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.
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.
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.
Using the time parameter in the time-dependent Schr\"odinger equation, we study the time of flight for a particle tunneling through a square barrier potential. Comparing the mean and variance of the energy and the flight time for transmitted and reflected particles, using both density and flux distributions, we find that, when accounting for momentum filtering, the suitably normalized transmitted and reflected distributions are identical in both the density and flux cases. In contrast to previous studies, we demonstrate that these results do not imply a vanishing tunneling time, but rather that the time it takes to tunnel through a square barrier is precisely given by the reflected phase time. For wide barriers, this becomes independent of the barrier width, as predicted independently by MacColl and Hartman. We show that these conclusions can be reached using a variety of arguments, including purely quantum mechanical ones. Analysis of the shapes of the distributions under consideration reveals that wave-packet reshaping is not an explanation for the MacColl-Hartman effect. The results presented here have direct implications for understanding recent experimental results in the study of the barrier crossing of rubidium atoms. The finite width of an incident wave packet significantly ``masks'' the tunneling time, and induces substantial asymmetry between the flight times of transmitted and reflected atoms.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.
Abstract Quantum tunnelling is a fundamental phenomenon in quantum physics, yet the question of how long it takes for a particle to tunnel through a barrier remains a topic of ongoing debate. Recent advances in attoclock experimental techniques in strong-field physics have renewed interest in this topic. While various definitions of tunneling time have been proposed, a consensus has yet to be reached. In this study, we adopt a definition of tunnelling time based on the ratio of the barrier width to the expected value of the velocity operator within the barrier. This approach reduces to the Büttiker-Landauer time (
τBL) in the case of an opaque barrier and offers a clear physical interpretation for very thin barriers, where
τBL may not be well-defined. Additionally, by utilizing a virtual detector at the tunnel exit, we calculate the time delay between the peak of the electric field and the moment of ionization for long wavelengths ($\geq$6
μm). Our findings indicate that the time delay stabilizes around 190 as, remaining independent of the laser wavelength. Furthermore, through our quantum tunnelling time definition, this time delay can be interpreted as the time required for the electron to tunnel from a point within the barrier to the exit.
The time of passage of the transmitted wave packet in a tunneling collision of a quantum particle with a square potential barrier becomes independent of the barrier width in a range of barrier thickness. This is the Hartman effect, which has been frequently associated with ``superluminality.'' A fundamental limitation on the effect is set by nonrelativistic ``causality conditions.'' We demonstrate first that the causality conditions impose more restrictive bounds on the negative time delays (time advancements) when no bound states are present. These restrictive bounds are in agreement with a naive, and generally false, causality argument based on the positivity of the ``extrapolated phase time,'' one of the quantities proposed to characterize the duration of the barrier's traversal. Nevertheless, square wells may in fact lead to much larger advancements than square barriers. We point out that close to the thresholds of new bound states, the time advancement increases considerably, while, at the same time, the transmission probability is large, which facilitates the possible observation of the enhanced time advancement.