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arXiv:0801.4308 · Physical Review A

Path-integral analysis of arrival times with a complex potential

J. J. Halliwell

Published Jun 13, 2008• 25 citations
arXivHTMLPDF

Abstract

A number of approaches to the arrival time problem employ a complex potential of a simple step function type and the arrival time distribution may then be calculated using the stationary scattering wave functions. Here, it is shown that in the Zeno limit (in which the potential becomes very large), the arrival time distribution may be obtained in a clear and simple way using a path integral representation of the propagator together with the path decomposition expansion (in which the propagator is factored across a surface of constant time). Crucially, this method shows that the resulting arrival time distribution, in the Zeno limit, is in fact independent of the form of the complex potential.

DOI
10.1103/physreva.77.062103

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A number of approaches to the arrival time problem employ a complex potential of a simple step function type and the arrival time distribution may then be calculated using the stationary scattering wave functions. Here, it is shown that in the Zeno limit (in which the potential becomes very large), the arrival time distribution may be obtained in a clear and simple way using a path integral representation of the propagator together with the path decomposition expansion (in which the propagator is factored across a surface of constant time). Crucially, this method shows that the resulting arrival time distribution, in the Zeno limit, is in fact independent of the form of the complex potential.
We carry out a decoherent-histories analysis of the arrival-time problem, taking advantage of a recently demonstrated connection between time-ordered strings of projection operators and evolution in the presence of a complex potential of step-function form. We concentrate on the limit of a weak potential, in which the resulting arrival-time distribution function is closely related to the quantum-mechanical current. We first consider the analogous classical arrival-time problem involving an absorbing potential, and this sheds some light on certain aspects of the quantum case. We use the path-decomposition expansion to give a derivation of the standard arrival-time distribution defined using a complex potential. This derivation is then used in the decoherent-histories analysis to obtain very simple and plausible expressions for the class operators (describing the amplitudes for crossing the origin during intervals of time). We show that decoherence of histories is obtained for a wide class of initial states (such as simple wave packets and superpositions of wave packets). We find that the decoherent-histories approach gives results with a sensible classical limit that are fully compatible with standard results on the arrival-time problem. We also find some interesting connections between backflow and decoherence.
The propagator approach combined with the multiple-scattering theory is applied to the particle time of arrival (TOA) problem. This approach allows us to naturally include in the consideration the components of the particle initial wavefunction (defined at t = t0) corresponding to the positive (forward-moving term) and negative (backward-moving term) momenta. For a freely moving particle it is shown that the Allcock definition of the ideal total TOA probability disregards the backward-moving and interference terms entirely. In the presence of a measuring apparatus modeled by an imaginary step potential with the amplitude V0, the general expression for the TOA rate is obtained, the forward-moving component of which coincides with that obtained by Allcock. It is shown that when the initial particle wavefunction is well separated from the point of arrival and has a well-defined average momentum, the contribution of the backward-moving and interference terms are small and can be neglected. For a small V0, except the well-known convolution result by Allcock–Kijowski, the exponential form of the TOA rate follows at the double limit condition V0 → 0, t − t0 ~ /2V0 → ∞ (2V0(t − t0)/ is finite) while the backward-moving and interference terms vanish. We show that the Allcock result for the TOA rate is valid in the entire range of V0 including the Zeno case (V0 → ∞) and the normalized TOA rate can be introduced for all values of V0 as a probability distribution. The latter is illustrated for the Gaussian wave packet.
It is of interest in a variety of contexts, and in particular in the arrival time problem, to consider the quantum state obtained through unitary evolution of an initial state regularly interspersed with periodic projections onto the positive x -axis (pulsed measurements). Echanobe, del Campo and Muga have given a compelling but heuristic argument that the state thus obtained is approximately equivalent to the state obtained by evolving in the presence of a certain complex potential of step-function form. In this paper, with the help of the path decomposition expansion of the associated propagators, we give a detailed derivation of this approximate equivalence. The propagator for the complex potential is known so the bulk of the derivation consists of an approximate evaluation of the propagator for the free particle interspersed with periodic position projections. This approximate equivalence may be used to show that to produce significant reflection, the projections must act at time spacing less than ℏ/ E , where E is the energy scale of the initial state.
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.