Polaritonic crystal formed of a tunnel-coupled microcavity array and an ensemble of quantum dots

Propagation of polariton excitations in a defect-containing one-dimensional lattice of microcavities with embedded ultracold atomic nanoclusters (quantum dots) is being considered. The virtual crystal approximation is used to study the properties of electromagnetic excitation spectrum resulting from random variations of the atomic subsystem composition and positions of micropores, as well as from a homogeneous elastic deformation of the considered one-dimensional structure. The group velocity dependence of polariton excitations on structural defect concentration and on deformation parameter is being numerically modeled.


Introduction
Some important features of photonic band-gap structures, which have been drawing an extensive attention [1] are related to the so-called 'slow' light, which is one of the promising fundamental physical phenomena that can be exploited in designing various quantum optical storage devices. Its correct understanding permits to effectively reduce the group velocity of electromagnetic excitation in coupledresonator optical waveguides [2,3] as well as in various types of solid-state semiconductor multilayer structures [4]. The key role in reducing the group velocity in these systems is played by the so-called bright and dark polaritons, which are linear superposition of photon states of the external electromagnetic field and macroscopic (coherent) perturbations of the two-level atomic medium.
In atomic systems, the lifetime of polaritons is limited by the lifetime of excited atoms and is usually of a nanoscale order [5]. At the present level of advancement of nanotechnologies and nanophotonics it is possible to study the "slow" light as well as phase transitions of polaritons by fabricating arrays of coupled microcavities with embedded two-level atoms [6][7][8]. Technologically, creation of data structures can be based on photonic crystals containing defects formed of microcavities doped with such atoms [9].
In order to gain theoretical understanding of the above systems, in the present paper we propose and investigate a modeling polaritonic crystal constituted by atomic nanoclusters (quantum dots) weakly interacting with the localized electromagnetic field in an array of tunnel-coupled microcavities. A remarkable feature of such a structure is the possibility of localization of polaritons, which is similar to the nonlinear optical phenomenon of light localization in photonic crystals (see e.g. [9]) as well as to exciton localization in quasi-periodic structures in solid state physics [10].
Basing on the previously developed description of ideal photonic structures [11], here we consider one such particular non-ideal system, namely a polaritonic crystal with an atomic subsystem formed of impurity atomic clusters. Several of our previous works have been devoted to designing microcavity structures where modulation of the dispersion of photon modes is achieved by introduction of  [12,13]. In technical applications, structural defects in supercrystals play a minor role as compared to temporary defects introduced by application of external fields and/or elastic strains. The present work analyzes in detail the effect of a uniform elastic strain on a one-dimensional microcavity array with embedded quantum dots. Such a system combines the advantages of an extreme optical non-linearity stemming from the coupling of quantum dots to photonic modes and a high sensitivity of optical eigen-modes to the applied strains. In other words, we dwell on a particular realization of a topologically ordered microcavity (resonator) system, which can have promising applications in optical integrated circuits.

Theoretical background
Basing on the approach developed in Refs. [12,13], let us first consider the dispersion of optical eigen modes in the most general case of a microcavity supercrystal composed of s sublattices. Each of tunnelcoupled microcavities is assumed to confine a single optical mode. The cavity photon Hamiltonian ( )Ĥ  is dependent on the deformation tensor  that is sensitive to the applied strain.
Adapting the Heitler-London approximation, the Hamiltonian ( )Ĥ  can be written as (see e.g. Ref. [14] for further details):    (1) are found by its diagonalization through the Bogolyubov-Tyablikov transformation [14]. This yields the following equation for elementary excitation spectrum and then we obtain the relation for the group velocity ( ) Below, on the basis of the above theory, the features of the dependence of the group velocity of elementary electromagnetic excitations on the concentration of structural defects and homogeneous deformation in a specific non-ideal porous 1D structure are investigated.

Results and discussion
As an example, let's consider polaritons in a one-sublattice quantum-dot-containing chain of unevenly spaced microcavities under a uniform elastic deformation (the corresponding component of the tensor  is  ). We consider the array of identical cavities with randomly embedded quantum dots of two types, whose concentrations are, correspondingly,  Here we also adopt the virtual crystal approximation [15,16] based on the diagonalization of the averaged Hamiltonian (1). The corresponding procedure yields a system of uniform linear equations, whose solvability condition is given by: Angular brackets in (4) denote the procedure of configuration averaging of the microcavity array over all possible positions of cavities (index "T") and compositions of quantum dots (index "C"). , Within the nearest-neighbor approximation, the quantities , ,, It follows from (4)

Conclusion
Theoretical investigation into photonic band structure of non-ideal lattices comprised of tunnel-coupled microcavities with embedded quantum dots shows that subjecting these systems to controllable elastic strains as well as introduction of structural defects provide effective tools for altering their eigen-mode structure and optical properties. Elastic strains and photonic crystal structure disorder have a direct effect on the magnitude of the group velocity of elementary excitations in the specified systems. This is illustrated, in particular, by the theoretical result that the slow light mode formation can be efficiently controlled by an externally applied strain. The presented results of numerical simulations contribute to modeling a new class of functional materials, namely photonic crystalline systems constituted of coupled microcavities, whose capabilities include controllable propagation of electromagnetic excitations. The obtained conclusions pave the way to applications of irregular microcavity arrays in optical integrated circuits as well as in classical and/or quantum optical switches.