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Physics Letters A

Exact solution of the density matrix equation of the coupled coherent and incoherent exciton motion on a linear chain

P. Reineker

Published Jul 1, 1973• 11 citations
Publisher
DOI
10.1016/0375-9601(73)90855-4

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Summary form only given. Effects of exciton-exciton correlations (e.g. bound biexciton, exciton-exciton scatterings) have been amply demonstrated in the /spl chi//sup (3)/ contributions to experiments measuring the interband polarization in semiconductor quantum wells. Increasing attention is currently being focussed on higher order excitonic correlation effects. One higher-order quantity is the incoherent exciton density (defined as total exciton density minus the interband polarization squared) which is created by excitons scattering each other out of the coherent component. Other than its role in the dephasing and relaxation of the system, the incoherent exciton density has not received much attention. In this contribution, we present results of our theoretical study on the behavior of this quantity and related correlation functions under conditions that inhibit relaxation. Our theory follows the general formulation of bosonic theories for excitons: the basic degrees of freedom are exact bosons, and the fermionic exchange effects are incorporated into the boson-boson and boson-photon effective Hamiltonians. We truncate the effective interaction at the two-boson level. No formal bosonization procedure is used: the matrix elements of our effective Hamiltonian are chosen in such a way that in the coherent /spl chi//sup (3)/ limit, our boson theory yields formally identical results as the standard fermionic dynamics controlled truncation formalism. Standard many-body techniques are used to derive equations of motion for densities, correlation functions, and measurable quantities such as the interband polarization.
The optical properties of a semiconductor pumped near the 1s exciton resonance are studied. The equation of motion for the density matrix in the Hartree-Fock approximation is traced back to the Bloch equation, well known from two-level systems. The essential difference, however, is the coupling of different wave vectors via the Coulomb potential. Principal features of the different stationary solutions are discussed. For a closer inspection of stability, we calculate the response to a smooth switchon of the pump field. For pump frequencies above the 1s exciton, the system follows first adiabatically the pulse shape, showing a blueshift and a slight increase of the oscillator strength as in the off-resonant case. The system jumps later into a second solution characterized by optical gain below the pump frequency, related to complete saturation (excitonic insulator). The major objective of our paper is to provide a numerical solution to the equation of motion in the Hartree-Fock approximation and to compare the results to those obtained by stationary considerations.
We introduce a method for calculating the stationary state of a translation invariant array of weakly coupled cavities in the presence of dissipation and coherent as well as incoherent drives. Instead of computing the full density matrix our method directly calculates the correlation functions which are relevant for obtaining all local quantities of interest. It considers an expansion of the correlation functions and their equations of motion in powers of the photon tunneling rate between adjacent cavities, leading to an exact second order solution for any number of cavities. Our method provides a controllable approximation for weak tunneling rates applicable to the strongly correlated regime that is dominated by nonlinearities in the cavities and thus of high interest.