Quantum stroboscopy for time measurements

Abstract

Mielnik's cannonball argument uses the Zeno effect to argue that projective measurements for time of arrival are impossible. If one repeatedly measures the position of a particle (or a cannonball!) that has yet to arrive at a detector, the Zeno effect will repeatedly collapse its wavefunction away from it: the particle never arrives. Here we introduce quantum stroboscopic measurements where we accumulate statistics of projective position measurements, performed on different copies of the system at different times, to obtain a time-of-arrival distribution. We show that, under appropriate limits, this gives the same statistics as time measurements of conventional ``always on'' particle detectors, that bypass Mielnik's argument using non-projective, weak continuous measurements. In addition to time of arrival, quantum stroboscopy can describe distributions of general time measurements. It can also be adapted to obtain the conditional probability distribution of arrival times, given that the particle was not previously detected at the detector.

Figures (2)

• Figure 1: Quantum stroboscopy for position. Divide the total measurement duration $T$ into $M$ intervals $\tau=T/M$. Then perform $L$ position measurements at times ${t}_m=m\tau+{t}_0$ (with $m=0,1,\cdots,M-1$) where $L\gg N$ is larger than the number $N$ of histogram bins at each time. Populate the $m$th histogram with the outcomes (vertical lines): add an event to the $n$th row, $m$th column for a position result $n\delta x$ (with $n=0,\cdots,N-1$) at time ${t}_m$ ($\delta x$ is the spatial resolution). Normalize the rows, to obtain the time-of-arrival probabilities at positions $n\delta x$ as $p(t_m|n)= \ell_{nm}/\sum_m \ell_{nm}$, where $\ell_{nm}$ is the number of shots with outcome $n$ at time ${t}_m$. By construction, this is normalized over time for all values of $n$. [It is undefined if there are rows $n$ with no events: the particle never "arrives" at location $n\delta x$]. A similar procedure can produce time probabilities of arbitrary measurements. Here we simulated a Gaussian packet propagating downward. The colormap and the lower graphs represent the probability $p(n|t_m)$ for each outcome at $t_m$. The right graphs show the time probability $p({t}_m|n)$ (histograms) and their compatibility (proved in the main body) with the quantum clock (continuous line).
• Figure 2: Monte Carlo simulation of non-instantaneous stroboscopic measurements. The upper panel shows the theoretical distribution (solid line), compared against the sequence of Gaussians that forms the POVM $\{\phi(m|t)\}$ (dashed line). Each vertical axis tracks a different stroboscopic bin. The lower panel compares the histograms of the ideal (projective) stroboscopic measurement against the non-instantaneous (POVM) one. (a) Stroboscopy for non-overlapping POVM. The real outcome closely resembles the projective one, with a $64\%$ probability of detection failure. (b) Stroboscopy with overlap. The real outcome significantly differs from the ideal one, with failure probability of $1\%$. (c) Worst-case scenario in which the POVM resolution is lower than the system time scale. Both the ideal and the non-instantaneous measurement collapse to a single bin.