Time evolution of the lowest wavepacket

Finding polariton states in disordered planar microcavities microscopically is a difficult task which do not attempt here. As a first excursion into the study of disorder effects on polariton dynamics, here we will follow (32) to explore the dynamics in a simpler microscopic model of a ID microcavity. Such microcavities are interesting in themselves and can have experimental realizations from the results known in the theory of disordered systems (39) one can also anticipate that certain qualitative features may be common for ID and 2D systems (38). 

It consists of a lattice of N molecular sites spaced by distance a and comprises the exciton part (an is the exciton annihilation operator on the site n), photon part (bk is the photon annihilation operator with the wavevector k and a given polarization) as well as the ordinary exciton-photon interaction. The cavity photon energy ek is defined by eqn (10.44), e represents the average exciton energy, while en are the on-site exciton energy fluctuations. 

We will use uncorrelated normally distributed en with zero mean and variance a  

The exciton-photon interaction is written in such a form that 2y yields the Rabi splitting energy in the perfect system. We chose to use the same number N of photon modes, and the wavevectors k are discrete with 2ir/Na increments. Our approach is to straightforwardly find the normalized polariton eigenstates Tj) (i is the state index) of the Hamiltonian (10.50) and then use them in the site-coordinate representation  

In (38) various numerical parameters have been tried in the model Hamiltonian with the results being qualitatively consistent the parameters exploited in these calculations have been chosen, on one hand, to be reasonably comparable with the experimental data in the output and, on the other hand, to better illustrate our point within a practical computational effort. It should be kept in mind though that we consider a model system and the numerical values of results may differ, likely within an order of magnitude, for various systems. 

---