Quantum Interference between a Single-Photon Fock State and a Coherent State

TL;DR

This paper analytically derives the quantum state transformations and Wigner functions for a beam splitter with a single-photon Fock state and a coherent state, applicable to various optical devices including interferometers.

Contribution

It provides a general analytical framework for the output states and Wigner functions when combining Fock and coherent states in passive optical devices, extending previous results.

Findings

Derived explicit output state expressions for beam splitter inputs.

Obtained analytical Wigner functions for the output states.

Extended results to Mach-Zehnder interferometers and general input states.

Abstract

We derive analytical expressions for the single mode quantum field state at the individual output ports of a beam splitter when a single-photon Fock state and a coherent state are incident on the input ports. The output states turn out to be a statistical mixture between a displaced Fock state and a coherent state. Consequently we are able to find an analytical expression for the corresponding Wigner function. Because of the generality of our calculations the obtained results are valid for all passive and lossless optical four port devices. We show further how the results can be adapted to the case of the Mach-Zehnder interferometer. In addition we consider the case for which the single-photon Fock state is replaced with a general input state: a coherent input state displaces each general quantum state at the output port of a beam splitter with the displacement parameter being the amplitude of the coherent state.