Questioning the adequacy of certain quantum arrival-time distributions

Abstract

It is shown that a class of exponentially decaying time-of-arrival probability distributions suggested by Włodarz, Marchewka and Schuss, and Jurman and Nikolić, as well as a semiclassical distribution implicit in timeof-flight momentum measurements, do not show the expected behavior for a Gaussian wave train. This casts doubts on the physical adequacy of these arrival-time proposals. In contrast, the quantum flux distribution (a special case of the Bohmian arrival-time distribution) displays the expected behavior.

When does a particle arrive?

“…The TOA probability distribution of Eq. ( 8) can also be derived in the framework of Bohmian mechanics [2,3,29]. In [2] the authors discuss the existence of free particle states with definite momentum sign for which the probabilistic interpretation of the quantum flux is still forbidden.…”

“…with Ψ(p) the components of the Fourier transform of Ψ(x, 0) and L the distance between the source of the wave and the detector [2,3]. Due to the nature of this definition, Eq.…”

“…We denote x k and p k respectively the mean position and momentum of the k-th Gaussian wave packet. An analysis similar to [3] revealed that all the TOA proposals are well-behaved for a Gaussian train that satisfies a no-spreading condition, namely when quantum diffusion is negligible and the propagation of each packet is approximately a rigid translation. In this regime, each TOA distribution yields an equiprobable sequence of peaks centred around the classical TOA |x k m/p k | [37].…”

“…We explore different regimes (see Fig. 1) where we can compare the Kijowski axiomatic construction [7] (which is in agreement with [4][5][6]), the quantum flux approach [3,8] and the quantum clock proposal [19,20]. Our analysis is not limited to predicting a specific experimental result but it highlights those regimes in which different theoretical approaches lead to contradicting predictions.…”

“…In experiments the TOA is usually computed using the semiclassical approximation [2], which holds only in regimes where quantum effects can be neglected, i.e. if the particle is approximated by a well localized wave packet with negligible spatial uncertainty compared to the detector dimension [3]. Due to the limited validity of this approach, several quantum time and quantum TOA formulations have been proposed.…”