First-Click Time Measurements

Abstract

There are two distinct perspectives on the quantum time-of-arrival: one can ask for the probability that a particle is found at the detector at a given time, regardless of whether it was previously detected, or for the probability that the particle is detected there for the first time. In this work, we analyze the latter by constructing the time-of-arrival distribution conditioned on the particle not having been detected at earlier times -- the first-click distribution. We work within the Page and Wootters formalism, where time is treated as a quantum observable, and introduce a memory mechanism that records the outcomes of successive detection attempts separated by the detector's finite time resolution. We apply this framework to a single Gaussian wave packet and to a superposition of two overlapping wave packets. We find that conditioning on non-detection redistributes probability toward earlier arrival times, producing narrower and sharper distributions compared with the standard unconditioned case. This effect persists in the presence of quantum interference, though coarser time resolutions broaden the distribution and shift it toward later times.

Figures (3)

• Figure 1: Time-of-arrival distributions for a single right-moving Gaussian wave packet propagating toward the detector. The packet parameters are $x_0 = 5l_0$ and $p_0 = 7\hbar/l_0$, corresponding to its initial average position and momentum, respectively. The detector’s time resolution is $\delta t = t_0$ and it extends between $a=10 l_0$ and $b=11 l_0$. The finite-size $p_{ML}$ is broader and exhibits a lower peak amplitude, consistent with its extended spatial range. The $p_{FC}$ distribution is narrower and shifted toward earlier times due to the conditioning on non-detection.
• Figure 2: Time-of-arrival distributions for the same Gaussian wave packet as in Fig. \ref{['image:GaussianWavePacket']}, with $p_{FC}$ evaluated for different time resolutions: $\delta t = \{t_0, 70t_0, \delta L/p_0\}$, as labeled. Increasing $\delta t$ broadens the distribution and shifts it toward later times, reflecting reduced temporal resolution. Even when $\delta t$ is chosen to match the time required for the particle to traverse the finite-size detector ($\delta t = m\delta L/p_0$ with $m=1$), $p_{FC}$ remains narrower and shifted toward earlier times than the finite-size $p_{ML}$.
• Figure 3: Time of arrival distributions for a superposition of $n=2$ right-moving Gaussian wave packets satisfying the overtaking condition. The parameters of each packet are $x_0 = -30 l_0$, $p_0 =10 \hbar /l_0$ and $p_1 =15 \hbar /l_0$ and $\sigma_0=\sigma_1=1 l_0$, which give $x_1=-45 l_0$, so that both packets arrive simultaneously at the left boundary of the detector, which extends between $a=0$ and $b=l_0$. The distributions display the interference fringe structure characteristic of the two-packet superposition. Compared with the point-like $p_{ML}$, $p_{FC}$ enhances the early peaks and suppresses the later ones, while the finite-size $p_{ML}$ partially smooths the fringes.