Continuous simultaneous measurement of position and momentum of a particle

TL;DR

This paper models a quantum particle's continuous measurement of position and momentum, revealing how different detection regimes influence its dynamics and the emergence of classical trajectories.

Contribution

It introduces a novel quantum measurement model using wavefunction collapse in a discrete set of coherent states and analyzes the resulting dynamics with semi-analytic methods.

Findings

Identification of regimes from stroboscopic to Zeno effect

Demonstration of classical trajectory emergence

Comparison with spatial filtering detection schemes

Abstract

We formulate a model of a quantum particle continuously monitored by detectors measuring simultaneously its position and momentum. We implement the postulate of wavefunction collapse by assuming that upon detection the particle is found in one of the meters' states chosen as a discrete subset of coherent states. The dynamics, as observed by the meters, is thus a random sequence of jumps between coherent states. We generate such trajectories using the Monte Carlo Wavefunction method. For sparsely distributed detectors, we use methods from renewal theory of stochastic processes to obtain some semi-analytic results. In particular, the different regimes of dynamics of the free particle are identified and quantitatively discussed: from stroboscopic motion in the case of low interrogation frequency, to delayed dynamics reminiscent of the Zeno effect if monitoring is frequent. For a semi-continuous spatial distribution of meters the emergence of classical trajectories is shown. Their statistical properties are discussed and compared to other detection schemes in which the effect of measurement corresponds to "spatial filtering" of the wavefunction.