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arXiv:quant-ph/0209027 · Physical Review A

Measurement-based approach to quantum arrival times

J. A. Damborenea, I. L. Egusquiza, Gerhard C. Hegerfeldt, Juan Gonzalo Muga

Published Nov 15, 2002• 103 citations
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Abstract

For a quantum-mechanically spread-out particle we investigate a method for determining its arrival time at a specific location. The procedure is based on the emission of a first photon from a two-level system moving into a laser-illuminated region. The resulting temporal distribution is explicitly calculated for the one-dimensional case and compared with axiomatically proposed expressions. As a main result we show that by means of a deconvolution one obtains the well-known quantum-mechanical probability flux of the particle at the location as a limiting distribution.

DOI
10.1103/physreva.66.052104

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For a quantum-mechanically spread-out particle we investigate a method for determining its arrival time at a specific location. The procedure is based on the emission of a first photon from a two-level system moving into a laser-illuminated region. The resulting temporal distribution is explicitly calculated for the one-dimensional case and compared with axiomatically proposed expressions. As a main result we show that by means of a deconvolution one obtains the well-known quantum-mechanical probability flux of the particle at the location as a limiting distribution.
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.
We present a calculation of the arrival-time probability distribution based on the extended probability, as proposed by Hartle (2008 Phys. Rev. A 78 012108). We show that two cases of the arrival-time probability distribution can be used. First, we use the path decomposition expansion to discuss a fine-grained history, for which a particle has a first crossing x = 0 at time τ. The arrival-time probability, related to the probability current, is obtained. Second, we consider that the arrival-time distribution is modeled by the first photon emitted from a two-level atom (the basic atom–laser model). The partition dynamic of this system is discussed. Each history is in the form of a coarse-grained history. We confirm that the arrival-time probability distribution computed by this method agrees with that computed via the probability of no-photon detection. Crucially, the resulting arrival-time distributions can be obtained without the restriction of the consistency condition.
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.
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 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 study the problem of computing the probability for the time of arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\"odinger current. We derive a well-defined uncertainty relation between time of arrival and energy; this result shows that the well-known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time representation'' of the quantum mechanics of a free particle, in which the time of arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics. \textcopyright{} 1996 The American Physical Society.