electrons find themselves at the base of a very thick energy wall, as it were. Consequently, the probability of tunnelling through the wall is virtually zero, and the atom is stable. However, when the physicists shoot a short laser pulse at the particles, their periodically oscillating electric field becomes superimposed on that of the nucleus. This momentarily reduces the width of the wall so that the probability of a bound electron tunnelling through the wall becomes very high. If the electron does cross the wall, it flies off, guided by the laser field, and can be detected by a sensor.

However, measuring the tunnelling time in such an experiment takes some effort. A group of theoreticians led by Karen Hatsagortsyan in Christoph Keitel’s Department investigated this process in purely theoretical terms. The simplest way of looking at the phenomenon, known as the simple-man model, assumes that tunnelling takes no time at all, so that the electron appears at the tunnel exit instantaneously and at zero velocity. A second possibility is that the electron takes a finite amount of time to tunnel through the energy barrier. The latter theory was published in 1955 by the physicist and Nobel laureate Eugene Wigner.

Tunnelling could take up to 180 attoseconds

Calculating how long an electron would take to tunnel in their experimental setup according to Eugene Wigner’s model was no easy matter. “The quantum mechanical calculations are very complex,” says Christoph Keitel. “We had to refine Wigner’s model and take very specific details of our experiment into consideration.”

As the results show, it takes a finite – albeit extremely short – amount of time for an electron to sprint across the barrier. “Our solutions show that if the Wigner model holds, an electron takes 80 to 180 attoseconds to tunnel through the barrier in the range of laser intensities we used,” says Enderalp Yakaboylu, who performed the calculations. An attosecond is one billionth of a billionth of a second.

Circularly polarized light rotates the “energy pot” of an atom

The tunnelling effect in a circularly polarized laser field: Left: The electron is located in a potential funnel produced by the electric field of the atomic nucleus and the laser pulse. Consequently, the funnel rotates (anticlockwise in this case). After laser stimulation (middle), the electron tunnels through the barrier. Right: The simple-man and Wigner models predict different electron trajectories.

Measuring such rapid processes is no less tricky than performing the calculations. It requires extremely accurate instruments, such as those which Robert Moshammer has constructed at the Max Planck Institute for Nuclear Physics in recent years. In addition, the researchers had to use a clever trick to determine whether the simple-man or the Wigner model describes the physical reality: they bombarded atoms with laser pulses that were circularly polarized. This means that the electric field of the light wave not only varies in a sinusoidal pattern but also rotates. When such a rotating laser field is superimposed on the electric field of an atom, the entire “energy pot” in which the electrons are located rotates.

At the same time, the laser pulse stimulates one of the electrons, thus starting the clock. The particle then tunnels through the energy barrier and exits at a certain point. If the electron takes time to cross the barrier, the energy pot will have rotated a bit further since the start of the tunnelling process, and the electron will emerge at a different location and at a different angle than if tunnelling occurred instantaneously. The object then was to measure this infinitesimal angular difference.

Trajectory angles differ for argon and krypton

“One of the biggest challenges was to measure the angle precisely,” says Thomas Pfeifer. It is extremely difficult to measure the angle of the electron trajectory in absolute terms. To overcome this problem, the physicists came up with a clever trick. They used a mixture of argon and krypton atoms, whose barrier height and tunnel length differ slightly.

In the simple-man model, this difference plays no role. Both atomic species would behave identically, since the tunnelling time is always zero. However, if the electrons obey the Wigner model, they would differ in the time it takes them to sprint through the quantum tunnel, and this, in turn, would be expressed in slightly divergent electron trajectory angles. The angle between these two trajectories can be measured much more accurately than the absolute angle of a single trajectory.

“We bombarded the gas mixture with 3000 pulses per second and were able to analyze the atomic species in which the tunnelling effect was triggered,” explains Nicolas Camus, who carried out the measurements. The results were unambiguous: the data quantitatively confirm the Wigner model and are inconsistent with the instantaneous tunnelling process predicted by the simple-man model. Thus, the physicists in Heidelberg have ended a decades-long debate and have closed a gap in quantum physics textbooks.


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