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Indeed, the most straightforward attempts to answer this question suggest that sometimes, particles can cross these "classically forbidden regions" faster than light -- a huge no-no according to Einstein. It took until the end of the 20th century for experiment and theory to reconcile themselves with the truth that "wave packets" (the probability distribution for where a quantum particle may be found, in light of Heisenberg's famous uncertainty relationship) may indeed appear to move faster than light, but with just the right loopholes to prevent any of the relativistic paradoxes physicists had worried about (see, for instance, "Faster Than Light").

In the end, the physics community came to understand that while the "peak" of a particle distribution at the exit of a barrier was certainly useful for determining the most likely time of detection, it needn't be directly connected to the peak of the input distribution. Even if we know when the particle left the barrier, because of the uncertainty principle, we're not sure exactly when that particle entered it. This reopened the question: how long do the particles actually spend in this region where they were never supposed to be at all?

In a new experiment to appear in the July 23, 2020 issue of Nature, Ramón Ramos, David Spierings, Isabelle Racicot, and Aephraim Steinberg of the University of Toronto have now measured how long ultracold Rubidium atoms spend tunneling through a 1-micron beam of laser light. They began by cooling their atoms down to one one-billionth of a degree above absolute zero (1 nanoKelvin); at such low temperatures, atoms' quantum wave-functions spread out to be several microns across, allowing them to tunnel a significant fraction of the time.

To measure how long they took, the experimenters made use of the fact that each atom has a "spin," a sort of compass needle -- or in this context, the hand of a clock. Using lasers and microwaves, they set things up so that these clock hands would spin only while the atoms were inside the laser beam. By looking at where the atomic spins "pointed" when atoms appeared on the far side, they were able to read off a time. In contrast to recent work which suggested tunneling might actually be "instantaneous" (see for instance https://www.iflscience.com/physics/quantum-tunneling-is-so-quick-it-could-be-instantaneous), they found the atoms traversed the barrier in about one leisurely millisecond, corresponding to a velocity of close to one millimetre per second. Particularly striking is that unlike in the familiar, classical, world, when atoms strike the barrier at lower velocities, it appears to take them less time to get across.

In future work, the Toronto team hopes to measure exactly where within the barrier the particles pass their time, testing predictions that they "hop" from one side to the other without really visiting the middle. They also hope to study the effects of observing a quantum system: what does an atom do if someone spies it while it's somewhere it isn't supposed to be? Not only does this work shed light on a century-old controversy about tunneling times, but it opens up a new window on the question of what quantum theory permits us to say about the past behaviour of a particle at all.

For Further Reading

The technical paper can be found on the arXiv or at Measuring the time a tunnelling atom spends in the barrier.

In addition to this MinutePhysics video, you may enjoy this Forbes post.

The wikipedia entry on quantum tunneling
Wikipedia on superluminality
BBC Horizon: "The Time Lords"
"Faster Than Light?," R.Y. Chiao, P.G. Kwiat, and A.M. Steinberg, Scientific American, vol. 269, (no. 2): 52-60, August (1993).
Tunneling Times and Superluminality, R. Y. Chiao and A. M. Steinberg, in Progress in Optics vol. XXXVII, Emil Wolf ed., Elsevier (Amsterdam: 1997), pp. 347-406.
No Thing Goes Faster Than Light
Speakable and Unspeakable, Past and Future, A.M. Steinberg, in SCIENCE AND ULTIMATE REALITY: Quantum Theory, Cosmology and Complexity, ed. John D. Barrow, Paul C.W. Davies, and Charles L. Harper, Jr., Cambridge University Press, 2003.

Acknowledgments

This experiment was made possible by the efforts and ingenuity of multiple generations of graduate students and other researchers who created our Bose-Einstein condensation apparatus and spent years developing it into a system for probing quantum tunneling: the authors wish to share the credit, and the joy of discovery, with Ana Jofre, Mirco Siercke, Chris Ellenor, Marcelo Martinelli, Rockson Chang, Shreyas Potnis, Alan Stummer, along with numerous others who have passed through the lab and contributed their perspective and their assistance.

This work was supported by NSERC and the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust; AMS is a fellow of CIFAR and RR acknowledges support from CONACYT.

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