Arrival Time -- Classical Parameter or Quantum Operator?

TL;DR

The paper tackles the unsettled question of how to describe arrival-time distributions in quantum mechanics by extending two canonical approaches—the time-operator and the time-parameter frameworks—to entangled two-particle systems. It derives the theoretical structures for both approaches (including the $\hat{t}_{AB}$ operator and the two-time wavefunction formalism) and proposes an experimentally feasible two-particle arrival-time setup to test their predictions. Numerical simulations show that, while the two frameworks converge in the far-field, they yield inequivalent joint distributions in the near-field, enabling experimental discrimination with current atom-optics technology; detector resolution and the quantum Zeno effect play crucial roles in shaping and distinguishing the outcomes. These results have implications for foundational quantum theory and for practical time-domain quantum technologies, such as non-local temporal interferometry and temporal state tomography, and they point to future work on relativistic extensions and interacting multi-particle scenarios.

Abstract

The question of how to interpret and compute arrival-time distributions in quantum mechanics remains unsettled, reflecting the longstanding tension between treating time as a quantum observable or as a classical parameter. Most previous studies have focused on the single-particle case in the far-field regime, where both approaches yield very similar arrival-time distributions and a semi-classical analysis typically suffices. Recent advances in atom-optics technologies now make it possible to experimentally investigate arrival-time distributions for entangled multi-particle systems in the near-field regime, where a deeper analysis beyond semi-classical approximations is required. Even in the far-field regime, due to quantum non-locality, the semi-classical approximation cannot generally hold in multi-particle systems. Therefore, in this work, two fundamental approaches to the arrival-time problem -- namely, the time-parameter and time-operator approaches -- are extended to multi-particle systems. Using these extensions, we propose a feasible two-particle arrival-time experiment and numerically evaluate the corresponding joint distributions. Our results reveal regimes in which the two approaches yield inequivalent predictions, highlighting conditions under which experiments could shed new light on distinguishing between competing accounts of time in quantum mechanics. Our findings also provide important insights for the development of quantum technologies that use entanglement in the time domain, including non-local temporal interferometry, temporal ghost imaging, and temporal state tomography in multi-particle systems.

Figures (4)

• Figure 1: Schematic diagram of the proposed setup. In this setup, two entangled particles are prepared in the initial entangled state $\Psi_0(x_1,x_2)$, composed of four Gaussian wave packets, as described in Eq. (\ref{['Initial-wave-function']}). The left detector $D_1$ and the right detector $D_2$ are waiting to measure the arrival times of the left-moving and the right-moving particles, respectively.
• Figure 2: Comparing arrival-time distributions in the near-field and far-field regimes. Panels (a), (c), and (e) (left column) represent two-particle arrival-time distributions resulted from the time-operator approach, whereas panels (b), (d), and (f) (right column) display the corresponding distributions computed using the time-parameter approach. The rows indicate different left-detector positions, arranged from the far-field at the top to the near-field regime at the bottom. As seen, the two approaches differ more clearly in the near-field regime. Each scatter plot is generated using $10^5$ sample points. The parameters of the initial wave function are chosen as $\sigma_0=10^{-6}$m, $u=10^{-1}$m/s, $l_A=8\sigma_0$, and $l_B=12\sigma_0$. The right detector is placed at $L_2=200\sigma_0$. In panels (a)-(b), (c)-(d), and (e)-(f), the left detector is placed at $L_1=-100\sigma_0$, $L_1=-50\sigma_0$, and $L_1=-20\sigma_0$, respectively. All the distributions are computed for Helium atoms with mass $m=6.64\times10^{-27}$kg. In Each panel, the colormap is normalized to $\Pi_{max}$, the maximum of arrival time probability density. In the right-column plots, the detection temporal resolution is fixed to be $\tau=10^{-6}$s.
• Figure 3: Detector temporal resolution effect on the arrival-time distribution. Panel (b) shows the two-particle arrival-time distribution obtained from the arrival time–operator approach, with its corresponding marginal shown in panel (a). Panels (d), (f), and (h) show the two-particle arrival-time distributions obtained from the time–parameter approach, with detector temporal resolutions $\tau = 10^{-6}\mathrm{s}$, $\tau = 5 \times 10^{-7}\mathrm{s}$, and $\tau = 10^{-7}\mathrm{s}$, respectively---their corresponding marginals appear in panels (c), (e), and (g). In all panels, the right detector is placed at $L_2 = 200\sigma_0$ and the left detector at $L_1 = -20\sigma_0$. The distributions are computed for Helium atoms using the same initial wave-function parameters as in Fig. (\ref{['fig2']}). All colormaps are normalized to the maximum arrival-time probability density of the time-operator distribution. In the time-parameter approach, higher temporal resolution lowers the total detection probability due to the quantum Zeno effect, reflected in the fewer sampled points in panels (d), (f), and (h).
• Figure 4: Conditional arrival-time probability density. The plot shows the normalized conditional arrival-time probability densities for the left detector, $\Pi(t_L|t_R)$, evaluated at the fixed right detection-time $t_R = 1.85\,\mathrm{ms}$ (corresponding to the vertical lines in Fig. \ref{['fig3']}). The solid curve depicts the arrival-time probability density obtained from the time-operator approach, while the dashed curves represent the results from the time-parameter approach. As seen, the results of two approaches coincide, when $\tau$ decrease.