It turns out that this is not a trivial issue at all. While that's a perfectly sensible question to ask from the point of view of classical physics, it's not easy to translate that question into the language of quantum mechanics. In lay terms, a spatial measurement tells you where a particle is, but doesn't say anything about where it was, and without such a measurement there is uncertainty in the initial position and momentum of the particle.
Some very clever people have thought about how to get at this issue. This review article by Landauer and Martin caught my attention when I was in grad school, and it explains the issues very clearly. One idea people had (Baz' and Rybochenko) is to use the particle itself as a clock. If the tunneling particle has spin, you can prepare the incident particles to have that spin oriented in a particular direction. Then have a magnetic field confined to the tunneling barrier. Look at the particles that did tunnel through and see how far the spins have precessed. This idea is shown below.
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| "Larmor clock", from this paper |
This is a cute idea in theory, but extremely challenging to implement in an experiment. However, this has now been done by Ramos et al. from the Steinberg group at the University of Toronto, as explained in this very nice Nature paper. They are able to do this and actually see an effect that Landauer and others had discussed: there is "back-action", where the presence of the magnetic field itself (essential for the clock) has an effect on the tunneling time. Tunneling is not instantaneous, though it is faster than the simple "semiclassical" estimate (that one would get by taking the magnitude of the imaginary momentum in the barrier and using that to get an effective velocity). Very cool.
I have to admit I didn't quite understand why this result is significant. Why can't the "obvious" experiment of timing a neutron or electron passing through a barrier (like a crystal) answer the question just fine?
ReplyDeleteAnon, it's not that simple to do such an experiment. When you think about firing a neutron at a barrier at some "start" time, in wavefunction language what you're doing is preparing a wavepacket with the peak localized at some initial position at some initial time, and having some momentum distribution localized around some initial momentum (as shown in the cartoon at the bottom part of the figure). While you can have a detector that goes ping after the barrier, you don't actually know the initial time when the particle hit the front side of the barrier. You only know when the centroid of the incident wavepacket would be expected to hit the barrier. This is exactly the issue that leads to counterintuitive issues about superluminal propagation. It's possible to have an incident gaussian wavepacket and a transmitted gaussian wavepacket, and if you just look at the positions of the peaks of those wavepackets, the peak can traverse the barrier faster than c; indeed, it's possible for the outgoing peak to emerge before the incoming peak has even hit the barrier yet, leading to the disturbing conclusion that there can be negative travel times. That first post I linked has some discussion of this.
ReplyDeleteThank you for the clarification, I am reading through the review.
DeleteFrom your perspective then, what would the physical interpretation be for the difference in time for a neutron packet passing from point A to B with and without the barrier? Thus time difference would always have the same sign, and, previously, I would have assigned it the meaning of "tunneling time" until I started reading the RMP article.
Anon, without the barrier that would be the time associated with the group velocity. Within the barrier the challenge is describing some kind of group velocity that is meaningful. This is not just an issue in quantum mechanics - Sommerfeld was worrying about these same issues in classical EM propagation. I confess, I still have a hard time keeping straight in my head the differences between group, "signal", and "front" velocities. See here: https://en.wikipedia.org/wiki/Front_velocity
ReplyDeleteIn classical EM, though, we're explicitly talking about waves, where in the quantum case we are talking about particles that can be (depending on your choice of terminology) spatially localized by a measurement. That distinction seems to make the interpretation more challenging to hold in one's head.
What is the effect of the back-action on the tunneling time? Does it speed the particle up or slow it down?
ReplyDeleteHi everybody!
ReplyDeleteI have created a classical work that extends special relativity in a natural way revealing a wider framework where quasiparticles may reach and even surpass the speed of light. Speaking about quantum tunneling, I thought it would be a welcome, however indirectly relevant distraction:
https://vixra.org/abs/2007.0027
See Fig.3 and Fig.4. The effective inertia of the quasiparticle decreases while its speed increases (even beyond the speed of light) having as lowest limit the zero.
Important Note: The wider framework cannot apply for bare particles e.g. electrons but just for quasiparticles e.g. electron trapped within a travelling standing wave. When one attempts to apply the wider framework to bare particles, it automatically reduces to Einstein's special relativity.
The work proves mathematically that Newton's 3rd law is incomplete by introducing the breaking of action-reaction symmetry through induced internal forces. Furthermore, it is proposed an experiment (a mechanical construction) to test the theory. Later, the findings are applied to special relativity and Lorentz transformation that lead to the discovery of a wider framework that is relevant just for quasiparticles.
I'm eager to receive your feedback.
Very interesting, nice summary, thank you.
ReplyDelete