Fisher information from quantum many-particle arrival time measurements

TL;DR

This work formulates a quantum time-of-arrival measurement for Bosonic many-particle systems via a dynamical absorption model in Fock space and derives arrival-time distributions $p_n({f t})$ for fixed, coherent, and quasi-free states. It then treats arrival times as a classical statistical model to obtain a tractable Fisher information for single-particle parameters such as the momentum $p_0$, with two key limits: a Dirac delta detector and a spatially uniform plane-wave beam. In the beam limit, arrival-time data encode momentum information even when spatial measurements are uninformative, and the authors obtain explicit analytic forms for the Fisher information in the sparse-beam regime, including distinct behaviors for coherent versus quasi-free states. The results advance understanding of temporal data from freely evolving quantum particles and provide foundations for metrology in many-particle quantum systems, with potential extensions to interactions and multi-parameter inference.

Abstract

We formulate a quantum arrival time measurement process for a Bosonic many-particle system, with the aim of extracting statistical information on single-particle properties. The arrival time is based on a dynamical multi-particle absorption model in the Fock space, and we consider systems in coherent and incoherent mixtures of $N$-particle states. We find the resulting probability distributions for arrival time sequences, which we consider as parametric models for the statistical inference of single-particle parameters, and derive a tractable expression for the associated (classical) Fisher information. Subsequently focusing on the concrete case of the momentum parameter of a 1D particle, we consider the idealized limits of a point (Dirac delta) detector and an infinite particle system forming a spatially uniform ``beam''. We observe that even though no information remains in the spatial distribution, the single-particle momentum is indeed identifiable from the arrival time data, even in the limit of ``sparse beams'' of vanishing particle density, where we obtain simple analytical form for the Fisher information, which, interestingly, coincides with the one obtained from a hypothetical time-stationary detection model. Our results contribute to the fundamental understanding of temporal measurement data arising from quantum systems consisting of freely evolving particles.

Figures (7)

• Figure 1: (a) An illustration of a beam of particles with arrival time detection. The detection takes place in blue region, mathematically described by the loss of the wavefunction normalisation giving rise to detection probabilities, and followed by particle annihilation in the Fock space. In the case of a Dirac delta detector, the annihilation takes place in a single point in space (red line). On the source side, suitably scaled position distributions of different initial single-particle states $\chi$ are shown depending on the mean particle number $\langle N\rangle$, and with the limit $\langle N\rangle=\infty$ corresponding to a spatially uniform beam localised at single momentum $p_0$. (b) A schematic figure showing the structure of the paper: we first present our general detection scheme, then work out the limiting cases illustrated in (a), and finally apply statistical inference to quantify information on $p_0$ obtained from the arrival time data.
• Figure 2: Probability distributions $p_1(t)$ for the first arrival times: single particle case (black, dashed line, reduced by a factor of $2$); many-particle case with average particle number $\langle N\rangle=100$: Fock (green, dotted lines, on top of red lines), coherent (red, solid lines), and quasi-free (blue, solid lines) beams. Finite beam with initial one-particle state \ref{['initialgaussian']} with $p_0/\bar{p}=1$, $x_0/l=-20$, $\Delta p^2/\bar{p}^2=0.5$, $m l/(\bar{p} \tau) = 1$. Detector state \ref{['gaussiandetector']} with $a \tau/l =0.1$, three detector state widths: (a) $\epsilon/l = 1$, (b) $\epsilon/l =0.5$, and (c) the delta detector limit $\epsilon = 0$.
• Figure 3: Convergence to the beam limit for the delta-detector: intensity function $\omega_{\rm delta}(t)$ for different values of the average particle number $\langle N\rangle$; analytically computed limit $\omega_{\rm delta}^{\rm beam}(t)$ (see \ref{['plane_wave_intensity']}; brown, solid line). Spatial particle density $r_0 l=56.42$, detector strength $a \tau/l=0.1$, the initial states $\chi_{{\langle N \rangle}}$ are given by \ref{['initialgaussian']} using \ref{['defpnn']} and $p_0/\bar{p}=1$, $x_0/l=-20$, $m l/(\bar{p} \tau) = 1$.
• Figure 4: (a) First-detection probability density $p_1 (t)$, (b,c) two-detection probability density $p_2 (t_1, t_2)$. In the delta detector and beam limit case: coherent state (red lines/surfaces), quasi-free state (blue lines/surfaces). Spatial particle densities $r_0 l=56.42$ (solid lines and (b)), $r_0 l=56.42/4$ (dotted lines and (c)). The other values are $a \tau/l=0.1$, $p_0/\bar{p}=1$, $m l/(\bar{p} \tau) = 1$.
• Figure 5: Fisher information $I_n$ for the (a) coherent state and (b) quasi-free state. In each figure, $a \tau/l=0.1$, fixed spatial particle density $r_0 l=56.42$. The result is shown for different values of the average particle number $\langle N\rangle$; the initial states $\chi_{{\langle N \rangle}}$ are given by \ref{['initialgaussian']} using \ref{['defpnn']} and $p_0/\bar{p}=1$, $x_0/l=-20$, $m l/(\bar{p} \tau) = 1$. In addition, the Fisher information based on the analytically computed limit $\omega_{\rm delta}^{\rm beam}(t)$, see \ref{['plane_wave_intensity']}, is shown ($N=\infty$, thick, black, solid lines).