«Numerical solutions to the Boltzmann equation have been

done for some time now.»

Numerical simulations of the polariton kinetic energy distribution in GaAs quantum-well microcavity structures. V. E. Hartwell and D. W. Snoke in Phys. Rev. B 82:075307 (2010).  What the paper says!?

This is one of the main texts on the relaxation of polaritons through the Boltzmann equations approach:

We are interested in directly comparing our experimen- tal data on the distribution function of the polaritons to our data for nonresonant pumping, which extends all the way up

to the polariton bottleneck region.

They provide a nice overview of previous works, detailed expositions of microscopic scattering rates, numerical integrations of the rate equations and fitting to experimental data.

Their rate equation absorbs populations into the scattering terms:

it cannot treat directly the coherence in the system. However, it can show the buildup of particles in low-energy states which

is the precursor for coherence in Bose-Einstein condensation.

They include «three types of interactions: polariton-polariton, polariton-phonon, and polariton-electron.»

One feature of our model in comparison to other models is that we can

simulate the entire set of polariton and excitons in one continuous, single band of energy, rather than treating the polaritons and excitons as two separate populations.

They define three regimes:

1. Short lifetimes: laser
2. Long lifetimes: BEC
3. intermediate case: non-equilibrium BEC

The latter regimes is

[...] evidenced by a buildup of particles in low-energy states due to collisions and spontaneous coherence in the ground state but

lack of complete thermalization in high-energy states.

They include renormalization of the dispersion relation:

Our code can take into account the energy shifts [...] and calculate a new dispersion relationship

They discuss (and explain) the phase-space filling effect as follows:

A second effect is phase space filling. As the density of particles increases, the states in the sample begin to fill up. Because the excitons are made of underlying fer- mionic constituents, there is an upper bound to the total num- ber of new exciton polaritons that can be created. As this limit is approached, the excitons start to decouple from the photons. Experimentally, this is seen as the polariton splitting becoming weaker. This leads to a blueshift of the lower po- lariton line and a redshift of the upper polariton line, i.e., a

closing of the line splitting.4,11

They point to two works that claim

the importance of exciton-exciton correlation in the effective exciton interaction in a two-dimensional (2D) system.

They provide detailed sections on the form of the interaction scattering terms.

In the introduction, they make interesting comments on previous works.

They describe the series of Work by Doan et al. as follows:

In a series of papers [33-36], Doan and co-workers showed the possibility of large accumulations in the polariton ground state. The first paper 33 showed that with the correct choice of parameters it is theoretically possible for acoustic phonons to overcome the bottleneck. They chose cavity lifetimes of 50 ps as opposed to typical lifetimes of current samples of a few picoseconds. In Ref. 36, a similar study was done for II-VI

materials. By treating the lowest states as discrete states instead of a continuum, this group was able to show a steady state Bose-Einstein distribution could occur.

They make this comment on the work by Porras et al.:

To further simplify the system they used a quantized area. As in the case of Refs. 33–36, this causes the wave vectors to be discrete. They also did the calculation in k space instead of energy space. This makes it easier to have a large number of low energies, the main region of interest. In three dimen- sions, the equations can be analytically reduced further in energy space than in k space. In two dimensions there is an integral that cannot not be simplified, which makes doing the calculation in k space equivalent to doing the calculation in

energy space.

Although the last part seems to criticize the benefits of energy quantization, that is the approach they also take. They give explicitly the shape of their quantization procedure (Fig. 1) The numerical integration seems to be a simple explicit forward Euler method (cf. Eq. 3).

we can only apply this method to the range of polaritons densities below the critical threshold for onset of coherence. Nevertheless, we can apply it to data just at the cusp of coherence. The simulations show that the Bose statistics of the particles play a crucial role in the buildup of particles in

low-energy states.

They fit their experiments and obtain several insights, for instance, chopping the exciting laser allows to fit with a constant lattice temperature as function of pumping, avoiding the heating of the sample. They cannot maintain the polariton scattering strength the same unless they include a free electron gas (in which case the « constant polariton-polariton cross section [has] a value about 20% greater than the literature value». They believe that «the stress induces a population of free carriers.» (stress of the sample)

They discuss the approach of Sarchi and Savona as «more elaborate models», which is described as follows:

In these studies not only do they keep track of occupation numbers but allow the dispersion relationship to shift in the energy due to particle-particle in-

teractions. Their method breaks the population up into three distinct regions

those being the condensate, polaritons and excitons.

Of these works, they say:

All of the numerical simulations discussed in this section have shown that the k=0 state can have orders of magnitude more population than more energetic polaritons below the bottleneck. None of these groups have directly compared numerically simulated distribution functions to experimental

data

but i) the first work from Tassone et al. did not reach macroscopic occupation of the ground state (it seems Porras et al. is the first to do that) and ii) the Savvidis rings appears to have been directly compared to experiments.