Duality relations in a two-path interferometer with an asymmetric beam splitter

TL;DR

This paper explores wave-particle duality in a Mach-Zehnder interferometer with an asymmetric beam splitter, revealing how asymmetry affects path knowledge and interference visibility, and deriving a trade-off relation.

Contribution

It introduces a quantitative analysis of duality relations considering asymmetry, showing how joint measurability of unsharp observables leads to a new trade-off relation.

Findings

Asymmetry affects which-path knowledge and fringe visibility.

Joint measurability constrains the trade-off between visibility and distinguishability.

Derived a new duality relation specific to asymmetric beam splitters.

Abstract

We investigate quantitatively the wave-particle duality in a general Mach-Zehnder interferometer setup with an asymmetric beam splitter. The asymmetric beam splitter introduces additional a priori which-path knowledge, which is different for a particle detected at one output port of the interferometer and a particle detected at the other. Accordingly, the fringe visibilities of the interference patterns emerging at the two output ports are also different. Hence, in sharp contrast with the symmetric case, here we should concentrate on one output port and distinguish two possible paths taken by the particles detected at that port among four paths. It turns out that two nonorthogonal unsharp observables are measured jointly in this setup. We apply the condition for joint measurability of these unsharp observables to obtain a trade-off relation between the fringe visibility of the interference pattern and the which-path distinguishability.