Our previous results [Phys. Rev. A 37, 2843 (1988)], based on the concept of sojourn time, for the transmission and reflection time delays in one-dimensional scattering by a potential barrier are compared to the corresponding results of E. H. Hauge, J. P. Falck, and T. A. Fjedly [Phys. Rev. B 36, 4203 (1987)], based on the idea of following the time evolution of the mean position of wave packets. It is shown that the mean-position approach agrees with the more general sojourn-time approach in the limit of well-defined momentum, and that, in general, the two approaches appear to be related in a classical-like manner.
We demonstrate that although the well-known analogy between the time-independent solutions for two-dimensional tunneling (e.g., frustrated total internal reflection) and tunneling through a one-dimensional potential barrier cannot, in general, be extended to the time domain, there are certain limits in which the delay times for the two problems obey a simple relationship. In particular, when an effective mass is chosen such that
mc2=\ensuremath{\Elzxh}\ensuremath{\omega}, the ``classical'' traversal times for allowed transmission become identical for a photon of energy \ensuremath{\Elzxh}\ensuremath{\omega} traversing an air gap between regions of index n and for a particle of mass m traversing the analogous square barrier of height
V0 in one dimension. The quantum-mechanical group delays are also identical, given this effective mass, both for E\ensuremath{\approxeq}
V0 (\ensuremath{\theta}\ensuremath{\approxeq}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$) and for E\ensuremath{\gg}
V0 (\ensuremath{\theta}\ensuremath{\ll}${\mathrm{\ensuremath{\theta}}}_{\mathit{c}}$). (For a smoothly varying potential or index of refraction, the agreement persists for all values of E where the WKB approximation applies.) The same relation serves to equate the quantum-mechanical ``dwell'' times for any values of E and
V0. On the other hand, in the ``deep tunneling'' limit, E\ensuremath{\ll}
V0 (\ensuremath{\theta}\ensuremath{\approxeq}\ensuremath{\pi}/2), one must choose
mc2=
n2\ensuremath{\Elzxh}\ensuremath{\omega} in order to make the group delays equal for the two problems. These equivalences simplify certain calculations, and the two-dimensional analogy may also be useful for geometrically visualizing the tunneling process and the anomalously small group delays known to occur in the opaque limit. We also demonstrate that the equality of the group delays for transmission and reflection for lossless barriers follows from a simple intuitive argument based on time-reversal invariance, and discuss the extension of the result to the case of lossy barriers.
The problem of time delay in one-dimensional quantum-mechanical scattering by a potential barrier is studied in the framework of time-dependent scattering theory. It is shown how the concept of sojourn time can be used to define three separate time delays: the time delay for transmission through the barrier, the time delay for reflection, and the total time delay (i.e., averaged over transmission and reflection). In addition to the anticipated dependence on the S matrix and its energy derivative, the time delays depend on certain parameters which can be interpreted as describing positions of detectors in a gedanken-experiment measurement. The problem of defining separate time delays for transmission and reflection appears to be rooted in the foundations of quantum theory and its interpretation.
Quantum tunneling in one spatial dimension in the presence of time-dependent potentials is investigated theoretically. First, a general multichannel formulation of the problem is given for harmonic time-dependence. The case of an oscillatory delta-function potential of constant strength is discussed at length and specific numerical calculations are presented for the reflection and transmission coefficients. Then other exactly solvable time-dependent potentials are obtained by carrying out successive unitary transformations. As an example of this class, a delta-function potential with time-dependent strength and boundary conditions is studied. Finally tunneling time-delay for time-dependent potentials is formulated in the multichannel situation, and also the concept of first passage time for the decay of a wave packet confined to one well of a double well potential is generalized for the case of time-dependent barrier and boundary conditions.
The analogy between quantum tunneling of particles and evanescent electromagnetic waves can be used to study particle tunneling. Possibilities for the measurement of the time delay resulting from the transmission of waves through an evanescent region, say with lower dielectric constant, are discussed. In contrast to particle tunneling, an electromagnetic pulse can consist of many photons and can be probed in a noninvasive way. We emphasize the delay of the centroid of an electromagnetic pulse transmitted below cutoff through a portion of a waveguide with the same cross section as the adjacent propagating guides. The boundary conditions at the interface between the propagating and the evanescent region lead to the same transmission and reflection coefficients as for a square-barrier tunneling problem. For a pulse restricted to a narrow frequency range, the time delay depends only on the frequency derivative of the phase shift associated with transmission. The delay of a centroid is just that; there is no deeper physical sense which links the incoming centroid to the outgoing centroid. For sufficiently long evanescent regions, the delay is independent of thickness.
Abstract We present a step by step introduction to the notion of time-delay in classical and quantum mechanics, with the aim of clarifying its foundation at a conceptual level. In doing so, we motivate the introduction of the concepts of “fuzzy” and “free-flight” sojourn times that we use to provide the most general possible definition for the quantum time-delay, valid for simple and multichannel scattering systems, with or without conditions on the observation of the scattering particle, and for incoming wave packets whose energy can be smeared out or sharply peaked (fixed energy). We conclude our conceptual analysis by presenting what we think is the right interpretation of the concepts of sojourn and delay times in quantum mechanics, explaining why, in ultimate analysis, they should not be called “times.”