The Schmidt modes of biphoton qutrits: Poincare-sphere representation

TL;DR

This paper derives explicit polarization Schmidt modes for biphoton qutrits, introduces a method for state factorization, and proposes an experimental scheme to measure entanglement parameters using Poincaré sphere representations.

Contribution

It provides a new explicit method for finding Schmidt modes of biphoton qutrits and links entanglement to operator commutation features, with a graphical Poincaré sphere approach.

Findings

Explicit polarization Schmidt modes are derived.

A simple factorization method for the state vector is proposed.

An experimental scheme for measuring entanglement parameters is suggested.

Abstract

For a general-form polarization biphoton qutrit, physically corresponding to a pair of arbitrarily polarized photons in a single frequency and wavevector mode, we explicitly find polarization Schmidt modes. A simple method is suggested for factorizing the state vector and the explicit expressions for the factorizing photon creation operators are found. The degrees of entanglement and polarization of a qutrit are shown to depend directly on the commutation features of the factorizing operators. Clear graphic representations for the Stokes vectors of the qutrit state as a whole, its Schmidt modes, and factorizing single-photon creation operators are given, based on the Poincar\'e sphere. An experimental scheme is proposed for measuring the parameters of the Schmidt decomposition as well as for demonstrating the operational meaning of qutrit entanglement.