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

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 nondiagonal, 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 in 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.

Analytical approach to semiconductor Bloch equations

“…In the case of large photon field for which such a W expansion is not valid, the relevant operators are no more exciton, but polariton operators. In a near future, we will address large photon fields using our work on interacting polaritons [22,23], along ideas similar to the ones we propose here. However, polaritons being far more complex composite bosons than excitons, this extension is at the present time beyond our scope.…”

Fundamental collapse of the exciton-exciton effective scattering

“…By contrast, it is worth noting that the closure relation for Frenkel excitons [24][25][26][27] , which are made of electron-hole pairs localized on the same ion site, has the same (1/N !) prefactor as the one of elementary bosons.…”

“…2 when the particle composite nature is kept while it is (1/N !) only when these particles are replaced by elementary bosons 23,24 . This prefactor difference proves that the formal replacement of Wannier excitons by elementary bosons is a dream, even in the extreme dilute limit of just N = 2 excitons, because all sum rules which result from closure relation, are going to be different, whatever the exciton-exciton effective scatterings.…”

Finite Temperature Formalism for Composite Quantum Particles

“…Such a closure relation has in fact been derived for Wannier excitons [19,20] made of one free electron k e and one free hole k h . The creation operator for a single exciton jii reads B y i ¼ P k e ;k h a y k e b y k h hk h k e jii.…”

“…appears with a n power is directly linked to the degrees of freedom of the composite particle at hand-as evidenced by Frenkel excitons, made of singleion-site excitations, which have the same closure relation as elementary bosons [20,21].…”