Wigner's Phase Space Current for Variable Beam Splitters -Seeing Beam Splitters in a New Light-

TL;DR

This paper introduces a novel phase space approach using Wigner's distribution to analyze the dynamics of variable beam splitters, providing new geometrical insights into mode interactions that are experimentally testable.

Contribution

It derives the Wigner current for variable beam splitters and offers a geometrical analysis of mode interactions in phase space, enhancing understanding of beam splitter dynamics.

Findings

Visualization of mode interactions via Wigner currents

Derivation of Wigner current for outgoing modes

Potential for experimental verification

Abstract

Beam splitters allow us to superpose two continuous single mode quantum systems. To study the behaviour of their strongly mode mixing dynamics we consider variable beam splitters and their dynamics using Wigner's phase space distribution, W, the evolution of which is governed by the continuity-equation $ \frac{\partial}{\partial \tau} W = - {\nabla } \cdot {J}$. We derive the form of the corresponding Wigner current, J, of each outgoing mode after tracing out the other. The influence of the modes on each other is analyzed and visualized using their respective Wigner distributions and Wigner currents. This allows us to perform geometrical analyses of the mode interactions, casting new light on beam splitter behaviour. Several of the presented results should be immediately testable in experiments.