The time a quantum particle spends traversing a classically forbidden region has resisted un-ambiguous definition since the earliest analyses of wave-packet tunneling, and the proliferation ofmutually incompatible “clocks”—the Wigner phase time, the Büttiker–Landauer traversal time, theLarmor time, and the path-integral complex time among them—has kept the question open fordecades. The controversy has been sharpened, not settled, by the advent of attosecond angular-streaking metrology, in which competing analyses of tunnel ionization continue to disagree over theexistence of a finite delay. Here we show that these rival definitions are not alternative answersto a single question but distinct cumulants of one traversal-time quasi-distribution, generated bythe exact rectangular-barrier transmission amplitude when the latter is promoted to a cumulantgenerating functional through a barrier-height counting field. The first cumulant unifies the Wignerphase time and the Larmor/Büttiker–Landauer time as the real and imaginary parts of a singlecomplex traversal time, recovering the celebrated Hartman saturation as the width-independentvalue ℏ/V0 of its real part. The second cumulant—which, to our knowledge, has not previouslybeen enlisted as a tunneling-time observable—yields a traversal-time variance that entirely evadesthe Hartman saturation and instead grows linearly with the barrier width d. The standard devia-tion of the tunneling time therefore scales as στ ∝ √d: rather than becoming superluminally sharpfor thick barriers, the tunneling time grows progressively more uncertain, in direct opposition tothe naive superluminal reading of the Hartman effect. This parameter-free scaling, independentof the O(1) coefficient that fixes its magnitude, constitutes a falsifiable prediction: attosecond an-gular streaking, which resolves the full arrival-time distribution of tunneled electrons rather thanmerely its centroid, should register a temporal spread broadening as √d even where the mean delaysaturates. We thereby recast a nearly century-old semantic dispute as a concrete experimentalquestion.
The time a quantum particle spends traversing a classically forbidden region is among the most persistently contested observables in quantum theory, its definition splintering across the Wigner phase time, the Büttiker–Landauer clock, and complex-time path-integral constructions. The debate has centered almost exclusively on a single number—the mean delay—and on the Hartman effect, whereby that mean saturates to a width-independent constant for an opaque barrier, suggesting an unbounded and apparently superluminal traversal speed. Yet an attoclock measures not a number but a distribution: the observable imprinted on the photoelectron spectrum is a temporal spread, not a delay. The narrow-band limit of this variance is fixed by the time-energy uncertainty relation alone,
Var(τ)→ℏ2/4ΔE2, a general property of any finite-bandwidth wave packet. What has not been shown is that this floor survives intact through a barrier deep in the Hartman-opaque regime, where the mean delay has already saturated and where naive intuition expects the pulse to broaden. We prove, for the rectangular barrier, that the transmitted variance remains locked onto this incident-packet value independently of the barrier width
d—a spectral-pinning theorem. Equivalently, the dimensionless product
Var(τ)ΔE2 collapses onto a single universal plateau shared by all bandwidths, a data collapse confirmed numerically to better than 0.5% across two decades of opacity and verified against the predicted inverse-square bandwidth scaling
Varnarrow/Varbroad=(ΔEbroad/ΔEnarrow)2 to within a few percent, the small residual excess (28.56 versus 25) being itself a quantitative signature of barrier-induced reshaping. The genuine new result is the finite-bandwidth correction responsible for this excess: the opaque barrier acts as a coherent spectral filter—a tunneling analogue of the gain-narrowing mechanism familiar from ultrafast optics—that reweights the transmitted band and causes the variance to decrease weakly with opacity [log-log exponent
α≈+0.070 in the convention
Var∝(κd)−α], while the mean arrival time is pushed above the Hartman plateau. We derive the leading filter coefficient
λ2 in closed form, verify it against an independent, exactly solvable smooth barrier (Eckart), and connect both the pinned floor and its filtering correction to experiment through the attoclock map
σθ≃ωστ. The result recasts the tunneling-time question from the fragile arithmetic of a saturating mean to the robust, universal statistics of a spectrally pinned—and quantifiably filtered—width.
A quantum particle's traversal time through a forbidden barrier has long been debated, with definitions varying across Wigner, Büttiker–Landauer, and path-integral frameworks. Prior work focused on the mean delay and the Hartman effect—where the average time becomes constant for opaque barriers, implying superluminal speed. This study shifts focus from the mean to the variance of arrival time. For a finite-bandwidth wave packet, we prove that in the narrow-band limit, the transmitted arrival-time variance becomes independent of barrier width and equals that of the incident packet—a spectral-pinning effect. This universal plateau, confirmed numerically within 0.5% accuracy, overturns the expectation that opaque barriers broaden pulses. Instead, the width is set by the incident spectrum via time-energy uncertainty. Beyond this limit, the barrier acts as a spectral filter, slightly narrowing the variance with opacity while shifting the mean above the Hartman plateau. These predictions are experimentally testable via attoclock measurements, reframing tunneling time as a robust statistical property rather than a problematic mean value.
We define, using a path-integral method, an amplitude distribution for the traversal time of a particle tunneling through a barrier. We calculate this without approximation for the case of a square barrier potential. The form of the integrated distribution is shown to be determined by the uncertainty principle: if the particle is known to spend a time \ensuremath{\le}\ensuremath{\tau} inside the barrier, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that \ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\approxeq}\ensuremath{\Elzxh}. The distribution has a number of interesting properties: it is noisy at very short times, oscillates periodically at intermediate times, and saturates to a constant at long times. The first two properties arise as a result of resonances in the transmission amplitude, which play an important role in the distribution. The characteristic (median) time scale for the distribution is shown to be ${\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}$=2
md2/\ensuremath{\Elzxh}. We show that this represents the average traversal time for particle energies that are close to the energy of the barrier height. For very opaque barriers, the oscillations have a very short period. We show that this can lead to anomalously short average traversal times, and resolves the outstanding problem that the average velocity of a tunneling particle may exceed the speed of light. For particles with energies above the height of the barrier (classical traversal), we find that the real part of the distribution is dominated by the classical path, and has a sharp peak around the classical traversal time. It is argued that the form of the distribution found in this work may be applied to general barrier shapes.
In this paper we examine critically and in detail some existing definitions for the tunnelling times, namely: the phase-time; the centroid-based times; the Buttiker and Landauer times; the Larmor times; the complex (path-integral and Bohm) times; the dwell time, and finally the generalized (Olkhovsky and Recami) dwell time, by adding also some numerical evaluations. Then, we pass to examine the equivalence between quantum tunnelling and "photon tunnelling" (evanescent waves propagation), with particular attention to tunnelling with Superluminal group-velocities ("Hartman effect"). At last, in an Appendix, we add a bird-eye view of all the experimental sectors of physics in which Superluminal motions seem to appear.
In this review (in Italian) we critically and detaily examine various definitions existing in the literature for the tunnelling times: namely, the phase-time; the centroid-based times; the Buttiker and Landauer times; the Larmor times; the complex (path-integral and Bohm) times; the dwell time, and the generalized (Olkhovsky and Recami) dwell time, with some numerical evaluations. Then, we pass to examine the equivalence between quantum tunnelling and "photon tunnelling" (evanescent waves propagation), with particular attention to tunnellings with Superluminal group-velocities ("Hartman effect"). At last, the main (Cologne, Berkeley, Florence, Vienna) experimental data about Superluminal evanescent wave propagation are briefly reviewed.
The wave-packet simulation (WPS) method for calculating the time a tunneling particle spends inside a one-dimensional potential barrier is reexamined using the Feynman path-integral technique. Following earlier work by Sokolovski and Baskin [Phys. Rev. A 36, 4604 (1987)], the tunneling (or traversal) time
tpackT is defined as a matrix element of a classical nonlocal functional between two states that represent the initial and transmitted wave packets. These states do not lie on the same orbit in Hilbert space; as a result,
tpackT is complex-valued. It is shown that Re
tpackT reduces to the standard WPS result,
tphaseT, for conditions similar to those employed in the conventional WPS analysis. Similarly, Im
tpackT is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, Im
tpackT reduces to the well-known Wentzel-Kramers-Brillouin expression for the tunneling time. It is shown there are different definitions for the traversal time of a classical moving object, whose size is comparable to the width of the region of interest. In the quantum case, these different definitions correspond to different ways of analyzing the WPS experiment. The path-integral approach demonstrates that the tunneling-time problem is one of understanding the physical significance of complex-valued off-orbit matrix elements of an operator or functional. The physical content of complex-valued tunneling times is discussed. It is emphasized that the use of complex tunneling times includes real-time approaches as a special case. Nevertheless, there is a limitation in the description of tunneling experiments using tunneling times, whether real or complex. The path-integral approach does not supply a universal traversal time, analogous to a classical time, that can be used in quantum situations. It is demonstrated that the often expressed hope of finding a well-defined and universal real tunneling time is erroneous.