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Physical review. B, Condensed matter

Path decomposition and the traversal-time distribution in quantum tunneling

H. A. Fertig

Published Jan 15, 1993• 33 citations
Publisher

Abstract

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${\mathit{md}}^{2}$/\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.

DOI
10.1103/physrevb.47.1346

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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 \ensuremathτQ{\mathrm{\ensuremath{\tau}}}_{\mathit{Q}}\ensuremathτQ​=2md2{\mathit{md}}^{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.
We derive the exact distribution of times spent in the classically forbidden region by a particle as it tunnels through a square barrier potential. We find that the integrated distribution has a simple interpretation. If one restricts the time spent in the barrier to be \ensuremath{\leqslant}\ensuremath{\tau}, then an effective uncertainty \ensuremath{\Delta}V is introduced in the barrier height, such that\ensuremath{\Delta}V\ensuremath{\tau}\ensuremath{\simeq}1. This uncertainty leads to unusual interference effects in the distribution. These interference effects explain why the average speed of a particle traversing a very opaque barrier may exceed the speed of light. \textcopyright{} 1990 The American Physical Society
The path integral formulation of the traversal time in tunneling utilizes a restricted Feynman amplitude. We present a derivation based on the Fisher Lee formula to compute the average time and the integrated distribution for arbitrary potentials. The (complex) tunneling time can be expressed as a derivative with respect to a potential which surrounds the region over which the particle is “detected”, and a correspondence with other tunneling time scales which depend on energy derivatives of the transmission coefficient is found. A semiclassical treatment confirms that there are two time scales associated with this measurement procedure. The imaginary component is related to the multiple instanton bounces in the inverted potential. The issue of positive times is found to depend crucially on the behavior of the transmission coefficient for asymptotic limits of the measurement potential. The calculation of higher moments of the distribution allows to define a positive barrier interaction time, without appealing to any regularization procedure.
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 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{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}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{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is complex-valued. It is shown that RetpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ reduces to the standard WPS result, tphaseT{\mathit{t}}_{\mathrm{phase}}^{\mathit{T}}tphaseT​, for conditions similar to those employed in the conventional WPS analysis. Similarly, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}tpackT​ is shown to contain information about the energy dependence of the transmission probability. Under semiclassical conditions, ImtpackT{\mathit{t}}_{\mathrm{pack}}^{\mathit{T}}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.
Barrier penetration is attributed to energy fluctuations expected from the uncertainty principle. Numerical simulations are made by calculating the traversal time and action for a large number of possible velocity profiles. Distributions of traversal time are determined by assuming that the probability of each velocity profile decreases exponentially with the action of the fluctuation it requires. Distributions of traversal times are reported for rectangular barriers having different sizes. For large barriers the distributions are leptokurtic and centered at the semiclassical traversal time T0 = d√m/[2(V0−E)], where d and V0 are the length and height of the barrier and m and E are the mass and energy of the particle. The kurtosis decreases and the mode shifts to shorter durations with decreasing barrier size.
A quantum mechanical transition path time probability distribution is formulated and its properties are studied using a parabolic barrier potential model. The average transit time is well defined and readily calculated. It is smaller than the analogous classical mechanical average transit time, vanishing at the crossover temperature. It provides a direct route for determining tunneling times. The average time may be also used to define a coarse grained momentum of the system for the passage from one side of the barrier to the other. The product of the uncertainty in this coarse grained momentum with the uncertainty in the location of the particle is shown under certain conditions to be smaller than the ℏ/2 formal uncertainty limit. The model is generalized to include friction in the form of a bilinear interaction with a harmonic bath. Using an Ohmic friction model one finds that increasing the friction, increases the transition time. Only moderate values of the reduced friction coefficient are needed for the quantum transition time and coarse grained uncertainty to approach the classical limit which is smaller than ℏ/2 when the friction is not too small. These results show how one obtains classical dynamics from a pure quantum system without invoking any further assumptions, approximations, or postulates.