How long the crossing takes

The wall that is not quite a wall answers how much gets through, and the answer is clean: a number between zero and one, computed by matching a wavefunction at two boundaries, agreeing with experiment across twenty orders of magnitude. The obvious next question is how long the crossing takes, and it is not clean at all.

The trouble is visible before any calculation. A stationary state has no time in it, so the transmission coefficient — which is what the matching gives — cannot contain a duration. A duration has to come from a wave packet, a packet is a bundle of energies, and different definitions of “when it arrived” pick out different features of a bundle whose shape the barrier has changed.

The delay that stops growing #

The phase time is the most natural definition available. Follow the peak of the incident packet, follow the peak of the transmitted one, and ask how much later the second arrived than a free particle would have. The answer is $\hbar$ times the energy-derivative of the transmitted phase, which is exactly what the group velocity argument in the packet that moves at another speed prescribes, applied to a barrier rather than a medium.

Applied to a rectangular barrier it gives a saturating function. The dependence on thickness enters as $\tanh(\kappa a)$, and once the barrier is more than a couple of decay lengths thick that hyperbolic tangent is one to within a rounding error and the thickness has left the expression.

That is the Hartman effect, and it has been known since 1962. It is not an approximation, an artefact of the rectangular shape, or a failure of the numerics: the figure computes the derivative by finite differences of the exact amplitude and checks the saturation rather than asserting it.

The uncomfortable part is the crossing. Past a certain thickness the peak of the transmitted packet leaves the far side of the barrier before a free particle travelling at the same speed would have got there. Extend the barrier far enough and the implied speed passes any bound.

An apparent speed with no ceiling #

For electrons the numbers stay comfortably below anything alarming, because an electron’s speed is so far below light’s that a barrier would have to be hundreds of nanometres thick — and utterly opaque — before the ratio mattered. The electromagnetic version does not have that cushion.

An evanescent microwave in an undersized waveguide obeys the same mathematics with the same solutions, so the same saturation happens, and there the free speed is already the speed of light. Measurements in the 1990s reported transmitted pulse peaks arriving at apparent speeds several times $c$, and the measurements were correct. The photonic analogue is exact enough that a refraction with no wave in it and the tunnelling problem share their equations.

So there is something to resolve, and the resolution is not that the calculation was wrong.

Why nothing has outrun anything #

The transmitted peak is not the incident peak.

A packet is a range of arrival times, and a barrier attenuates by a factor that depends on energy — but more importantly, in the time domain, the barrier is still busy transmitting the early part of the packet while the late part is still arriving. The exponential suppression multiplies the whole packet, and the parts of the packet that were already inside get out; the parts that arrive later have to cross a barrier that is already ringing with the field the early part left behind.

What emerges is a small, reshaped copy of the front of the incident packet, and its peak sits earlier within it than the incident peak sat within the incident packet. Comparing the two peaks compares two different features of two different shapes, and the difference is a reshaping rather than a propagation.

The test that settles it is the front. The leading edge of the incident packet — the first instant at which the amplitude is non-zero at all — is what carries information, and it is exactly this that the speed that carries no signal distinguishes from a group velocity. The transmitted front never precedes the light cone of the incident front. Every calculation of it, in the quantum case and the optical one, gives the same answer, and the front’s arrival is not affected by the barrier’s thickness in the way the peak’s is.

And the transmission is the clue that was there all along. At the thicknesses where the apparent speed is impressive, the transmission is smaller than a part in a million. An effect that requires throwing away all but a millionth of the signal in order to see a peak arrive early is a filtering effect, and filtering has been known to advance a peak since long before quantum mechanics.

The same behaviour at other numbers #

A saturation found at one choice of energy and height is worth checking against another, because a flat curve is exactly the shape a numerical accident produces.

The two figures differ in every number and agree in every feature. The phase time is smaller, because a deeper barrier has a shorter decay length and the saturated value scales with it. The free-flight line is shallower, because the electron outside is faster. The crossing has moved inward, to a thickness of a fraction of the first one.

What has not changed is that there is a crossing, that the phase time flattens once the barrier is a couple of decay lengths thick, and that the flattening is checked by doubling the barrier and measuring the change rather than by inspection of the curve.

That is worth stating because the natural suspicion — that the effect is an artefact of a badly chosen range — is a good suspicion and this is how it gets addressed. The saturated phase time is set by the decay length and not by the thickness, so a barrier deep enough to have a short decay length saturates sooner and at a smaller value. Nothing in that depends on the particular electron.

Four clocks, four answers #

The deeper problem is not the paradox; it is that there is no agreed quantity to be paradoxical about.

Büttiker and Landauer proposed reading the time off a clock carried by the particle. Put a small magnetic field inside the barrier and nowhere else, send in a spin polarised across the field, and measure how far the transmitted spin has precessed. That is a physical measurement with a definite answer, and it has two components: the spin rotates in the plane perpendicular to the field, and it also acquires a component along it, because the two spin states see barriers of slightly different height and are transmitted in different amounts.

The in-plane rotation gives $\tau_z$ and the alignment gives $\tau_y$, and the figure shows them behaving completely differently. $\tau_y$ grows in proportion to the thickness — it is $m a/\hbar\kappa$ in the opaque limit, which the figure verifies against the numerical derivative — while $\tau_z$ saturates like the phase time.

Neither is wrong. They are answers to different questions: one asks how much the barrier’s height affects the size of what gets through, the other how much it affects the phase. A clock is a physical system, and different physical clocks couple to different features of the amplitude. There is no reason for them to agree, and they do not.

The Larmor times combine into a single quantity — the square root of the sum of their squares — that grows with thickness and is the one most often called the traversal time. That combination is a choice too.

What a time would have to be #

It is worth being clear about why the question is harder than it looks, rather than treating the disagreement as a temporary state of ignorance.

A duration in quantum mechanics is normally the difference between two times, and a time is normally read from a clock, which is a physical system with its own Hamiltonian. Time is not an observable in the sense the questions that can be asked together uses — there is no operator whose eigenvalues are instants — so “the time spent in a region” cannot be defined the way position and momentum are, by naming an operator and taking its expectation.

What can be defined is the expected value of the time a particular clock accumulates, and that depends on the clock. The dwell time — the integrated probability density inside the barrier divided by the incident flux — is the closest thing to a definition that needs no clock at all, and it is the one quantity everybody agrees on. It has the disadvantage of not distinguishing transmitted particles from reflected ones, which is precisely the distinction the question was about.

So the honest summary is that the barrier problem has one well-defined time that answers the wrong question, and several well-defined times that answer specific experimental questions and disagree. The disagreement is a feature of the question rather than of the answers, and this is a good deal more common in quantum mechanics than the textbook presentation of it suggests.

The one time everybody agrees on #

There is a quantity in the problem that has no ambiguity at all, and it is instructive that it does not answer the question.

The dwell time is the amount of probability sitting inside the barrier at any moment, divided by the rate at which probability arrives. It is a ratio of two things a stationary state defines exactly, it needs no packet and no clock, and every treatment of the problem gets the same number for it. In the opaque limit it behaves like the Larmor $\tau_z$: it saturates.

Its defect is that it counts everything in the barrier without asking where it is going. Almost all of the probability under an opaque barrier belongs to particles that will be reflected — the transmission is a millionth — so the dwell time is overwhelmingly a statement about the reflected population. Splitting it into a transmitted part and a reflected part is exactly the step that cannot be taken without a further assumption, because a particle inside the barrier has not yet been sorted into the two categories and no measurement can sort it without destroying the interference that makes the problem what it is.

That is the crux, and it is a clean example of a general difficulty. Asking “how long do the transmitted ones spend inside” requires assigning a history to particles classified by their future, and quantum mechanics does not supply histories. The clocks work because each of them physically records something during the crossing and is then read afterwards, and what each records is a different functional of the amplitude — which is why they cannot be reconciled by thinking harder.

The unambiguous quantity is unambiguous because it refuses to make the distinction the question is about. That trade is worth recognising, because it recurs: a definition that survives every objection has often survived by not saying anything.

Where the experiments are #

The most-quoted modern measurements come from strong-field ionisation, where an intense laser pulse bends an atom’s potential into a barrier and the electron leaves through it. The rotating field acts as a clock: the direction the electron comes out in encodes the phase of the field when it left, so a delay becomes an angle.

The technique is beautiful and the interpretation is exactly the difficulty above. Converting a measured angle into a time under the barrier requires a model of the electron’s motion after it emerges, and different models implement different definitions. Groups analysing similar data have reported delays of tens of attoseconds and delays consistent with zero, and the disagreement has been traced to the conversion rather than to the measurement.

Solid-state versions are cleaner in one respect: the barrier is a real object with a known thickness, which the last atom does all the seeing exploits for a quite different purpose. They are worse in another, because the electron never travels freely on either side and the comparison with a free particle stops being available.

There is also a purely optical measurement that avoids atoms altogether. Frustrated total internal reflection is the same barrier with light instead of an electron, the gap between two prisms is the thickness, and a pulse crossing the gap can be timed by interference against a reference arm. It has the great advantage that the free-flight comparison is unambiguous and the great disadvantage that the barrier is only ever a wavelength or two thick, so the saturation is barely established before the transmission has gone. Every version of the experiment has confirmed the saturation and none has needed a new principle to explain it.

Where the model stops #

The barrier is rectangular and one-dimensional. A real barrier has a shape, and the saturating $\tanh$ that produces the Hartman effect is specific to the rectangle. The saturation survives for smooth barriers, but its numerical value does not, and the crossing thickness in the second figure belongs to this shape only.

The packet is treated as narrow in energy. The phase time is the first term of an expansion, and it describes the transmitted peak only when the transmission does not vary much across the packet’s bandwidth — which is precisely the condition that fails when the barrier is opaque enough for the effect to be dramatic. That is not a coincidence; it is the same reshaping, arriving as a caveat.

Two-particle effects are absent. The electron does not interact with the barrier’s charges, does not lose energy, and does not care about anything else in the material. Real tunnelling in a solid is inelastic often enough that the elastic problem is a limiting case.

And the relativistic version is a different subject. The Dirac equation’s barrier problem has the Klein paradox in it, the front behaves differently, and none of the timing arguments above transfers unchanged.

What the pictures cannot show #

Every figure here draws a time computed from a stationary-state amplitude, and a stationary state has no arrival in it. The times are derivatives — of phase with respect to energy, of magnitude and phase with respect to barrier height — and reading them as durations is an interpretation the figures cannot supply or check.

Nothing drawn shows the transmitted packet’s shape, and the shape is where the resolution of the paradox lives. A figure of the reshaping would show a small transmitted pulse sitting under the early part of the incident one, and would make the “early peak” look like what it is; it would also need a propagation rather than an amplitude, and would be a different generator’s job.

Where the ladder goes next #

The tunnelling ladder began with the wall that is not quite a wall, passed through a wall that a factor of two makes impassable and the exponential sensitivity that makes an instrument of it, and reached two walls that let more through than one, where interference inside the gap beats the product of the two barriers. This rung asks for a duration and finds that the question needs a clock before it has an answer. The rungs after it: dissipative tunnelling, where a coupling to an environment supplies the clock and suppresses the rate; and macroscopic quantum tunnelling, where the coordinate under the barrier is a current in a circuit rather than a particle.

The habit worth carrying away is that some questions are not answered by a better calculation. “How long did it take” presupposes a clock, and when the answer depends on which clock is used, the right response is to say which one — not to look for the true value that the clocks are approximating.