Continuous simultaneous measurement of position and momentum of a particle

Abstract

We formulate a model of a quantum particle continuously monitored by detectors measuring simultaneously its position and momentum. We implement the postulate of wave-function 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 quantum Monte Carlo wave-function method. For sparsely distributed detectors, we use methods from renewal theory of stochastic processes to obtain some semianalytic 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 semicontinuous 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 operation on the system due to measurement corresponds to ``spatial filtering'' of the wave function.