This paper shows the necessity of introducing a quantum object, the ``coboson,'' to properly describe, through a fermion scheme, any composite particle, such as the exciton, which is made of two fermions. Although commonly dealt 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. As a direct consequence of this composite nature, there is no correct way to describe the interaction between cobosons as a potential
V. This is rather dramatic because, with the Hamiltonian not written as
H=H0+V, all the usual approaches to many-body effects fail. In particular, the standard form of the Fermi golden rule, written in terms of
V, cannot be used to obtain the transition rates of two cobosons. To get them, we have had to construct an unconventional expression for this Fermi golden rule in which
H only appears. Making use of this expression, we give here 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 Hamiltonian in which the excitons are considered as elementary bosons with effective scatterings between them, these scatterings resulting from an elaborate mapping between the two-fermion space and the ideal boson space. We show that the relation between the inverse lifetime and the sum of the transition rates for elementary bosons differs from the one of the composite bosons by a factor of
1∕2, so that it is impossible to find effective scatterings between bosonic excitons giving these two physical quantities correctly, 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.
We derive a bosonized Hamiltonian describing two-exciton correlation of semiconductor-photon coupled systems with a bosonization method, which takes into full account an effect of deviation of the excitons from ideal bosons. This deviation effect stems from the fact that excitons are composite particles, whose character appears clearly in the case where the excitons overlap each other. We call this effect a composite-particle effect (CPE). To our knowledges this effect was not considered completely in previous theoretical works on exciton-exciton interaction. The Hamiltonian introduced in this paper includes the results of the previous works as low-order terms of the CPE. After the introduction of a general theory of the bosonization method for arbitrary dimension and electron-hole mass ratio, we also demonstrate an application to a semiconductor bulk system coupled with a photon field in the heavy-hole limit. The bosonized Hamiltonian shows that the CPE brings about an enhancement of the exciton-exciton scattering strength and a qualitative change of the photo transition amplitude. It is also shown that the Hamiltonian describes two-exciton bound and scattering states.
We revisit the approach proposed by Mukamel and coworkers to describe interacting excitons through infinite series of composite-boson operators for both, the system Hamiltonian and the exciton commutator -- which, in this approach, is properly kept different from its elementary boson value. Instead of free electron-hole operators, as used by Mukamel's group, we here work with composite-exciton operators which are physically relevant operators for excited semiconductors. This allows us to get \emph{all} terms of these infinite series explicitly, the first terms of each series agreeing with the ones obtained by Mukamel's group when written with electron-hole pairs. All these terms nicely read in terms of Pauli and interaction scatterings of the composite-exciton many-body theory we have recently proposed. However, even if knowledge of these infinite series now allows to tackle
N-body problems, not just 2-body problems like third order nonlinear susceptibility
χ(3), the necessary handling of these two infinite series makes this approach far more complicated than the one we have developed and which barely relies on just four commutators.
We propose an exact bosonization scheme for two two-fermion composites with identical internal structures, that is, a mapping of the two-exciton (four-fermion) subspace to a two-boson subspace. We obtain analytical, exact expressions of the boson-boson interactions and the boson-photon ones taking into full account that the commutation relation of the composite bosons deviates from the ideal-boson commutation due to internal motions of the composite. We can distinguish the "composite-particle effects' from the Coulomb interactions among the fermions in the interactions. With this method, origins of optical nonlinearity in a system with two 1s excitons are studied in terms of the mutual excitonic correlations and exciton-photon interactions.
We revisit the approach proposed by Mukamel and co-workers to describe interacting excitons through infinite series of composite-boson operators for both the system Hamiltonian and the exciton commutator---which, in this approach, is properly kept different from its elementary-boson value. Instead of free-electron-hole operators, as used by Mukamel's group, we here work with composite-exciton operators which are physically relevant operators for excited semiconductors. This allows us to get all terms of these infinite series explicitly, the first terms of each series agreeing with the ones obtained by Mukamel's group when written with electron-hole pairs. All these terms nicely read in terms of Pauli and interaction scatterings of the composite-exciton many-body theory we have recently proposed. However, even if knowledge of these infinite series now allows us to tackle
N-body problems, not just two-body problems like third-order nonlinear susceptibility ${\ensuremath{\chi}}^{(3)}$, the necessary handling of these two infinite series makes this approach far more complicated than the one we have developed and which barely relies on just four commutators.
The electron-hole pair operator commutation relations for a semiconductor \nare explicitly derived showing that the pairs are quasi-bosons \nwith statistics intermediate between bosons and fermions. These pair \noperators are transformed into exciton operators, that incorporate the \nintra-pair Coulomb interaction, and it is found that the commutation \nrelations remain quasi-bosonic. The exciton dynamics written in terms \nof the exciton operators are derived in the Heisenberg picture and \ncompared with the traditionally used semiconductor Bloch equations.