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arXiv:1807.01814

A probability distribution for quantum tunneling times

Jose Tadeu Lunardi, Luiz A. Manzoni

Published Jul 4, 2018• 0 citations
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Abstract

We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. This general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state after the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong field ionization.

DOI
10.48550/arxiv.1807.01814

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We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. This general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state after the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong field ionization.
We propose a general expression for the probability distribution of real-valued tunneling times of a localized particle, as measured by the Salecker-Wigner-Peres quantum clock. This general expression is used to obtain the distribution of times for the scattering of a particle through a static rectangular barrier and for the tunneling decay of an initially bound state after the sudden deformation of the potential, the latter case being relevant to understand tunneling times in recent attosecond experiments involving strong field ionization.
Abstract We consider a simple model for strong field ionization and use the Salecker-Wigner-Peres clock to obtain a distribution of (real) times associated with this phenomenon. By coupling the Salecker-Wigner-Peres clock to the particle in different regions, we are able to distinguish between the ionization (exit) time, the tunneling time and the time the particle spends in the well before entering the barrier (the well time). We show that the distribution of ionization/exit times, which are the times usually measured in attoclock experiments, may differ significantly from the distribution of tunneling times, suggesting that the ionization times cannot be generally interpreted as tunneling times.
We formulate quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles passing through a barrier at a detector located a distance L from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time of arrival determined in C. Anastopoulos and N. Savvidou, J. Math. Phys. 47, 122106 (2006). This only depends on the initial state, the Hamiltonian, and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that for a class of localized initial states, the detection probability allows for an identification of tunneling time with the classic phase time. We also establish limits to the definability of tunneling time. We then generalize these results to a sequential measurement setup: the phase-space properties of the particles are determined by an unsharp sampling before their attempt to cross the barrier. For such measurements the tunneling time is defined as a genuine observable. This allows us to construct a probability distribution for its values that is definable for all initial states and potentials. We also identify a regime in which these probabilities correspond to a tunneling-time operator.
We reexamine 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 nonrelativistic wave packet, sharply concentrated around a tunneling energy, incident on a symmetric double-barrier potential. In order to deepen the discussion about the generalized Hartman effect, we consider the case in which the clock runs only when the particle can be found inside the region between the barriers and show that, whenever the probability to find the particle in this region is non-negligible, the corresponding time (which in this case turns out to be a dwell time) increases with barrier spacing.
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-detector approach, one finds that these two complementary methods give very similar results. Due to the asymmetric distribution of the tunneling time, there is a nonnegligible discrepancy between the mean tunneling time and the most probable tunneling time.
The time-of-flight measurement approach of Peres based on the Salecker-Wigner quantum clock is applied to the one-dimensional scattering of a wave packet from a rectangular barrier. By directly evaluating the expectation value of the clock-time operator in the asymptotic states of wave packet long after the scattering process, we derive an average wave-packet clock time for the barrier interaction, which is expressed as an average of the stationary-state clock time over all possible initial scattering states of the wave packet. We show that the average wave-packet clock time is identical to the average dwell time of a wave packet.