[Paper Review] Immanants for Three-Channel Linear Optical Networks

Hubert de Guise, Si-Hui Tan+2 more|arXiv (Cornell University)|Feb 11, 2014

Photonic and Optical Devices3 citations

TL;DR

This paper develops a group-theoretic framework using permutation and SU(3) symmetries to describe how a three-channel passive linear optical interferometer transforms single-photon pulse inputs. It shows that coincidence rates as a function of photon delay are governed by linear combinations of immanants of the interferometer's unitary matrix, linking partial distinguishability to quantum interference patterns via representation theory.

ABSTRACT

We use permutation-group methods plus SU(3) group-theoretic methods to determine the action of a three-channel passive optical interferometer on single-photon pulse inputs to each channel. Our mathematical description connects partial distinguishability of input photons with linear superpositions of immanants of the interferometer SU(3) matrix. By delaying identical photons, partial distinguishability is controllable, and landscapes of coincidence rates vs delay times between pairs of photons are explained in terms of immanants.

Motivation & Objective

• To model the quantum optical response of a three-channel passive interferometer under single-photon inputs.
• To connect the degree of partial distinguishability among photons to the mathematical structure of immanants of the interferometer's SU(3) matrix.
• To explain experimentally observable coincidence rate patterns as functions of photon delay using group-theoretic tools.
• To provide a unified theoretical description of quantum interference in multi-channel linear optics using representation theory.

Proposed method

• Employing permutation group theory to analyze the symmetry structure of multi-photon states in three-channel systems.
• Applying SU(3) group-theoretic methods to describe the unitary transformation of the interferometer on single-photon inputs.
• Deriving the output state as a superposition of immanants of the interferometer's unitary matrix, weighted by photon indistinguishability.
• Using time delays between identical photons to control their partial distinguishability, enabling experimental tuning of interference behavior.
• Expressing coincidence detection rates as functions of delay times through linear combinations of immanants.
• Mapping the resulting interference landscapes to the representation theory of SU(3) and symmetric groups.

Experimental results

Research questions

• RQ1 How does partial distinguishability of input photons affect the coincidence rates in a three-channel linear optical network?
• RQ2 What is the mathematical structure of the output state in terms of immanants when three single photons enter a passive interferometer?
• RQ3 How do time delays between identical photons modulate the interference patterns observed in coincidence rates?
• RQ4 In what way do SU(3) and permutation group representations characterize the quantum state evolution in this system?
• RQ5 Can immanants fully describe the dependence of coincidence rates on photon delay in three-channel linear optics?

Key findings

• The output state of three single photons in a three-channel interferometer is expressed as a linear superposition of immanants of the interferometer's unitary matrix.
• Coincidence rates as a function of time delay between photons are directly determined by the coefficients of these immanants.
• Partial distinguishability, controlled by time delays, modulates the interference pattern through the immanant decomposition.
• The SU(3) representation theory provides a natural framework to classify and compute the interference contributions.
• The method explains complex coincidence rate landscapes in terms of group-theoretic invariants, offering a predictive model for experimental setups.
• The immanant-based formalism unifies the description of quantum interference across different degrees of photon distinguishability.