Handling an unsettled problem carefully, while leaving it unsettled ── the conclusion that "the time cannot be defined" is probably the deeper one
The problem Episode 1 set aside. When a particle slips through a wall, how many seconds was it inside? It looks like a plain and reasonable question. But do the calculation and something odd happens right away ── thicken the wall and the time does not increase. Read Hartman's 1962 observation straight and the effective speed rises with thickness until it exceeds light. Relativity is not broken, so something must be wrong. What is wrong is the question. There are several ways to measure "the time spent inside the wall," and they return different answers. Worse, quantum mechanics has no observable (no Hermitian operator) corresponding to "time" at all. This episode treats a question with no answer, including why it has no answer ── one episode ending in "we don't know" seems only fair.
The most natural definition is to measure how much later the wave packet arrives. Compare when the transmitted peak shows up with and without the wall. This comes from differentiating the phase of the transmission amplitude (Wigner's group delay).
Compute this for a rectangular barrier and it saturates as the wall thickens. In the opaque limit
$$\tau_\varphi\ \longrightarrow\ \frac{2m}{\hbar\kappa k}\qquad(\text{independent of the thickness }a\text{!})$$That is the Hartman effect. Double the thickness or multiply it by ten and the delay does not grow. So the "effective speed" \(a/\tau_\varphi\) can be made arbitrarily large ── and eventually exceeds light.
The conclusion first: you cannot send information faster than light this way. There are two reasons.
① The emerging wave has been remade.
The wall acts as a filter that passes only the leading edge of the incoming packet, weakly, and cuts off the rest. The peak of the emerging packet is not the incoming peak that has moved; it is a reshaped fragment of the leading edge. So "the peak arrived early" does not mean "something travelled fast."
② The front of a signal always travels at exactly \(c\).
As Sommerfeld and Brillouin showed in 1914, the very leading edge of a waveform (a discontinuous onset) always propagates at \(c\), whatever the medium. Only that front can carry information. The speed of a peak is not a signal velocity.
For electrons there is also a much more concrete circumstance ── entering the apparently superluminal regime requires an extremely thick wall, where the transmission probability is of order \(10^{-1000}\). The exponential effectively seals off the superluminal region. The figure lets you check this.
Now the real subject. There are several ways to measure "the time spent inside the wall," and each measures a different quantity and returns a different answer.
| Clock | What it measures | For a thick wall |
|---|---|---|
| phase time (Wigner) | the delay of the transmitted packet's peak | saturates (Hartman) |
| dwell time (Smith) | probability present inside the wall ÷ incident flux | saturates, but includes reflection |
| Larmor clock (Baz', Rybachenko) | put a weak magnetic field only inside the wall and read the time from how far the spin precessed | two components appear; one saturates, one grows |
| Büttiker–Landauer | oscillate the barrier and infer the time from the frequency at which it can no longer keep up | grows with thickness |
| weak measurement (weak values) | measure weakly and average without disturbing | gives a complex time |
One wants to ask "which is the right one," but the question is probably posed wrongly.
In quantum mechanics, position and momentum have corresponding operators. But there is no Hermitian operator corresponding to "time" (Pauli's argument, 1933). Time is a parameter that orders states, not a thing to be measured.
So "how many seconds was it inside the wall" has no determinate meaning on its own. Meaning appears only when a specific clock is coupled to the system ── precess a spin, oscillate the barrier, track the peak of the packet. Different clocks measure different quantities.
This is exactly the measurement story from the sister series "Quantum That Clicks." "What was the particle doing inside the wall" has no answer unless you measure ── and once you measure, that measurement becomes part of the answer.
| Year | Experiment | Result |
|---|---|---|
| 1990s | Microwave and photon experiments by Nimtz, Chiao and others | Observed wave-packet peaks propagating "superluminally." Interpretation settled, however: the signal velocity does not exceed \(c\) |
| 2008 | Eckle and colleagues, the "attoclock" (Science). Ionise helium with circularly polarised light and read the time from the direction the electron emerges | The tunneling delay is consistent with zero |
| 2014 | Landsman and colleagues | Claimed a non-zero delay |
| 2019 | Sainadh and colleagues (Nature). Repeated in atomic hydrogen, a system theory can handle accurately | With the Coulomb field treated properly the delay is consistent with zero. The earlier "non-zero" may have been an artefact of the analysis model |
| 2020 | Ramos, Spierings, Steinberg and colleagues (Nature). Send a Bose–Einstein condensate of cold atoms through an optical barrier and measure with a Larmor clock | A finite value of about 0.6 milliseconds. And they confirmed the prediction that it gets shorter at higher energy |
The 2020 experiment matters. Rather than "can tunneling time be measured at all," it produced a clean answer in the form "the Larmor clock returns this value." Specify a clock and you get an answer. Do not specify one and you do not. That is where the problem currently stands.
Established: that the phase time (group delay) for a rectangular barrier saturates with thickness (Hartman 1962); that the formal effective speed can exceed \(c\) while the signal velocity cannot (the front of a waveform always propagates at \(c\) ── Sommerfeld–Brillouin 1914); that the transmitted packet is a reshaped version of the incident packet's leading edge; that several definitions of tunneling time (phase, dwell, Larmor, Büttiker–Landauer, weak values) exist and disagree; that no Hermitian operator corresponding to time exists (Pauli); and the results of the attoclock experiments (Eckle et al. 2008, Sainadh et al. 2019) and the Larmor-clock measurement in a Bose–Einstein condensate (Ramos et al. 2020).
Unsettled, and caveats: (1) Whether there is a unique correct definition of "tunneling time" is itself unsettled. The majority position is that it has no determinate meaning without specifying a measurement, but some researchers regard a particular definition as fundamental. (2) The interpretation of attoclock experiments depends strongly on the theoretical model. The 2019 atomic-hydrogen result indicated that earlier non-zero claims may have been model artefacts, but that does not close everything. (3) The 1990s microwave "superluminal" experiments do not contradict relativity; some popular coverage was misleading. (4) The phase time in the figure is a rectangular-barrier, stationary-state, non-relativistic calculation and does not correspond to the real experimental systems (atomic Coulomb fields, optical lattices). (5) The figure's "apparently superluminal point" is a formal crossing, and the transmission probability there is displayed at the same time ── it is not an experimentally accessible region. (6) The 0.6 ms of Ramos et al. 2020 is the value that system's Larmor clock returned, not a general proposition that "an electron spends 0.6 ms inside a wall."
The phase time for a rectangular barrier saturates as the wall thickens (the Hartman effect, 1962), so the formal effective speed can exceed light. Relativity is nonetheless intact ── the transmitted peak is not the incident peak that moved but a reshaped leading edge, and the front of a waveform, which carries information, always travels at \(c\). Moreover, for electrons the apparently superluminal region has a transmission probability of order \(10^{-1000}\) ── the exponential seals off the superluminal region.
The deeper problem is that the definition of "the time it took" is not unique. Phase time, dwell time, Larmor time, Büttiker–Landauer time, weak values ── all return different answers. At bottom lies the fact that quantum mechanics has no time operator (Pauli). Time is a parameter, not an observable.
So the current position is this ── specify a clock and you get an answer; do not, and you do not. In 2020 Steinberg and colleagues measured about 0.6 milliseconds with cold atoms and a Larmor clock, matching the theoretical prediction. Ninety years polished the question, not the answer.
Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, move the barrier height and energy to watch the phase time saturate and cross a/c (the formally superluminal point). The transmission probability there is displayed alongside. "See the answer" opens each solution.