Time-of-arrival probabilities and quantum measurements: II Application to tunneling times

TL;DR

This paper formulates quantum tunneling as a time-of-arrival problem using POVMs, linking tunneling time to classical phase time and defining it as a genuine observable in sequential measurements.

Contribution

It introduces a POVM-based approach to define and interpret tunneling times, including a regime where tunneling time becomes a well-defined quantum observable.

Findings

Detection probability aligns with classical phase time for localized states.

Limits are established for the definability of tunneling time.

Sequential measurements enable a genuine observable for tunneling time.

Abstract

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 quant-ph/0509020 [JMP 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 set-up: 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.