Time of arrival in the presence of interactions

Abstract

We introduce a formalism for the calculation of the time of arrival t at a space point for particles traveling through interacting media. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t in the context of the positive-operator-valued measures. We then compute the probability distribution in the times of arrival at a point for particles that have undergone reflection, transmission or tunneling off finite potential barriers. For narrow Gaussian initial wave packets we obtain multimodal time distribution of the reflected packets and a combination of the Hartman effect with unexpected retardation in tunneling. We also employ explicitly our formalism to deal with arrivals in the interaction region for the step and linear potentials.

Manufacturing time operators: Covariance, selection criteria, and examples

“…On the theory side, an open question is how to adapt the proposed framework, possibly in combination with previous investigations [18][19][20][22][23][24][25], to arrival times when a particle moves in a potential.…”

“…If the original incoming trajectory requires a certain time τ to arrive at the origin (with free motion), the reversed trajectory is outgoing, and departed from the origin at −τ . These operators provide in summary information of the free-motion dynamics of incoming and outgoing asymptotes of the state, and scattering time delays [19,20]. Thus, although the operatorT A Θ is unique when one applies the criteria of the previous section it does not supersedeT A ± since it does not describe the same physics, and all three operators have their own legitimacy.…”

Time-of-arrival distributions for interaction potentials

“…These distributions previously appeared in [20] (later superseded by [21]). It is important to notice that this is not the way in which the original distributions in [20] were presented; they have been adapted here to the unified notation we are using in order to achieve a better handle for comparison.…”

“…These distributions were first introduced and discussed by León et al in [21], as justified by quantization through quantum canonical transformations of the classical time of arrival (see also [22] for further analysis of this proposal). As previously stated, the rewriting in terms of crossing states is intended to clarify the physical consequences of these distributions.…”

“…It is important to notice that this is not the way in which the original distributions in [20] were presented; they have been adapted here to the unified notation we are using in order to achieve a better handle for comparison. Their physical content is made evident by making use of (14), (15), and the isometry of the Möller operators,…”