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arXiv:quant-ph/0311107 · Physical Review A

Operator-normalized quantum arrival times in the presence of interactions

Gerhard C. Hegerfeldt, D. Seidel, Juan Gonzalo Muga, B. Navarro

Published Jul 22, 2004• 38 citations
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Abstract

We model ideal arrival-time measurements for free quantum particles and for particles subject to an external interaction by means of a narrow and weak absorbing potential. This approach is related to the operational approach of measuring the first photon emitted from a two-level atom illuminated by a laser. By operator normalizing the resulting time-of-arrival distribution, a distribution is obtained which for freely moving particles not only recovers the axiomatically derived distribution of Kijowski for states with purely positive momenta but is also applicable to general momentum components. For particles interacting with a square barrier the mean arrival time and corresponding ``tunneling time'' obtained at the transmission side of the barrier become independent of the barrier width (Hartman effect) for arbitrarily wide barriers, i.e., without the transition to the ultraopaque, classical-like regime dominated by wave packet components above the barrier.

DOI
10.1103/physreva.70.012110

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We model ideal arrival-time measurements for free quantum particles and for particles subject to an external interaction by means of a narrow and weak absorbing potential. This approach is related to the operational approach of measuring the first photon emitted from a two-level atom illuminated by a laser. By operator normalizing the resulting time-of-arrival distribution, a distribution is obtained which for freely moving particles not only recovers the axiomatically derived distribution of Kijowski for states with purely positive momenta but is also applicable to general momentum components. For particles interacting with a square barrier the mean arrival time and corresponding ``tunneling time'' obtained at the transmission side of the barrier become independent of the barrier width (Hartman effect) for arbitrarily wide barriers, i.e., without the transition to the ultraopaque, classical-like regime dominated by wave packet components above the barrier.
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.
The theoretical analysis of time observables in quantum mechanics has been controversial and is still subject to debate. In the present work, a measurement-based approach to quantum arrival times is studied in detail. It is based on the emission time of the first spontaneous photon from a a two-level atom moving into a laser-illuminated region. This quantum-optical model is shown to give an operational understanding of various ideal proposals for quantum arrival-time distributions as well as for local kinetic energy densities. As a main result, a first measurement-based procedure to Kijowski`s arrival-time distribution is given and the approach is used to generalize this distribution to the case of particles interacting with an external potential.
A recent approach to arrival times used the fluorescence of an atom entering a laser illuminated region, and the resulting arrival-time distribution was close to the axiomatic distribution of Kijowski, but not exactly equal, neither in limiting cases nor after compensation of reflection losses by normalization on the level of expectation values. In this paper we employ a normalization on the level of operators, recently proposed in a slightly different context. We show that in this case the axiomatic arrival-time distribution of Kijowski is recovered as a limiting case. In addition, it is shown that Allcock's complex potential model is also a limit of the physically motivated fluorescence approach and connected to Kijowski's distribution through operator normalization.
We introduce a formalism for the calculation of the time of arrival t at a detector of particles traveling through interacting environments. We develop a general formulation that employs quantum canonical transformations from the free to the interacting cases to compute t. We interpret our results in terms of a Positive Operator Valued Measure. We then compute the probability distribution in times of arrival at a detector of those particles that, after their initial preparation, have undergone quantum tunneling or reflection due to the presence of potential barriers. We obtain the expected retardation or advancement for transmitted wave packets, and non-foreseen double bump structures for some cases of reflection.
Detector-based first-arrival models and operator-based arrival-time observables provide two conceptually distinct approaches to the quantum time-of-arrival problem. Here we connect these approaches by using the positive unidirectional first-arrival current of Marchewka and Schuss (MS) as the basis for constructing a family of positive arrival-time operators and, after normalization, an arrival-time POVM. The first-arrival character is inherited from the dynamics with a Dirichlet boundary, while the detector response is introduced at the amplitude level through a non-negative spectral function λ(k)λ(k)λ(k). For a one-dimensional free particle, resolution of the identity uniquely selects λ(k)=π4kλ(k)=\fracπ{4k}λ(k)=4kπ​, yielding an arrival-time POVM. The normalized unidirectional current can therefore be interpreted as a generalized Born-rule probability density rather than as a hazard rate, as in the original MS formulation. We further show that the normalized one-sided MS amplitude coincides, up to an irrelevant phase, with the corresponding momentum-sector amplitude in Kijowski's arrival-time distribution. Consequently, the two directional contributions in Kijowski's distribution can be reproduced by two first-arrival problems defined on opposite sides of the boundary. This establishes an equivalence between the resulting arrival-time statistics while emphasizing the different physical interpretations: MS describes a first-arrival process with coherent momentum components, whereas Kijowski's POVM treats the two momentum sectors as separate directional contributions. Thus, the normalized unidirectional current provides a POVM description of first arrival at a point, combining detector-based dynamics with a positive-operator measurement structure.