Although it would not get you past a brick wall and onto Platform 9¾ to
catch the Hogwarts Express, quantum tunneling—in which a particle “tunnels”
through a seemingly insurmountable barrier—remains a confounding,
intuition-defying phenomenon. Now Toronto-based experimental physicists
using rubidium atoms to study this effect have measured, for the first time,
just how long these atoms spend in transit through a barrier. Their findings
appeared in
Nature
on July 22.
The researchers have showed that quantum tunneling is not instantaneous—at
least, in one way of thinking about the phenomenon—despite recent headlines
that have suggested otherwise. “This is a beautiful experiment,” says Igor
Litvinyuk of Griffith University in Australia, who works on quantum
tunneling but was not part of this demonstration. “Just to do it is a heroic
effort.”
To appreciate just how bizarre quantum tunneling is, consider a ball rolling
on flat ground that encounters a small, rounded hillock. What happens next
depends on the speed of the ball. Either it will reach the top and roll down
the other side or it will climb partway uphill and slide back down, because
it does not have enough energy to get over the top.
This situation, however, does not hold for particles in the quantum world.
Even when a particle does not possess enough energy to go over the top of
the hillock, sometimes it will still get to the opposite end. “It’s as
though the particle dug a tunnel under the hill and appeared on the other
side,” says study co-author Aephraim Steinberg of the University of Toronto.
Such weirdness is best understood by thinking of the particle in terms of
its wave function, a mathematical representation of its quantum state. The
wave function evolves and spreads. And its amplitude at any point in time
and space lets you calculate the probability of finding the particle then
and there—should you make a measurement. By definition, this probability can
be nonzero in many places at once.
If the particle confronts an energy barrier, this encounter modifies the
spread of the wave function, which starts to exponentially decay inside the
barrier. Even so, some of it leaks through, and its amplitude does not go to
zero on the barrier’s far side. Thus, there remains a finite probability,
however small, of detecting the particle beyond the barrier.
Physicists have known about quantum tunneling since the late 1920s. Today
the phenomenon is at the heart of devices such as tunneling diodes, scanning
tunneling microscopes and superconducting qubits for quantum computing.
Ever since its discovery, experimentalists have strived for a clearer
understanding of exactly what happens during tunneling. In 1993, for
example, Steinberg, Paul Kwiat and Raymond Chiao, all then at the University
of California, Berkeley, detected photons tunneling through an optical
barrier (a special piece of glass that reflected 99 percent of the incident
photons; 1 percent of them tunneled through). The tunneling photons arrived
earlier, on average, than photons that traveled the exact same distance but
were unimpeded by a barrier. The tunneling photons seemed to be traveling
faster than the speed of light.
Careful analysis revealed that it was, mathematically speaking, the peak of
the tunneling photons’ wave functions (the most likely place to find the
particles) that was traveling at superluminal speed. The leading edges of
the wave functions of both the unimpeded photon and the tunneling photon
reach their detectors at the same time, however—so there is no violation of
Einstein’s theories of relativity. “The peak of the wave function is allowed
to be faster than light without information or energy traveling faster than
light,” Steinberg says.
Last year Litvinyuk and his colleagues published results showing that when
electrons in hydrogen atoms are confined by an external electric field that
acts like a barrier,
they occasionally tunnel through it. As the external field oscillates in intensity, so does the number of
tunneling electrons, as predicted by theory. The team established that the
time delay between when the barrier reaches its minimum and when the maximum
number of electrons tunnel through was, at most, 1.8 attoseconds (1.8 x
10-18 second). Even light, which travels at about 300,000 kilometers per
second, can only travel over three ten-billionths of a meter, or about the
size of a single atom, in one attosecond. “[The time delay] could be zero,
or it would be some zeptoseconds [10-21 second],” Litvinyuk says.
Some media reports controversially claimed that the Griffith University
experiment had shown tunneling to be instantaneous. The confusion has a lot
to do with theoretical definitions of tunneling time. The type of delay the
team measured was certainly almost zero, but that result was not the same as
saying the electron spends no time in the barrier. Litvinyuk and his
colleagues had not examined that aspect of quantum tunneling.
Steinberg’s new experiment claims to do just that. His team has measured how
long, on average, rubidium atoms spend inside a barrier before they tunnel
through it. The time is of the order of a millisecond—nowhere close to
instantaneous.
Steinberg and his colleagues started by cooling rubidium atoms down to about
one nanokelvin before coaxing them with lasers to move slowly in a single
direction. Then they blocked this path with another laser, creating an
optical barrier that was about 1.3 microns thick. The trick was to measure
how much time a particle spent in the barrier as it tunneled through.
To do so, the team built a version of a so-called
Larmor clock
using a complicated assemblage of lasers and magnetic fields to manipulate
atomic state transitions. In principle, here is what happens: Imagine a
particle whose spin points in a certain direction—think of it as a clock
hand. The particle encounters a barrier, and inside it is a magnetic field
that causes the clock hand to rotate. The longer the particle stays within
the barrier, the more it interacts with the magnetic field, and the more the
hand rotates. The amount of rotation is a measure of the time spent in the
barrier.
Unfortunately, if the particle interacts with a strong enough magnetic field
to correctly encode the elapsed time, its quantum state collapses. This
collapse disrupts the tunneling process.
So Steinberg’s team resorted to a technique known as weak measurement: An
ensemble of identically prepared rubidium atoms approaches the barrier.
Inside the barrier, the atoms encounter, and barely interact with, a weak
magnetic field. This weak interaction does not perturb the tunneling. But it
causes each atom’s clock hand to move by an unpredictable amount, which can
be measured once that atom exits the barrier. Take the average of the
clock-hand positions of the ensemble, and you get a number that can be
interpreted as representative of the correct value for a single atom—even
though one can never do that kind of measurement for an individual atom.
Based on such weak measurements, the researchers found that the atoms in
their experiment were spending about 0.61 millisecond inside the barrier.
They also verified another strange prediction of quantum mechanics: the
lower the energy, or slower the movement, of a tunneling particle, the less
time it spends in the barrier. This result is counterintuitive, because in
our everyday notion of how the world works, a slower particle would be
expected to remain in the barrier for a longer stretch of time.
Litvinyuk is impressed by the measurements of the rotation of the clock
hand. “I see no holes in this,” he says. But he remains cautious. “How,
ultimately, it relates to the tunneling time is still up for
interpretation,” he says.
Irfan Siddiqi, a quantum physicist at the University of California,
Berkeley, is impressed by the technical sophistication of the experiment.
“What we are witnessing now is quite amazing, in that we have the tools to
test all of these philosophical musings [of] the last century,” he says.
Satya Sainadh Undurti, a co-author of Litvinyuk’s 2019 study who is now at
Technion-Israel Institute of Technology, agrees. “The Larmor clock is
certainly the right way to go about asking tunneling time questions,” he
says. “The experimental set up in this paper is a clever and clean way to
implement it.”
Steinberg admits that his team’s interpretation will be questioned by some
quantum physicists, particularly those who think weak measurements are
themselves suspect. Nevertheless, he thinks the experiment says something
unequivocal about tunneling times. “If you use the right definitions, it’s
not really instantaneous. It may be remarkably fast,” he says. “I think
that’s still an important distinction.”