How long does it take a particle or wave packet to tunnel through a barrier? This question has occupied physicists since the early days of quantum mechanics, when it was suggested that the process is instantaneous.1 In tunneling, a particle lacking the energy to go over a classically impenetrable potential barrier can nevertheless end up on the other side, albeit with small probability. This process is the basis for devices such as the scanning tunneling microscope, which enables imaging with atomic-scale resolution. It is also at the origin of phenomena such as alpha decay, which sets off nuclear fission. Resolving the issue of tunneling time is thus of both fundamental and practical importance.
Several experiments to determine tunneling time have been carried out using single photons or classical electromagnetic pulses that can tunnel through forbidden regions in the form of evanescent waves.2–4 These measurements show that the peak of the much attenuated tunneling wave packet appears at the exit of a barrier sooner than the peak of a wave packet that traversed an equal distance in free space. Since the wave packet in free space travels with the speed of light c, the widely accepted conclusion from these experiments has been that tunneling wave packets travel with superluminal (faster-than-light) group velocities. An even more surprising result is that the tunneling time, as measured by the arrival of the peak, becomes independent of barrier length for thick enough barriers (see Figure 1). This phenomenon, predicted by Hartman in 1962 (and termed the Hartman effect), has been observed in several experiments and is at the crux of the tunneling time conundrum.5 How can the wave packet know the barrier length increased and thus speed up to cover the increased distance in the same time? Our work solves this mystery by introducing a new paradigm: the group delay in tunneling is the lifetime of stored energy escaping through both ends of the barrier.6,7 It is not the transit time from input to output, as has been assumed for decades.
Figure 1. The Hartman effect, in which the tunneling time (group delay) of a wave packet saturates with barrier length.
A barrier is a filter (e.g., a multilayer dielectric mirror) that rejects a range of frequencies that lie within its ‘stop band’ while transmitting frequencies outside this band. For a pulse to tunnel, its spectrum must be narrow enough to fit neatly within the stop band, as shown in Figure 2. A narrow spectrum means that the pulse is long in time and has a spatial extent greater than the barrier length. As a result, the barrier is essentially a lumped element with respect to the pulse envelope, and the interaction is quasi-static.8,9 When such a long pulse is incident on a barrier, a standing wave is formed in front of the barrier as a result of the interference between incident and reflected waves. Inside the barrier is an evanescent wave that is also a standing wave with an exponentially decaying amplitude. In a standing wave, the field at every spatial point oscillates up and down with the same phase, and the entire distributed structure acts as if it were a lumped element. In reality, because this evanescent standing wave is in contact with transmitting boundaries, the energy it stores can leak through the boundaries. This storage and release of energy leads to a time delay and thus a phase shift between output and input. For a long enough barrier, the amplitude transmission function is
The group delay is the derivative of the transmission phase shift with respect to frequency:
and it tells us the time at which the transmitted field attains a peak at the exit. The transmission and group delay are shown in Figure 2. Note that inside the stop band the group delay is less than that of a pulse in free space.
Figure 2. Black curve: Transmission of a barrier as a function of frequency detuning from the center (Bragg) frequency. The stop band is the region of low transmission (between –5 and 5). Blue curve: Normalized group delay, which is the same as the normalized stored energy. The normalized free-space delay and stored energy is 1. In the stop band, the barrier delay and stored energy are less than the free-space values.
An interesting result is that the group delay is exactly equal to the stored energy in the barrier U divided by the incident power Pin:
where Ω c is the width of the stop band. For a pulse whose center frequency is tuned to the Bragg frequency, the stored energy is proportional to tanh κ L, a quantity that saturates with length. Since the group delay is proportional to stored energy, the Hartman effect or the saturation of the group delay with barrier length is thus explained by the saturation of stored energy.10 In the limit of a very long barrier, this delay saturates at the value τg = 1/Ωc, which is just the inverse of the filter bandwidth.
Since the group delay is proportional to the stored energy, we can now explain why the delay in the presence of a barrier is shorter than in its absence. Figure 3(a) shows the stored energy in a barrier-free region of length L when the peak of the input pulse is at z=0. The group delay is the time it takes to move all that stored energy out of the region in the forward direction. When a barrier of the same length is introduced, the stored energy in the region is reduced below the free-space value as a result of destructive interference. The group delay is the time it takes for this energy to leave the barrier in both directions: it is a cavity lifetime. Since the stored energy in the barrier is less than that in free space, the delay will be shorter.
Figure 3. (a) Blue curve: Snapshot of incident pulse in free space at the instant its peak reaches z=0. The green shaded area represents the energy stored in a region of length L for unity input power. (b) Blue curve: Snapshot of incident pulse and intensity distribution in the presence of barrier at the instant the peak reaches z=0. The green shaded area representing stored energy in the barrier is much smaller than the free-space value.
Because the delay is not a transit time, ‘group velocity’ is not a valid concept, whereas group delay remains a useful quantity that tells us how long it takes for stored energy to be released out of both ends of the barrier, most of it leaving in the backward direction. Because the group delay is a cavity lifetime, it does not imply superluminal velocity. In all cases, the delay is much shorter than the pulse length. For this reason, a better measure of the duration of the tunneling event is just the length of the pulse. Finally, in the middle of the stop band the transmission is flat, and there is no dispersion in the group delay. This means that the phase is linear in frequency and there is a pure delay without distortion or reshaping.11
Since tunneling is a phenomenon that is universal to all waves, whether matter, electromagnetic, water, or sound, a resolution of this problem has far-reaching implications. Our work has introduced a new paradigm that treats the group delay in tunneling as a lifetime rather than a transit time. This interpretation resolves the paradox of the Hartman effect and explains the anomalously short tunneling times observed in experiments without appealing to superluminal velocities. This opens the way to a deeper understanding of tunneling dynamics and to new methods to measure and control tunneling time.
Electrical Engineering and Computer Science Department
University of Michigan
Ann Arbor, MI
Herbert G. Winful received an SB from MIT in 1975 and a PhD in electrical engineering from the University of Southern California in 1981. From 1980 to 1986 he was a member of technical staff at GTE Laboratories, Waltham, MA. He joined the University of Michigan in 1987 and is currently an Arthur F. Thurnau Professor of Electrical Engineering. He is a fellow of the IEEE, the American Physical Society, and the Optical Society of America.