Dear Sofia

Quantum tunnelling or tunneling (see spelling differences) refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun. It has important applications to modern devices such as the tunnel diode,quantum computing, and the scanning tunnelling microscope. The effect was predicted in the early 20th century and its acceptance as a general physical phenomenon came mid-century.

Tunnelling is often explained using the Heisenberg uncertainty principle and the wave–particle duality of matter. Pure quantum mechanical concepts are central to the phenomenon, so quantum tunnelling is one of the novel implications of quantum mechanics.

So follow this paper to understand quantum tunneling well

https://courses.physics.illinois.edu/phys485/fa2015/web/tunneling.pdf

Reagds

Saeed Al Rashid,

My dear friend, thanks for the material. But I don't see in it the issue on which I focus, the time-interval needed for crossing the barrier. If you think that this issue is addressed, please guide me.

Look what I found in another article:

arXiv:0907.1792v1, "Tunneling times with covariant measurements"

I quote"

"after a long rectangular barrier and for a wave function of narrow momentum distribution, in leading order the transmitted pulse appears at the end of the tunnel instantaneously. Therefore, it has been suggested that this effect means superluminal signal transport"

But, it's not clear to me whether indeed the pulse jumps over the barrier in a time-interval equal to zero, I see only some complicated mathematics without a general outline to guide me through that mathematics.

With kind regards

   Dear Sofia, just remember that at one time light was also thought to move instantaneously...

Hi Sofia

My background (antennas and EM bioeffects) concerns an unconventional mathematical physics called Self-Field Theory (SFT) (file attached) which is a new deterministic version of quantum mechanics. I have been studying this new formulation for about 20 years and have found much to support the new mathematics (results are surprisingly similar to Bohr's early work and much more). However depending on your own studies to date you may not want to go any further with it. And I will understand. However let's say you are a fairly recent (post)graduate and are studying quantum tunneling as a first project in quantum theory. In this case I would suggest you look at what I say below. because it is much simpler than the learning required for quantum theory. It is a new field theory that follows the field theories of classical times but with a twist that makes it more fundamental than quantum field theory. The main feature of SFT is that it says the electron moves as a 'bispinor', two orthogonal vectors that obey a complex exponential function (exp(j*w*t). So it performs  two motions simultaneously and  deterministically.

We talk about the way in which the electron pierces the classical barrier as shown in the first figure shown in the paper (Figure 1) https://courses.physics.illinois.edu/phys485/fa2015/web/tunneling.pdf

But as shown the potential barrier is in reality a sinuosoid so it goes between high and low amplitude levels.Imagine the electron is coming in from BELOW the 'roller coaster', say from the second paragraph of writing heading up to the first paragraph of writing i.e in a northerly direction (south-north) whereas the potential barrier is in the West to East direction.

As we know from quantum theory the bound electron has an orbital which defines its probability density (SF finds motions that are equivalent to these orbitals. But these orbitals are probabilistic and depending on the actual motion of the atomic electron in its sinuosoidal motions the other electron may be able to 'sneak' through the barrier while it is not there.

There is also a book available from Pan-Stanford Press called Self-Field Theory, A New Mathematical Description of Physics by Tony Fleming 2012 (there is a 2nd Edition available in about 2018)

regards Tony

https://courses.physics.illinois.edu/phys485/fa2015/web/tunneling.pdf

There is more than one way to analyse it. The attached link to a conference proceeding, entitled Can one speak about tunnelling times in a polite society?, discusses some approaches. I seem to remember that one conclusion of the whole research project was that you cannot tunnel information faster than the speed of light, but you can perhaps use tunnelling to speed up information transfer in electronic devices.

PS. We need not go into the question of whether RG is a polite society or not :=D

Added:

> for a wave function of narrow momentum distribution, 

A narrow distribution in momentum space implies a wide distribution in position space, hence it becomes ambiguous to assign a precise tunnelling speed, since the shape of the distribution also changes in the process.

There is also the question of accounting for reflected particles in a probabilistically honest way. If every particle at the head of the spatial distribution is let through the barrier, and every particle at the tail is reflected, then the mean position of the transferred particles can be advanced very fast relative to the mean position of the incoming particles. One can make statistical conjuring tricks with that even within classical mechanics.

http://cds.cern.ch/record/316823/files/SCAN-9701012.pdf?version=1

I am not a physicist, but couldn´t tuneling imply non-locality?

Dear Dina,

It's a bit difficult to answer your question. Even the wave-function of a single particle displays non-locality. Usually, the wave-function occupies some non-zero volume, but when sent upon a photographic plate, only a tiny spot is left. So, the question is, does the wave-packet suddenly contracts?

About tunneling, we have a wave-packet which impinges on the barrier and a wave-packet exiting the barrier. It seems that if a particle were located in the peak of the impinging wave-packet, moves to the peak of the exiting wave-packet in null time. But, from what explains Kåre, the uncertainty in position inside the initial wave-packet is extremely big, s.t. it's not clear whether so is the situation.

Now, the most typical case of nonlocality is offered as you know, by the entanglements.

Best regards!

Dear Kåre,

I understand your explanation. But I would ask/object to it in two ways:

1. The probability that a (hypotetical) particle, initially located in the peak region, passes to the peak region of the exiting wave-packet, is equal to zero? Even if there is a small probability for that, then the crossing in null time is true.

2. What if the wave-packet impinging on the barrier has a small waist - small distribution in space? Of course, it would have a big distribution in velocity. But people speak here of infinite velocity, i.e. instantaneous crossing the barrier. I saw claims that no matter how wide is the barrier, the crossing time is the same.

Bottom line, can you indicate references with experiments? I don't like the treatment with weak measurements - maybe I am unjust, but I just don't subscribe to that.

Best regards!

@Tony,

It's not the formalism that interests me - as I said in the question - I am looking for experimental evidence.

Best regards!

Hi Sofia

Can I ask you: do you have any ideas about how it might be happening? Would my explanation satisfy your requirement of 'infinite speed'?

Regarding the theory, I'm not sure any other (quantum) theory would predict a probability of success of tunneling in any experimental case. If you do find any 'strong' experimental evidence I'd be most interested too but from a theoretical standpoint. 

Hi, Tony!

Please see, I posted this question, because I don't know experimental reports that satisfy me. If I knew some, I wouldn't have asked - I don't have time to waste. Theories, we can make many, explanations we can give many. I am interested to see experiments, not theories. I said it also in the question itself.

If I find experiments that answer my question, whether positively, or negatively, I don't need anymore this question.

Sofia> can you indicate references with experiments?

I add links below to two experimental papers published in PRL. By looking at the lists of citing articles, with respectively 274 and 495 entries, you should find pointers to more recent works, both experimental and theoretical.

I have not worked in this field myself; the knowledge I have arise from seminars and lunch conversations in my group during the '80s and '90s. The abstract of PRL 71 says

The peak of the photon wave packet appears on the far side of the barrier 1.47±0.21 fs earlier than it would if it were to travel at the vacuum speed of light c. Although the apparent tunneling velocity (1.7±0.2)c is superluminal, this is not a genuine signal velocity, and Einstein causality is not violated.

The abstract of PRL73 says

The measured transit time is found to be paradoxically short (implying superluminal tunneling). ... Although in apparent conflict with causality and the uncertainty principle, neither of these general principles is violated because of the strong attenuation suffered by the transmitted signals.

The last abstract hints to a statistical selection effect. Like good universities produce better candidates on average, simply by luring the best students to study there...

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.73.2308

http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.71.708

Dear Sofia,

you are right when you say: no matter how wide the barrier, the tunnelling time is the same. That is called the Hartman effect, and it has been measured often. If it is a fact, and I believe it is, then the tunnelling time is not instantaneous, but the short duration no matter how thick the barrier, is the important part.

 Low, F. E. (1998). "Comments on apparent superluminal propagation". Ann. Phys. Leipzig. 7 (7–8): 660–661. Bibcode:1998AnP...510..660L. doi:10.1002/(SICI)1521-3889(199812)7:7/83.0.CO;2-0.

Nimtz, G. (2011). "Tunneling Confronts Special Relativity". Found. Phys. 41 (7): 1193–1199. arXiv:1003.3944Freely accessible. Bibcode:2011FoPh...41.1193N. doi:10.1007/s10701-011-9539-2.

No. Quantum tunneling is just the name given to the effect that distinguishes a classical potential from a quantum potential. The quantum particle doesn't tunnel-and moves in a completely different potential from the classical potential, whose form would seem to imply what's called tunneling.

Stam,

"quantum potential" is Bohmian mechanics. And Bohmian mechanics was proved wrong by Partha Ghose, and recently by myself.

In the Schrodinger equation appears the potential energy provided by the physical field, a classical field, electric, magnetic, nuclear, etc.

Tunneling is due to the fact that the wave-function doesn't vanish inside a potential barrier, even if the energy of the quantum system is lower than the peak of the barrier, s.t. the particle may be detected outside the barrier.

The question here is whether crossing the barrier takes time, or not. I am reading now some experimental reports.

Best regards!

The question is not quite precise in to what kind of entity "involve crossing... at infinite velocity" refers.

If the entity is, say, the peak of an electromagnetic pulse or the center of energy of its transmitted part (i.e. neglecting that most of the whole wave packet is reflected), then the answer is yes. For the peak or the partial center of energy can both move faster than light (without violating special relativity!) and there is then a frame of reference, in which this means infinite speed, due to the relativity of simultaneity.

If the entity, to which the statement refers, is either energy or (controlled) information, the answer is no. These entities cannot move faster than light.

Consider a rectangular pulse of 1 light second length, traveling at the speed of light. Have it run through a device that cuts off the trailing half of the pulse, without changing its velocity. Let the process take 1 second. What is the average speed of the center of energy during this process? Since the pulse is rectangular, its center of energy is at its middle, i.e. 1/2 light second behind the front at the beginning of the cutting process and 1/4 light second behind it at the end, because then the pulse has a length of 1/2 light second only. Since the cutting took a second, the front of the pulse traveled a light second during the operation. But the center traveled from front position minus 1/2 light second to front position minus 1/4 light second, i.e. 5/4 light seconds. So its average speed was 5/4 c. Of course, no information traveled faster than c -- the front moved at c only.

The process of tunneling of particles within an electromagnetic pulse is very similar, except that these pulses are not rectangular and both their height and width diminish, but in a way that the small pulse that gets through remains inside the envelope of the big pulse, had it been able to continue its path unhindered. This can be seen experimentally, when you have two transmission lines, one with a tunneling barrier and another one without, making the two pulses race against each other. The front of the pulse without barrier is as fast as the front of the pulse with barrier, but the maximum of the latter moves faster than that of the former.

Dear Prof. Kassner:

   I like your example, only it is a bit surprising that the length of the pulse is 300.000 km! Is this an essential feature?

@Simon:

No, it is just to make the calculation easy. Take a pulse of 30 cm length and replace second by nanosecond, then you have a scaled-down example.

Thank you.

There is a fairly large literature on the definition of tunneling time in QM (not the same as for waves, since in the latter case no measurement issues arise). I do not know much about the subject, but putting ``tunneling time problem'' on Google scholar leads to interesting results.

as relativity says, space is curvable; why don`t you curve it in the way YOU like!? not yet possible? get a black hole!

The velocity of information is not the phase velocity, nor is it the group velocity in some cases, that is a classical, early result of quantum mechanics. Moreover, if the considered velocity is the one of quantum correlation, it has been proven that it carries no information. An experimental result should demonstrate the transmission of a bona fide signal, and not merely be hand waving. I only faintly remember an experiment where it was attempted to transmit a Mozart's composition, but it couldn't even be recognised at the other end because of a strong quantum noise. As quantum tunnelling happens only with a very low probability, that can't in any way be a reliable transmission, and more so because the rare bits that are transmitted are totally random (i.e. even a pure sinusoidal signal becomes noise.)

I think a question dosn't has a sence in quantum-mechanical theory. A position of the quantum-mechanical wave is a great question from the beginning. What you mean about the crossing some distance for a wave?

Dear Gennadiy,

Your question is smart. So, let me explain: the wave-packet shows a peak. I saw in some papers that the peak appears beyond the barrier at the same time as it appears at the barrier internal wall. Worse than that, it may appear beyond the barrier before it touched the internal wall of the barrier.

Now, some users called the attention on the fact that the barrier absorbs a very big part of the wave-packet and re-shapes the exiting wave-packet, placing therefore the time issue under question mark.

But, there are experiments with light - classical electromagnetism - in which the medium through which the light passes does not absorb the light. So, the wave-packet is not reduced, but it is reshaped, and the peak appears as said above. The following article reports such an experiment:

==> S. Cialdi, I. Boscolo, F. Castelli and V. Petrillo, "Superluminal advancement of a single photon far beyond its coherence length", arXiv:0805.1315v1 [physics.optics].

Best regards!

Remi my friend,

Please see my reply to Ghennadiy. I also indicate here an article which shows the appearance of super-luminal group velocity in an experiment which is not at all quantum, just classical electromagnetism. A DC pair is produced, one photon passes through a birefringent medium, and the other photon through vacuum. One gets that the former photon propagates with a group-velocity > c. 

==> S. Cialdi, I. Boscolo, F. Castelli and V. Petrillo, "Superluminal advancement of a single photon far beyond its coherence length", arXiv:0805.1315v1 [physics.optics].

In particular saying, I believe that this article may be of interest for you.

With kind regards!

Sofia, the wave packet you speak about is a calculation from the Schrödinger equation, that is, unitary evolution. But Quantum Mechanics doesn't reduces to that, there is the projection postulate, and the associated probabilistic interpretation. The peak across the barrier means that if the particule were detected there, it would be at that location. But it also means that the probability for the particle to be detected there is tiny. If it isn't, the peak disappears altogether (the projection postulate applies to a negative measurement outcome as well) and the particle can only be detected before the barrier. It is that random process that destroys completely the information transfer.

Incidently, the evanescent wave isn't a quantum phenomenon, but an interference one. Far from the interface, the approximation is valid that results in the cancellation of the amplitude, but near it isn't. That's not a prediction of geometrical optics, but of classical wave propagation which is also able to explain refraction and (total) reflection.

Dear Claude Pierre,

I understand what your line of thought, though there is no issue of wave-packet reduction here. The peak does not cover a point on the x-axis, there is a whole region under the peak, and the probability to get a detection is this region, is much bigger than on the rest of the x-axis.

But, to make simple the issue, let me recommend the following article, which is not at all about quantum mechanics, it is about classical electromagnetism

==> S. Cialdi, I. Boscolo, F. Castelli and V. Petrillo, "Superluminal advancement of a single photon far beyond its coherence length", arXiv:0805.1315v1 [physics.optics].

A DC pair is produced, one photon passes through a birefringent medium (equivalent to a potential barrier), and the other photon passes through vacuum. The experiment showed that the former photon propagates quicker than the latter.

Best regards!

Sofia, without detection, there can't be transfer of information, and with detection, there is always reduction. There is not only one peak, but two: the transmitted one, and the reflected one, which is much bigger.

In the experiment you describe, there is say 1/1000 photon that propagate faster than light. Moreover, you are speaking about a photon, not information. If you want to understand better, try and figure out a process for transmitting a signal when the photons have a probability of 1/1000 of being received. In particular, find out how much time this process will take. Is it still faster than light now?

Dear Claude Pierre,

I all the time speak of a photon and the question is about achieving superluminal velocity. It's you who speak of information, not I. Next, I don't see in the article the 1/1000 probability. Where from did you take it? The photon is a wave-packet most of which laying in the peak region.

As to the retarded and advanced wave-packets, it is the advanced one that reaches the detector at super-luminal group-velocity.

But, I'd better explain you my purpose: I am preoccupied by the problem whether the QM can admit a substructure, particles following trajectories. The Bohmian mechanics was proved as clashing with the experiment, s.t. what I investigate is whether trajectories of GENERAL form are possible, i.e. NOT according to Bohm's formalism. Since no object carying mass can travel faster-than-light, or, worse, backward in time, I am try to see what I can infer from tunneling and similar effects.

(By the way, since you speak of information, what benefit may bring us information carried faster-than-light over a distance of c×100femtoseconds (30microns)? If we want to transmit information, the distances are kilometers.)

Best regards!

If you are not interested in the speed of information, then take the phase velocity of a material particle. It is always greater than c, and can be as large as wanted with a particle almost at rest. But phase velocity can't be assimilated to a trajectory, and that is also true for every process that doesn't carry information, otherwise that would violate causality.

Hi, Claude-Pierre!

I am not sure whether I understand you. How is the phase-velocity greater than c? Please explain.

Now, phase-velocity doesn't interest me, it's exactly the group-velocity which I am interested in. This is why I look at what happens to the peak region of these wave-packets, and don't rummage in the Fourier components.

As to causality, I don't put the things another way: a physical object carying mass cannot surpass the light velocity, moreover, won't travel backward in time. That places the idea of continuous trajectory under question mark.

You see, Ghirardi and Bassi wrote in the past an article, with a very beautiful mathematical treatment, launching the idea of spontaneous localization of the wave-packet - in short, no trajectories, just a discontinuous series of localizations. It was interesting, though I can't say that I believe in it. Neither do I believe that the wave-function "collapses", i.e. that some parts of the wave-function disappear during the measurement.

I opened a whole project on the "collapse" topic. But, one step, at one time, first I want to elucidate the issue of continuous trajectories.

Best regards! 

In the tunnel effect, the group velocity isn't well defined, since the wave packet splits in two parts, a small one that passes through the barrier, and a big one that is reflected.

Dear Claude Pierre,

Yes, of course, the group velocity isn't well defined. But, I suppose that you read what I am after. Since the possibility exists that the (supposed) particle travels with the advanced wave-packet, I am interested in the group velocity of this one.  

Dear Remi,

I am not sure whether I understand what you try to say. Please ellaborate a bit.

What I said is that while some part of the wave-function gives a response, the rest of the parts are "there", do not disappear, just don't produce a response. Under certain circumstances, a particle may be injected into them, and they do produce response. I know such a case.

Kind regards!

The group velocity can only be defined for a wave packet. In the barrier, there is no packet, all the frequencies are imaginary, and the wave function is then monotonically decaying. You can't skirt the corpuscle aspect that is inherent of quantum mechanics, and all its interpretations and reformulations. The corpuscle aspect is just the statement that the wave function, or any ensemble that could substitute it, must collapse, since a corpuscle can't be detected at two different locations.

Claude Pierre,

The wave-packet yes passes through the barrier - continuity equation.

But, what interests me is the difference between the arrival time of the wave-packet that travelled through vacuum, and the wave-packet that travelled through the medium (see the article that I recommended). The experimenters claim that the former is longer.

Remi,

You didn't read attentively my reply. I'll try to put it more clearly. However, come with me to the project, I'll answer you there because this question is related to the project.

Infinity is not the exact ward, but yes tunneling phenomenon acts like that, physical interpretation is it is a statistical possibility of De-groggily waves of particles.