A probability distribution for quantum tunneling times

In this work we revisit the Salecker-Wigner-Peres clock formalism and show that it can be directly applied to the phenomenon of tunneling. Then we apply this formalism to the determination of the tunneling time of a non relativistic wavepacket, sharply concentrated around a tunneling energy, incident on a symmetric double barrier potential. In order to deepen the discussion about the generalized Hartmann effect, we consider the case in which the clock runs only when the parti...

The Salecker-Wigner-Peres quantum-clock approach is applied in order to determine the tunneling time of an electron in strong-field tunnel ionization via a time-dependent electric field. Our results show that the ionization of the electron takes a nonvanishing period of time. This tunneling time is of the order of the Keldysh time but strictly larger than the Keldysh time. Comparing the quantum-clock tunneling time to the mean tunneling time as obtained by the virtual-detecto...

The quantum clock of Salecker-Wigner-Peres is used, by performing a post-selection of the final state, to obtain average transmission and reflection times associated to the scattering of localized wave packets by static potentials in one dimension. The behavior of these average times is studied for a gaussian wave packet, centered around a tunneling wave number, incident on a rectangular barrier and, in particular, on a double delta barrier potential. The regime of opaque bar...

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...

I propose to consider photon tunneling as a space-time correlation phenomenon between the emission and absorption of a photon on the two sides of a barrier. Standard technics based on an appropriate counting rate formula may then be applied to derive the tunneling time distribution without any {\em ad hoc} definition of this quantity. General formulae are worked out for a potential model using Wigner-Weisskopf method. For a homogeneous square barrier in the limit of zero tunn...

We discuss the propagation of wave packets through interacting environments. Such environments generally modify the dispersion relation or shape of the wave function. To study such effects in detail, we define the distribution function P_{X}(T), which describes the arrival time T of a packet at a detector located at point X. We calculate P_{X}(T) for wave packets traveling through a tunneling barrier and find that our results actually explain recent experiments. We compare ou...

Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: One class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like i.e the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which c...

Asymptotic time evolution of a wave packet describing a non-relativistic particle incident on a potential barrier is considered, using the Wigner phase-space distribution. The distortion of the trasmitted wave packet is determined by two time-like parameters, given by the energy derivative of the complex transmission amplitude. The result is consistent with various definitions of the tunneling time (e.g. the B\"{u}ttiker-Landauer time, the complex time and Wigner's phase time...

Quantum particles interacting with potential barriers are ubiquitous in physics, and the question of how much time they spend inside classically forbidden regions has attracted interest for many decades. Recent developments of new experimental techniques revived the issue and ignited a debate with often contradictory results. This motivates the present study of an exactly solvable model for quantum tunneling induced by a strong field. We show that the tunneling dynamics can d...

We calculate a tunneling time distribution by means of Nelson's quantum mechanics and investigate its statistical properties. The relationship between the average and deviation of tunneling time suggests the exsistence of ``wave-particle duality'' in the tunneling phenomena.