A.A. Galkin Donetsk Institute for Physics and Engineering, National Academy of Sciences, Ukraine
Received Date: November 05, 2014; Accepted Date: December 29, 2014; Published Date: January 09, 2015
Citation: Rumyantsev V, Fedorov S, Sychanova M (2015) Peculiarities of Light- Matter Coupling in Imperfect Lattice of Coupled Microresonators. J Laser Opt Photonics 2:113. doi: 10.4172/2469-410X.1000113
Copyright: © 2015 Rumyantsev V, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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We study exciton-like electromagnetic excitations in non-ideal microcavity lattice with the use of the virtual crystal approximation. The effect of point defects (vacancies) on the excitation spectrum is being numerically modeled for a quasi-two-dimensional non-ideal binary microcavity supercrystal. The adopted approach permits to obtain the dispersion dependence of collective excitation frequencies and the energy gap width on defect concentrations in a microcavity lattice. Based on the representations of the ideal photonic structures, the non-ideal polaritonic crystal, which is a set of spatially ordered cavities containing atomic clusters, is considered too. The analytical expressions for polaritonic frequencies, effective mass and group velocities, as a function of corresponding quantum dots and vacancies concentrations is obtained.
Microcavity supercrystal; Admixture elements; Virtual crystal approximation; Band gap width
A number of recent theoretical and experimental works indicate that microcavity supercrystals may have interesting applications. Photonic structures and metamaterials are in the focus of interdisciplinary studies, which span laser physics, condensed matter physics, nanotechnology, chemistry and information science [1,2]. Many papers have been devoted to realization of light-emitting devices based on polariton crystals [3,4]. In this context, semiconductor microcavities represent quantum confined optical systems  featured by strong coupling of elementary crystal excitations (excitons) and the optical field. Photonic supercrystals can be built from spatially-periodic systems of coupled microcavities . Semiconductor microcavities are widely used in optoelectronic devices nowadays [7,8]. Nanocavities in photonic crystals [9,10] represent a particular case of microcavities characterized by a discrete photonic spectrum. Nanocavities with embedded quantum dots have been used to demonstrate the strong light-matter coupling regime in Ref. [3,4,11] indicate also that chains of microcavities may be used for practical realization of quantuminformation processing.
Here we study dispersions of localized electromagnetic excitations in an array of coupled microcavities which form a non-ideal supercrystal rich by point-like defects. The effect of point defects (vacancies) on the excitation spectrum is being numerically modeled. The adopted virtual crystal approximation (VCA) permits to obtain the dispersion law and the energy gap width of the considered quasiparticles and to analyze the dependence of their density of states on defect concentrations in a microcavity supercrystal.
Based on the representations of the ideal photonic structures, the non-ideal systems of this class - polaritonic crystal, which is a set of spatially ordered cavities containing atomic clusters, is considered too. Moreover, in this part of the work we believe that the spatial distribution of cavities (resonators) is translation invariant, and the atomic subsystem has randomly distributed defects (impurity atomic clusters: quantum dots or a vacancies). Numerical modeling of dependence of the dispersion of polaritons in this imperfect lattice of associated microresonators on impurity concentration is completed. The analytical expressions for polaritonic frequencies, effective mass and group velocities, as a function of corresponding quantum dots and vacancies concentrations, is obtained. It turned out that even with a small number of vacancies in the lattice (one position for a thousand resonators) weight polaritons increases by three orders of magnitude. These results enable to extend the possibility of creating a new class of functional materials - polaritonic crystal systems.
Unlike in Refs [3,11,12] devoted to coupled resonators with dopant atoms let us here pose a somewhat different problem. Namely, we intend to examine a 2D array of tunnel-coupled randomly distributed microresonators of different types at the total absence of an atomic subsystem (Figure 1). Each resonator is assumed to possess just one optical mode. We also account for the overlap of optical fields, which enables photons to move along the chain. Since the VCA consists in replacement of configurationally dependent Hamiltonian parameters with their averaged values, Hamiltonian of a “virtual” crystal in our case reads as follows:
Here angular brackets denote configurational averaging. Quantity defines the overlap of optical fields of the nα -th and mβ -th cavities and the transfer of the corresponding excitation, are bosonic creation and annihilation operators describing the photonic mode. Subscripts n and m are two-dimensional integer lattice vectors, α and β numerate sublattices, whose total number is σ . is the frequency of photonic mode localized in the nα -th site (cavity). Hamiltonian (1) is formally identical to the tight- binding excitonic Hamiltonian in a semiconductor crystal [13,14], for which reason the studied electromagnetic excitations can naturally be referred to as exciton-like. It is worth stressing that we discuss photonic super-crystal excitations and no electronic transitions are involved. Nevertheless, it will be seen below that the dispersion relations of purely electromagnetic crystal excitations in the system we study are quite similar to the Frenkel exciton bands in molecular crystals [13,15]. Here we consider a topologically ordered non-ideal lattice of microcavities with point-like defect: vacancies and non-typical microcavities. In such a system, Hamiltonian Hph is no more translation invariant, hence the quantities are configurationally dependent. A convenient tool to study the quasiparticle excitation spectrum in a system with randomly distributed defects consists in configurational averaging of the solutions of corresponding Hamiltonians . An averaged solution is translation invariant, hence the corresponding elementary excitation spectrum can be characterized by a wave vector K. A widespread method of computation of quasiparticle states in disordered media is the virtual crystal approximation (VCA) [16,17]. It proves sufficient to elucidate the transformations of elementary excitation spectra under varying defect concentrations. In what follows we rely on this method to compute and analyze the spectrum of electromagnetic excitations as well as the corresponding optical characteristics of the considered non-ideal super crystal.
Configurational averaging “restores” the translation invariance of the considered super crystal system. Eigenvalues of Hamiltonian (1) are found via its diagonalization by means of the Bogolyubov-Tyablikov transformation [13,14], and are ultimately found from the system of algebraic equations of the order σ :
are eigen functions of the σ × σ matrix whose elements are expressed through the corresponding characteristics of the Hamiltonian (1):
Here are concentrations of the V-th and μ-th types of cavities, being the radius-vector of the resonator located in the α th sublattice of the nth elementary cell. The solvability condition of the system (2)
yields the dispersion law of electromagnetic excitations in the considered photonic super crystal.
Consider localized electromagnetic excitations in a twosublattice (σ = 2 ) system of cavities. The left-hand side of Equatation. (4) is then a second-order determinant, which being equated to zero gives the following dispersion of photonic excitations:
Here , and are the matrix elements of the operator
To be more specific, let us consider a spectrum of electromagnetic excitations in a binary system where each sublattice contains only two types of cavities. In such a case, the quantities and are given by ,
Being applied to the supercrystal lattice of microcavities where the only defects are vacancies, these expressions take the form
where is the cavity concentration in the first sublattice, is the cavity concentration in the second sublattice, is vacancy concentration in the 1st and/or 2nd sublattices. Concentrations must obviously satisfy the relations . In (6) matrix elements , , characterize the overlap of optical fields of cavities pertaining to the same sublattice but different cells (Figure 1). The energy spectrum of exciton-like electromagnetic excitations is defined by the type of the considered sublattices and the quantities and . Below we carry out a nearest-neighbor calculation for the case of a square Bravais lattice of period d . Location of cavities is defined by the radius-vector hence their location in the zero elementary cell, rn=0, is defined by vectors r01=0 and where (Figure 1). In the adopted approximation the matrix elements can with reasonable accuracy be written as:
In (7) the overlap characteristic of optical fields defines the transfer probability of electromagnetic excitation between the nearest neighbors in the first (second) sublattice, and is the excitation transfer probability between cavities in the first (second) and second (first) sublattices in an arbitrary cell. Substitution of expressions (7) into Equatation. (5) gives the dispersion law for electromagnetic excitations (Figure 2). We performed calculation for modeling frequencies of resonance photonic modes in the cavities of the first and second sublattices and respectively and for the overlap parameters of resonator optical fields and . The lattice period was set equal to . Figure 2 gives the example of surfaces depicting the dispersion dependence of collective excitation frequencies in the considered non-ideal microcavity lattice. Surfaces in Figure 2, plotted for . The presence of two dispersion branches (see Equatation. (5)) reflects a two-sublattice structure of the resonator system. For molecular crystals with two molecules in a cell an analogous occurrence of two branches in the dispersion law is referred to as the Davydov splitting of exciton zone  (Figures 2 and 3). Vacancies concentration dependence of the photonic gap width in the studied microcavity supersystem. Concentration dependence of the energy gap width is shown in Figure 3. The surface is non-monotonic and turns to zero in a certain range of . Therefore in a certain region of electromagnetic excitations pass unhindered through the binary two-sublattice microcavity system.
One way to create a polaritonic crystal is capture of two-level atoms in the photonic structure consisting of an array of tunnelcoupled microcavities (CROW) . As in [19,20] we study 1D lattice microcavities containing one optical mode, each of which interacts with a neighbor in the chain. Thus, each resonator contains the macroscopic cluster of ultracold two level atoms of the same type with the levels and interacting with the quantized electromagnetic field directed (along the Z axis) perpendicular to the chain, which oriented along the X axis (Figure 4). In such a configuration the overlap of optical field and the wave functions of atoms are taken into account and there is a possibility photon tunneling along the chain resonators (Figure 4). A schematic model of polariton crystal: microresonators lattice containing macroscopic ensembles of two-level atoms and interacting with resonators electromagnetic modes. The Hamiltonian H of the system considered is:
Here Hat corresponds to an ensemble of two-level atoms (quantum dots) in the trap-resonator, Hph corresponds to the propagation of the light field, Hint describes the atom-optical interaction in the cavity. Proceeding from the concepts developed, for example, in , the Hamiltonian H for ideal photonic structures can be reduced to the form:
where the annihilation (creation) operators in equation (9) characterize the dynamical properties of atomic ensembles (atomic quantum modes) at the lower and upper levels in the n-site of array n=1,2…..M, and characterize the energy of atom sat the levels respectively. The coupling coefficients are the nearestneighbor hopping constants that depend on overlapping integrals of the atomic cloud wave function. Wave functions (these functions are real Wannier functions) are responsible for spatial distribution of ultracold n-site atoms under the so-called strong-bonding approximation . The annihilation (creation) operators in (10) describe the temporal behaviour of a single photonic mode with frequency located at the n-th cavity. The parameter α characterizes a spatial field overlapping between the neighboring cavities. The interaction of twolevel atoms with the quantized electromagnetic field in equation (11) is considered under the rotating wave approximation and determined by the constant Gn. It is assumed in framework of the polarization crystal model  that all the cells are identical to each other and have the same number of atom N=N…., as well as the coefficients of atomic-optical interaction in (11) are the same in all cells, that g=g1=g2=…..gM. Let’s consider the non-ideal system of this class-polaritonic structure–with atomic subsystem containing the impurity atom clusters. Moreover, the spatial distribution of traps resonators remains, as before, is translation invariant (with the lattice constant l) but the atomic subsystem contains randomly distributed foreign (relative to the ideal system) quantum dots (each cavity contains only one atomic complex of a certain sort). Therefore, the parameters of the problem related to the atomic subsystem are configuration-dependent variables. Numerical simulation of such systems can be completed within the VCA, which is to replace the configuration-dependent hamiltonian parameters Hat on their configuration averaged values. In this case, the configurationdependent values are . After configuration averaging of these quantities we obtain:
Here Cv is the concentration of atomic clusters (quantum dots) type V, which satisfies the equality are the characteristics relating to atomic components of V-type. The procedure for configuration averaging allowed to “restore” (), the translational invariance of the Hamiltonian and, therefore, to use the scheme of calculations [3,12]. With the help of the Bogolyubov transformations Hamiltonian, which describes the light and dark polaritons , is reduced to a diagonal form:
The annihilation operators in equation (13) characterize two types of quasi particles (due to the atom–field interaction) i.e. upper and lower branch polaritons, respectively. These quasi particles propagate along the X-direction of the periodical structure. The characteristic frequencies Ω1,2 in (13) define a dispersion relations, as well as the polaritonic band structure of the crystal.
To concretize the non-ideal polaritonic system, let’s consider the case of the system containing only one type of atoms with the concentration C1 and vacancies with concentration Cv C1 + Cv=1. In this case we obtain:
Here the dispersion relations for the atomic and photonic subsystems in a neighbor-hood dot Kl = 0 defined as follows:
and the Rabi frequency and atomic-optical detuning are:
In expression (15) appeared term , which is extremely large in comparison, for example, with the average frequency of transition for rubidium D-line which is . This term, even for a very small concentration of vacancies ( ~ 10 ) amounts to several THz. For comparison, the parameter α characterizing the photon tunneling, is from a few hundred GHz to THz, and the parameter of atomic-optic communication g consists of from tens to hundreds of GTz. It are these parameters, α and g , even in the absence of vacancies are the main contributors to the features of the dispersion curve. The expression for the mass of polaritons has the form:
In formulas (21) and (22):
Figure 5 shows the dependence of the mass m2 of the polariton on concentration of vacancies Cv introduced by the method described above. In this case, we used the following numerical values of the parameters: the size of the resonators is l = 3μm , the effective mass of the atoms in the lattice without vacancies is , the effective mass of the photon is , the atom-optical the detuning Δ = 0 is independent on the wave vector, the average frequency of the rubidium D-line is , the parameter of atomic-optic coupling is . In the case of a lattice without vacancies the mass of polaritons is
Analysis of the graph shows that even for small number of vacancies in the lattice (one vacancy on 10-4 of resonators) the mass of polaritons increases by an order. The last circumstance testifies an essential role vacancies in effectively reducing of excitation velocity jumping between cavities.
A number of recent experimental works indicate that microcavity supercrystals may have interesting applications, in particular for creating of optical clockworks of unprecedented accuracy [23-25]. In  authors considered spectrum of exciton-like electromagnetic excitations in a quasi-2D binary lattice of coupled microcavities. Here we have used the virtual crystal approximation to model the effect of lattice point defects (vacancies) on the spectrum and the photonic gap width dependence on the vacancies concentration for the specific case. Based on the representations of the ideal photonic structures, the nonideal systems of this class-polaritonic crystal, which is a set of spatially ordered cavities containing atomic clusters, is considered in the paper. The cflculations show that even a minute presence of vacancies (one per 104 of resonators) increases the mass of polaritons by an order. This illustrates the crucial role of vacancies in reducing the effective excitation velocity in chains of microcavities. Obtained here dispersions of electromagnetic excitations are noticeably more complex than those of primitive lattices. This complexity is due to the non-ideality of the structure and to the presence of two sublattices. The latter entails multiple manifestations in experimentally observable integral characteristics of optical processes. Evaluation of excitation spectra in more complex photonic systems requires the use of more sophisticated computational methods. Depending on particular cases such can be the one- or multiple-node coherent potential method , the averaged T-matrix method  as well as their various modifications. Our study contributes to the modeling of novel functional materials with controllable propagation of electromagnetic excitations.
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