On the concept of the tunneling time

We study the temporal aspects of quantum tunneling as manifested in time-of-arrival experiments in which the detected particle tunnels through a potential barrier. In particular, we present a general method for constructing temporal probabilities in tunneling systems that (i) defines `classical' time observables for quantum systems and (ii) applies to relativistic particles interacting through quantum fields. We show that the relevant probabilities are defined in terms of spe...

The phenomenon of quantum tunneling is reviewed and an overview of applying approximate methods for studying this effect is given. An approach to a time-dependent formalism is proposed in one dimension and generalized to higher dimensions. Some physical examples involving the resulting wavefunction which is determined are presented.

We present a proposal for the estimation of B\"uttiker-Landauer traversal time based on the visibility of transmission current. We analyze the tunneling phenomena with a time-dependent potential and obtain the time-dependent transmission current. We found that the visibility is directly connected to the traversal time. Furthermore, this result is valid not only for rectangular potential barrier but also for general form of potential to which the WKB approximation is applicabl...

The stationary phase method is often employed for computing tunneling {\em phase} times of analytically-continuous {\em gaussian} or infinite-bandwidth step pulses which collide with a potential barrier. The indiscriminate utilization of this method without considering the barrier boundary effects leads to some misconceptions in the interpretation of the phase times. After reexamining the above barrier diffusion problem where we notice the wave packet collision necessarily le...

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients $R$ or $T$ are then given by the total weight of all classical phase space trajectories co...

We theoretically study the tunneling time by investigating a wave packet of Bose-condensed atoms passing through a square barrier. We find that the tunneling time exhibits different scaling laws in different energy regimes. For negative incident energy of the wave packet, counterintuitively, the tunneling time decreases very rapidly with decreasing incident velocity. In contrast, for positive incident energy smaller than the barrier height, the tunneling time increases slowly...

Semiclassical approximations for tunneling processes usually involve complex trajectories or complex times. In this paper we use a previously derived approximation involving only real trajectories propagating in real time to describe the scattering of a Gaussian wavepacket by a finite square potential barrier. We show that the approximation describes both tunneling and interferences very accurately in the limit of small Plank's constant. We use these results to estimate the t...

We develop a new quantum-mechanical approach to scattering a particle on a one-dimensional (1D) system of two identical rectangular potential barriers, which implies modelling the dynamics of its subprocesses -- transmission and reflection -- at all stages of scattering. On its basis we define, for each subprocess, the dwell time as well as the local (exact) and asymptotic (extrapolated) group times. Our concept of the asymptotic transmission group time confirms the validity ...

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 i...

New time-dependent treatment of tunneling from localized state to continuum is proposed. It does not use the Laplace transform (Green's function's method) and can be applied for time-dependent potentials, as well. This approach results in simple expressions describing dynamics of tunneling to Markovian and non-Markovian reservoirs in the time-interval $-\infty<t<\infty$. It can provide a new outlook for tunneling in the negative time region, illuminating the origin of the tim...