Light propagation in disordered media: From Maxwell equations to a spherical p-spin model and light condensation effects

Summary form only given. The well-known phenomenon of the formation of a Bose-Einstein condensate (BEC), a striking consequence of the Bose-Einstein statistics, has been traditionally linked to an ensemble of ultra-cold gas molecules. However, classical systems can also exhibit condensation effects; in the field of photonics, for example, signatures of this condensation in the mode dynamics (“light condensation”, LC) have been theoretically investigated and experimentally observed in various types of multimode lasers [1,2 and ref. therein]. In contrast to the previous efforts, we developed a novel theory for LC in the presence of disorder, which takes into account the overlapping mode structure and shows that light condensation is naturally sustained by the governing equations in such systems. We begin with the Maxwell equations in a generic, disordered medium with non-orthogonal eigenmodes and complex eigenfrequencies. By employing the Feshbach projection technique [3], we separate the resonator space and the channel space and for the modal amplitudes we obtain ∂an ∂t + n'n n'fn n' (1) where |an|2 represents the energy carried by the nth mode inside the resonator, Jn'n are the coupling amplitudes containing the interaction between the cavity and the environment and fn are the noise generators, calculated exactly in our model. We can show that the experimentally relevant dynamics can be described by the following Hamiltonian and noise generators, a rigorous reduction from the Maxwell eq. to a spherical 2-spin model: H = - 1 2 Jn'nan'an fn = 0 fn(t) fn'(t') = 2Tδnn'δ(t - t') n'n (2) where is the thermodynamic average and T is a dimensionless temperature. The contribution of disorder is contained in the noise generators and in the distribution of the eigenvalues of J. From here, we study the photon thermodynamics in the regime of strong localization of light, which corresponds to small fluctuations in the noise and, therefore, to low temperatures. We show that the photonic system experiences a phase transition, a condensation process - in good analogy with BEC systems - and, below the critical temperature, the energy of a single mode grows linearly as the temperature further decreases. For further investigation, we have also performed an ab initio, massively parallel 2D FDTD simulation campaign (reaching over 12 million core hours) on the IBM Blue Gene/P supercomputer “Shaheen”, using our custom code. We considered positional disorder of a photonic lattice, launched a Gaussian pulse and measured the local density of states inside the structure for decreasing localization length (increasing refractive index or disorder). One of our main observation is that the energy of a single, dominant mode (normalized by the total energy) had a substantial increase in the strongly localized regime.