Closure relations for composite bosons: difference between polaritons and Wannier or Frenkel excitons

TL;DR

This paper derives and compares closure relations for polaritons made of different excitons, revealing how their composite nature influences their mathematical description and the effective bosonic behavior.

Contribution

It provides a unified derivation of closure relations for polaritons with various excitons, highlighting differences due to their internal structure and fermionic components.

Findings

Closure relations vary significantly depending on exciton type.

Wannier exciton polaritons can reduce to elementary boson closure relations.

The composite nature affects the effective bosonic behavior and internal degrees of freedom.

Abstract

We derive the closure relation for $N$ polaritons made of three different types of excitons: bosonized excitons, Frenkel or Wannier excitons. In the case of polaritons made of Wannier excitons, we show how this closure relation, which appears as non-diagonal, may reduce to the one of $N$ elementary bosons, the photons, with its $1/N!$ prefactor, or to the one of $N$ Wannier excitons, with its $(1/N!)^2$ prefactor. Widely different forms of closure relations are thus found depending on the composite bosons at hand. Comparison with closure relations of excitons, either bosonized or kept composite as Frenkel or Wannier excitons, allows us to discuss the influence of a reduction of the number of internal degrees of freedom, as well as the importance of the composite nature of the particles and the existence of fermionic components.