Numerous studies of one-dimensional scattering systems have failed to provide a satisfactory theory for tunneling times. Although there is some support for the idea that such interaction times should be complex, the physical meaning of complex interaction times has defied explanation. Here we demonstrate an explicit relation between several proposed complex times and a Hermitian sojourn-time operator. We have previously shown that this operator describes the effect of a perturbation of the scattering system on observable quantities.
Numerous studies of one-dimensional scattering systems have failed to provide a satisfactory theory for tunneling times. Although there is some support for the idea that such interaction times should be complex, the physical meaning of complex interaction times has defied explanation. Here we demonstrate an explicit relation between several proposed complex times and a Hermitian sojourn-time operator. We have previously shown that this operator describes the effect of a perturbation of the scattering system on observable quantities.
A general formulation of the sojourn-time operator approach is presented, valid for arbitrary scattering systems and free of restrictions to one-dimensional systems present in the previous work of the authors. We show how first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. Both time-dependent and time-independent perturbations, including coupling between translation and internal degrees of freedom, are considered. The mean values of these Hermitian sojourn-time operators give the mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system defined by sets of commuting observables. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, to a first-order approximation, fully expressible in terms of these operators. We also show that this approach provides a unified treatment of the various idealized one-dimensional systems, which have frequently been employed in the debate on tunneling and interaction times. In particular, we provide some insight into the role of complex times, which appear as combinations of matrix elements of the sojourn-time operators.
The system studied is a two-level particle whose translational motion is one dimensional and restricted to the half line (0,\ensuremath{\infty}). The internal degrees of freedom of the particle are weakly coupled to the translational motion via a coupling including both adiabatic and nonadiabatic parts. We explore the idea of describing the interaction times of this scattering system in terms of Hermitian sojourn-time operators. Treating the coupling as a perturbation, we show that the first-order perturbation theory can be consistently formulated in terms of appropriate sojourn-time operators. The mean values of these operators give mean sojourn time (dwell time) of the particle in spatial regions or, more generally, in certain subspaces of the state space of the system. The perturbed S operator and also the effect of the perturbation on the change of observables due to the scattering are, in first-order approximation, fully expressible in terms of the sojourn-time operators. In particular, these operators fully determine the first-order approximation to the inelastic scattering probabilities. The relation to the case of scattering on the whole line (-\ensuremath{\infty},\ensuremath{\infty}) is briefly discussed.
A ready comparison of the two methods noted in the title is obscured by the very different formalisms that they employ. Our sojourn-time approach [Jaworski and Wardlaw, Phys. Rev. A 45, 292 (1992)] is based on a Hermitian sojourn-time operator in the framework of time-dependent scattering theory whereas Sokolovski and co-workers [Sokolovski and Baskin, Phys. Rev. A 36, 4604 (1987); Sokolovski and Connor, ibid. 42, 6512 (1990)] have developed an approach based on the Feynman path-integral formulation of quantum mechanics. By translating the path-integral approach into the language of wave functions and linear operators, we reveal the similarities of the two approaches and show that the results for the interaction times are closely related, becoming equivalent in the asymptotic limit.
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 discuss some aspects of the time picture of tunneling for open quantum systems described by non-Hermitian (NH) Hamiltonians. The concept of sojourn time for such systems is introduced in the framework of the biorthonormal formalism. Due to the various definitions of probability density in the non-Hermitian case, we get three different sojourn times, two real and one complex. We consider as model of a dissipative NH system the complex, generalized parametric oscillator, for which we derive the exact expressions of the three sojourn times in terms of the Wei-Norman characteristic functions entering the non-unitary evolution operators. The special case of the inverted Caldirola-Kanai oscillator with complex friction parameter is investigated for an initial extended wavepacket. We also discuss the Landau-Zener-like transitions of the NH parametric oscillator, i.e. the dissipative tunneling through a dynamical barrier due to the perturbative effect of the damping.
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.