New Results in Quantum Tunneling vs. The Speed of Light (Script) (Patreon)

Paradoxically, the most promising prospects for moving matter around faster than light may be to put a metaphorical brick wall in its way. New efforts in quantum tunneling - both theory and experiment - show that superluminal motion may be possible, while still managing to avoid the paradox of superluminal signaling.

Quantum tunneling is one of the weirder phenomena in the generally very weird world of quantum mechanics. It describes how quantum particles are able to move across seemingly impenetrable barriers - for example, when atomic nuclei decay.  But it’s not just the Houdini-like power that makes the quantum world weird - it’s also that the tunneling motion may move particles faster than they could travel if the barrier wasn’t there - and even faster than light could traverse the same distance.

Now we covered quantum tunneling a long LONG time ago - in fact it was the first video we did on quantum mechanics. But things have actually evolved in the 5 years since. I mean, we’ve learned a lot on this show and so we can dig deeper into the FTL aspect of quantum tunneling. But there’s also new science, and so today we’re going to look at a new theoretical result and a new experiment that are bringing us closer to understanding the superluminal prospects of quantum tunneling.

First, a quick recap of quantum tunneling. Imagine you’re driving a car towards a steep hill when the engine cuts out. You can still make it over the hill if you have enough speed - enough kinetic energy to see you to the top. But if the car isn’t moving fast enough then it’ll inevitably slow down and roll back. There’s nothing in the laws of physics that could allow you to reach the other side of the hill. Well, nothing in the laws of classical physics anyway.

A similar thing happens in the world of quantum mechanics, where particles are pushed and pulled by the fundamental forces, forming energetic hills and valleys - a landscape of so-called potential barriers. For example, the protons and neutrons in an atomic nucleus are held in the potential barrier of the strong nuclear force. If one of these particles had enough energy it could punch through that barrier. Fortunately for the stability of atoms, nucleons mostly remain trapped. Mostly. In radioactive decay, particles that should never have enough energy to escape the nucleus are found to leak out.

This is quantum tunneling. The key to the escape is quantum uncertainty. Between observations, quantum particles don’t have well defined properties - and that includes their positions. We represent the location of, say, a proton in a nucleus as a wavefunction. It’s an abstract wave that encodes the information of where the proton might be. Upon measurement, or upon interaction with another particle, the proton can end up anywhere within that wavefunction, with some locations more likely than others.

To understand what happens when a proton bounces around inside a nucleus, we need to see how its wavefunction evolves according the the Schrodinger equation - which is just the equation of motion of wavefunctions. This equation tells us that the wavefunction is mostly reflected or scattered back by the wall of the nucleus. But the Schrodinger equation is very clear that this isn’t the only thing that can happen. Due to the blurred-out nature of the wavefunction, a small part of it leaks through to the other side. The proton ends up being simultaneously reflected back AND transmitted through the barrier. Now the latter is very improbable - only as likely as the tiny fraction of the wavefunction that peaks through the barrier. But improbable isn’t impossible, and so if you then observe the nucleus, it’ll “collapse” into one of those two states - either business as usual, or a nuclear decay.

We see tunneling everywhere, Radioactive decay of course, but this quantum tunneling drives many other important processes. It’s necessary for the nuclear fusion reactions that power the sun, in some biological processes, and it’s even a critical part of the working of transistors and other electronic components. But even with the ubiquity of this phenomenon, we know very little about what happens during the tunneling event itself. For example, is the transition of the particle from one side of the barrier instantaneous, or does it take some time?

It turns out it’s very hard to determine the so-called tunneling time because, in the fuzzy world of quantum mechanics, it’s hard to even define what we mean by tunnelling time or time for that matter.

One thing is clear at least - for a number of definitions of tunneling time, faster-than-light movement, really does seem to be a thing. This was first shown by the physicist Thomas Hartman in 1962, who found that for one definition, the time taken to tunnel can become independent of the thickness of the barrier. In other words, you can double the length of your barrier, and your particle will take the same amount of time to travel all the way through. For a thick enough barrier, this ‘Hartman effect’ can effectively teleport real, physical matter between locations faster than it would take to travel that distance sans barrier - even at the speed of light.

Now, our old pal Professor Einstein is not a fan of faster-than-light motion. As we’ve said before, his theory of special relativity explains that if you can move faster than light, you can send signals into the past, and create a whole bunch of paradoxes. So in our previous tunneling episode, we offered an explanation for why this effect doesn’t break relativity. It came down to the definition of tunneling time. If the position of the tunneling particle isn’t perfectly known, how do we know when to start and stop our tunneling stopwatch? It seems natural to define those times as whenever the center of the wavefunction passes the start and end points. But what if the wavefunction changes while moving through. In a sense, the leading edge of the old wavefunction becomes the center of the new.

It’s like if you measured a train’s travel time through a tunnel by clicking a stopwatch as the center of the train passed the entrance to the tunnel, and then again when the front of the train reaches the exit. You’ll get a shorter time than if you clicked for the same point at both tunnel ends.

Now if this was a quantum-tunneling train, then only the front carriage would make it through the tunnel, while the rest of the train would be reversed and travel back the way it came. And then when you observe the train, all but one of the carriages would vanish!

It’s hard to measure the travel time of a quantum train OR a quantum wavefunction because it’s hard to define the start and end points. Certain definitions seem to imply faster than light motion. But that’s also true of motion without a barrier. Launch a particle through empty space with a well defined starting position, and it’s position wavefunction will spread out before the finish line. The center of that wavefunction can’t travel faster than the speed of light, but upon measurement, the particle may appear to be at the leading edge of its wavefunction - potentially nudging it above light speed.

So you can see how the question of tunneling time is a bit messy. If we want to answer this, we need to define a better question. Let’s instead ask the following: is it possible to send a message between two points that are separated by a barrier faster than you can transmit the same message through empty space? Fortunately a recent paper helps us answer exactly that from a theoretical standpoint. Most previous work on tunneling time relied on the Schrodinger equation, which doesn’t incorporate Einsteins’s special theory of relativity and so has no speed limit baked in. These guys use the Dirac equation, which properly incorporates special relativity and so we can take any emerging FTL motion more seriously.

The explanation boils down to a thought experiment. Imagine you try to send a message encoded in a collection of particles to a friend, and you want it to arrive as soon as possible. Should you send the message through empty space, or through a barrier? Can quantum tunneling really speed up the transmission of the information contained in that message? Well it turns out that the answer depends on what it means to “receive the message”. If you can count the message received at the instant the first particle arrives, then the new study finds that the tunneling message really does arrive first. And the thicker the barrier, the bigger the difference in arrival time. That’s just what Hartman calculated using arguably the wrong equation back in 1962.

The study also finds that the tunneling wave packet isn’t necessarily “reshaped” all that much - it’s not clear that  we can really think of it as the cut-off front end of the wavefunction, or the first carriage of the quantum train.

So FTL verified, right? Not quite. The study finds that the average travel time for tunneling particles is shorter than the time the free-flying particles. But that’s only for the tunneling particles that make it through. Most get reflected by the barrier, and as the barrier gets thicker, exponentially more get reflected until only a miniscule number pass through. If you try to send your message over and over, your friend will most likely receive a free-flying particle long before they receive a tunneling particle - staggeringly more likely for any meaningful distance, and that’s just because the former is much more likely to make it.

So does this really save causality? In order to violate causality, your friend would need to send a return message that was influenced by your message to them, which could then cause a paradox loop. The authors say that more work is needed to verify that this is ruled out, but in general it looks like a lifeline for causality.

OK, all this theoretical stuff is good and fun, but what does experiment have to say? Efforts from the early 80s and on seemed to agree that the Hartman effect is real, but interpretation of the results suffer from some of the same problems as the theoretical calculations - how do we define the tunneling time? And even trickier, how do we define a tunneling time that we can actually measure?

For a real, physical experiment, we need a clock that’s physically measurable. In 2020, a paper was published in the journal Nature that used the swiveling axis of a particle’s quantum spin as a clock hand. The phenomenon is called Larmor precession, in which a particle’s dipole magnetic field, which is defined by its spin axis, precesses like a top in an external magnetic field. The rate of rotation can be used as an internal clock.

In their experiment, they fired ultracold rubidium atoms at a laser field that was spread out over a small area. That field was strong enough to deflect the atoms completely, and so provided an insurmountable barrier. Some particles, naturally, did manage to tunnel through anyway. For those ones, their spins were altered by the magnetic field of the laser, and the longer they spent inside the barrier, the more their spins changed.

So, ok, what did they find? Did their particles travel faster than light? Well… no, but they weren’t trying to make them do that. They were just trying to verify whether using spins as an internal clock would work at all, and they were successful. The spins were altered by pretty much the same amount that theory predicted they would be.

However the results are still relevant for the faster-than-light Hartman effect, because the spin-based clocks that they were working with should still show the effect under FTL circumstances, with faster particles and a thicker barrier. The point is that the theory and the experimental tools are now converging on a way to answer our questions once and for all. Is faster-than-light motion or influence possible? Perhaps yes, but it seems only in cases where faster-than-light signaling is IMpossible. Because as we’ve discussed many times before, when it comes to the speed of light, the house always wins. All signals in our universe, whether via quantum tunneling or quantum entanglement, seem to be bound by the same limits imposed by relativity. The universe insists that we take the long way around, and as fast as we can find them it seals up any new shortcuts through spacetime.