Exciton-exciton scattering: Composite boson versus elementary boson

TL;DR

This paper develops a new framework for describing excitons as composite bosons, called cobosons, highlighting their differences from elementary bosons and demonstrating the importance of their composite nature in many-body interactions.

Contribution

It introduces a new approach to analyze exciton interactions using cobosons, revealing fundamental differences from elementary boson models and providing a rigorous proof of the significance of their composite nature.

Findings

Inverse lifetime differs from transition rate sum by a factor of 1/2 for cobosons.

Standard potential-based interaction descriptions are inadequate for composite bosons.

Cancellation of two-exciton transition rate at finite momentum transfer in Born approximation.

Abstract

This paper introduces a new quantum object, the ``coboson'', for composite particles, like the excitons, which are made of two fermions. Although commonly dealed with as elementary bosons, these composite bosons -- ``cobosons'' in short -- differ from them due to their composite nature which makes the handling of their many-body effects quite different from the existing treatments valid for elementary bosons. Due to this composite nature, it is not possible to correctly describe the interaction between cobosons as a potential $V$. Consequently, the standard Fermi golden rule, written in terms of $V$, cannot be used to obtain the transition rates between exciton states. Through an unconventional expression for this Fermi golden rule, which is here given in terms of the Hamiltonian only, we here give a detailed calculation of the time evolution of two excitons. We compare the results of this exact approach with the ones obtained by using an effective bosonic exciton Hamiltonian. We show that the relation between the inverse lifetime and the sum of transition rates for elementary bosons differs from the one of composite bosons by a factor of 1/2, whatever the mapping from composite bosons to elementary bosons is. The present paper thus constitutes a strong mathematical proof that, in spite of a widely spread belief, we cannot forget the composite nature of these cobosons, even in the extremely low density limit of just two excitons. This paper also shows the (unexpected) cancellation, in the Born approximation, of the two-exciton transition rate for a finite value of the momentum transfer.