1. Introduction

An optical material can be regarded as being composed of a large number of polarizable particles (atoms or molecules) embedded in the vacuum and coupled to a propagating electromagnetic field. At low particle densities, it is sufficient to consider the interaction of a single particle with the field when developing equations of motion, as each particle interacts with the electromagnetic field independently. However, at high densities, an electromagnetic field mediates interactions between the polarizable particles of a material in what is known as the Lorentz local-field condition (LLFC). In the linear regime, the LLFC leads to the familiar Clausius-Mossotti-Lorentz-Lorenz (CMLL) relation for linearly polarizable particles and to a renormalization of the resonance frequency by the plasma frequency in the Drude model in which atoms are modeled as harmonic oscillators. While it has long been known that the LLFC applies to linear media and to nonlinear media with static fields, a recent field theoretical derivation has shown that the LLFC applies to nonlinear media and propagating fields, as well [1]. For atoms that are modeled quantum mechanically as consisting of discrete energy levels connected by electric dipole transitions, the near dipole-dipole (NDD) interactions between the atoms due to the LLFC are responsible for intrinsic frequency modulation and give rise to a wide array of novel and interesting phenomena in nonlinear and quantum optics including intrinsic optical bistability [2], quasiadiabatic following and inversion [3], and piezophotonic and magnetophotonic switching [4].

Optically nonlinear condensed matter often contains more than one polarizable component, Figs. 1a, 1b, 1c, 1d. For example, ions, which provide nonlinear optical response, are present as impurities in a crystal host material. It is typically assumed that the density of the nonlinear impurity is sufficiently low that NDD interactions between the impurity particles are negligible. Then all local field effects are due to the host and can be accounted for by a renormalization of the dipole moment if the host dielectric is modeled as a dense collection of linearly polarizable particles.[5, 6] Recent experiments [7] by Hehlen, Güdel, Shu, Rai, Rai, and Rand (HGSR³) demonstrated intrinsic optical bistability (IOB) due to near dipole- dipole (NDD) interactions between Yb³⁺ ions in a Cs₃Y₂Br₉ crystal. In light of this example in which more than one species of particles is sufficiently dense to require the application on the LLFC, we investigate the local-field effect for optically nonlinear materials which have more than one dense polarizable component. We find that local field effects in dense multicomponent nonlinear media lead to local-field enhancement effects, local cooperative decays, and coherence exchange processes.

2. The Lorentz local-field condition

In the atomistic view of an optical material, the presence of the polarizable particles results in large local variations in the field in the interior of the material. Then the field acting on a particle, the microscopic Lorentz local field, must be distinguished from the macroscopic field that is obtained by performing an average of the microscopic field over a region of space that contains a great number of particles. For media composed of more than one component, the average must explicitly account for the multiple species of particles present. To do this we generalize the field theoretical derivation of the LLFC by Bowden and Dowling in Ref. [1].

Employing the Lorentz program [8], the local field E _L = E+E _i is written as the sum of the average macroscopic field E and E _i, the internal field due to the polarization of the near dipoles within a volume of the order of a cubic wavelength. The internal field within this volume is calculated in two pieces. At the smallest length scale d, of the order of the typical intermolecular spacing, the effects of retardation are negligible and the near field component E _near of the internal field is calculated by taking the actual contribution of individual dipoles in the static limit, Fig. 2. It is well known that E _near is identically zero for a cubic lattice as a consequence of symmetry. However, it can be seen in Fig. 2 that cubic symmetry is broken if more than one species of polarizable particle is present. To account for the non-cubic symmetry, we write E _near = ∑_i s_Li P _i, where s_Li is the structure factor and P _i is the partial polarization for species i.

 

Figure 1: (a) Vapor cell: Dilute atomic vapors near resonance can be modeled as two-level systems in a vacuum. At low densities, the interaction between atoms is negligible, and it is sufficient to consider the interaction of a single particle with the field when developing equations of motion. (b) Dense medium: At high densities, the particles interact via the electromagnetic field. The Lorentz local-field condition (LLFC) leads to the Clausius - Mossotti - Lorentz - Lorenz (CMLL) relation for linearly polarizable particles and to near dipole-dipole interaction for two-level systems. (c) Multicomponent medium with dilute nonlinear component: We consider optically nonlinear condensed mater comprised of two polarizable components. If the density of nonlinear particles is sufficiently low then interaction between the nonlinear particles is negligible. (d) Dense nonlinear multicomponent media: When near dipole-dipole interactions are significant due to a high density of two-level atoms, local-field effe associated with the presence of an optically linear material component lead to local-field enhancement of the NDD interaction, local cooperative decays, and coherence exchange processes. [Media 1] [Media 2] [Media 3] [Media 4]

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Figure 2: Calculation of the near field: Individual dipoles are used in the calculation of E _near. In cubic symmetry E _near = 0, but here the symmetry of the lattice is altered by the presence of a second species. [Media 5]

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The second contribution - E _P to the internal field arises from the transition from microscopic variables to macroscopic variables. This transition occurs on length scales of order l, where d ≪ l ≪ λ. The procedure is detailed in Ref. [1], but in the present case the integration must be carried out over all of the polarizable particles in the volume. As shown in Fig. 3, the volume of integration contains particles of more than one species and one obtains E _P = - (4π/3)∑_i P _i upon performing the integration.

The result is that the Lorentz local field condition for a material composed of several species of polarizable particles is

where P _i is the partial polarization of species i, s_Li are structure factors for the different species, and η_Li = 1 + 3s_Li/4π. We see that the polarization that appears in the Lorentz local field condition is the sum of the partial polarizations of the constituents of the material. This simple result has profound implications for the nonlinear dynamics of dense multicomponent media because the macroscopic polarization couples the nonlinear interaction of one component with the electromagnetic field to the dynamics of other components.

3. Dense two-level atoms embedded in a linearly polarizable host

Near resonance laser interaction with a dense collection of two-level atoms has been investigated and a number of effects associated with induced NDD interactions have been identified, such as intrinsic optical bistability [2],nonlinear spectral shifts [9], intrinsic optical switching [10], and quasiadiabatic following and inversion [3]. Studies of these NDD effects have focused on two-level atoms embedded in vacuo. However, citing the example of the (HGSR³) experiment, we note that materials for which the resonant atoms are sufficiently dense for NDD effects to be significant often contain more than one polarizable component, even though only the one near resonance with the applied field is of primary interest. Such a material can be described macroscopically as a dense collection of two-level atoms embedded in a linear dielectric. But in order to self-consistently apply the LLFC, one adopts the atomistic description of two species of atoms embedded in vacuo; one species has a dipole-allowed transition that is nearly resonant with the incident laser field and is modeled as a two-level quantum system while the other species is far from resonance and is modeled as a linearly polarizable particle. Then the total polarization is the sum of the partial polarizations of the two-level atoms and the linearly polarizable particles and Eq. 1 becomes [6]

 

Figure 3: Transition from microscopic to macroscopic variables: The transition from microscopic to macroscopic variables is obtained by integrating over a large number of particles. For a dense multi-component medium, all species of particles are included in the volume of integration. [Media 6]

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in terms of envelope functions ε_L , ε, 𝑝, 𝑝^lin, and 𝑝^res where $E_L=\frac{1}{2}({\epsilon }_L{e}^{-i{\omega }_{p}t}+c.c.)$, $P=\frac{1}{2}\left({𝑝}_L{e}^{-i{\omega }_{p}t}+c.c.\right)$, etc. Here, 𝑝^lin and 𝑝^res are the envelope functions of the partial polarization due to the linear particles and the two-level atoms, respectively. Also, α is the linear polarizability and N _α is the number density of linear systems. The Clausius-Mossotti-Lorentz-Lorenz (CMLL) relation

relates the microscopic polarizability and number density of linear systems to the linear dielectric function ε = n ² of the background. Employing the CMLL relation, the local field can be written as

where

The dynamics of a two-level atom interacting with a microscopic electromagnetic field are described by the optical Bloch equations in the rotating-wave approximation [11].

Here, r ₂₁ = ρ ₂₁ ^iω_pt, ρ ₂₁ is the off-diagonal density matrix element, ω = ρ ₂₂ - ρ ₁₁ is the inversion, μ is the transition dipole moment, and Δ = ω_p - ω ₀ is the detuning from resonance.

Now we make a transition from the microscopic regime of equations of motion for an atom interacting with the local field to the macroscopic equations of motion of a collection of atoms interacting with the macroscopic Maxwell field. Performing a local spatial average of the atomic variables, in the same sense as for the Scully-Lamb laser equations [12], while replacing the microscopic local field with the macroscopic Maxwell field and polarization using the above relation, we obtain

where 𝑝^res = 2NμR ₂₁ is the nonlinear polarization and ∊ = 4πηNμ ²/(3ħ) is the NDD interaction parameter, a measure of the interaction between near dipoles. We have adopted the convention of using italicized upper-case Roman letters for the macroscopic, spatially averaged, atomic variables in the rotating frame of reference: R ₂₁ = 〈r ₂₁〉_sp, R ₁₂ = 〈r ₁₂〉_sp, and W = R ₂₂ - R ₁₁ = 〈r ₂₂〉_sp - 〈r ₁₁〉_sp. Here, 〈⋯〉_sp corresponds to a spatial average over a volume of the order of a resonance wavelength cubed. Damping has been introduced phenomenologically, where γ _∥ = 1/T ₁ is the population relaxation rate, γ _⊥ = 1/T ₂ is the dipole dephasing rate, and W _eq is the population difference at thermal equilibrium. T ₁ and T ₂ are the familiar longitudinal and transverse relaxation times, respectively.

Equations (8) and (9) are referred to as the generalized Bloch equations because they can be written as optical Bloch equations generalized to media of densities such that there are many atoms within a cubic wavelength. Allowing for dispersion and absorption, we assume that the dielectric function of the host can be represented by a complex constant as a consequence of the Kramers-Kronig relations. Separating the local-field enhancement factor l = l_r + il_i into real and imaginary components, the generalized Bloch equations (8) and (9) can be written as

Equations (10) and (11) display several local-field effects that create fundamental differences between these equations and the usual equations of motion of a two-level atom: i) The term involving the product l_rWR ₂₁ is bilinear in the atomic variables and can be interpreted as an intrinsic frequency modulation or inversion-dependent detuning from resonance. The presence of the host medium enhances the inversion-dependent detuning (nonlinear Lorentz frequency shift) that is due to the NDD interaction. This is especially significant because NDD effects are only important at sufficiently high densities and large oscillator strengths. For example, Friedberg, Hartmann, and Manassah [9] derived a threshold condition for intrinsic optical bistability in a dense vapor, namely ∊ > 4γ _⊥. This density threshold condition cannot be controlled in a vapor of two-level systems because the dephasing rate increases in direct proportion to the density due to collisional broadening. For two-level systems in a dispersionless dielectric, the threshold condition l∊ > 4γ _⊥ can be satisfied by a smaller value of ∊, because the inversion- dependent detuning is enhanced in condensed matter and the homogeneous linewidth is no longer restricted to the formula for a collisionally broadened vapor. Significantly, it is in a similar case that intrinsic optical bistability, analyzable by a two-level model, was observed experimentally [7]. ii) There is an enhancement of the magnitude of the field that drives the atoms. This effect is present for a dilute embedding of atoms in a dielectric [5, 6]. iii) The imaginary part of the enhancement factor appears as a coefficient, along with the NDD parameter, of bilinear products of the macroscopic atomic variables in two terms of Eqs. (10) and (11). These terms correspond to local cooperative decay effects representing the interaction of near dipoles mediated by the imaginary component of the dielectric function of the host medium. Because the effects of NDD interactions can be manifested in films that are significantly thinner than a vacuum wavelength, the detrimental effects, absorption and heating, associated with the imaginary part of the index of refraction can be mitigated for dense media. Then, for a thin film of a strongly dispersive dielectric containing a dense collection of two- level atoms, one can expect local cooperative decay effects to play a significant role in the dynamics.

4. The generalized Bloch-Drude model

To this point we have assumed that a constitutive relationship exists for the interaction of the host with the field, allowing us to obtain representative results with the simplest physical model for the host, which is of secondary importance. Let us now assume that a more realistic model for the host is a dense collection of harmonic oscillators. Applying the LLFC, we obtain generalized Bloch-Drude equations

where the plasma frequency ω_p is defined by the relation ${\omega }_p^2$ = 4πN_be ²/m, N_b , is the number density of oscillators, ω_b is the resonance frequency of the oscillators, and X̄ = mμX/eħ is a normalized coordinate. The coupling between the equations of motion for the two species of particles represents a coherence exchange that occurs because the dynamics of the two species of particles are coupled through the electromagnetic field as described by the LLFC.

If the field is sufficiently slowly varying and sufficiently far from resonance of the harmonic oscillators, Eq. (14) can be adiabatically eliminated to yield

which is the usual Drude result for a dense collection of oscillators but with a dynamic field renormalization arising from coherence transfer due to the LLFC. The conditions for adiabatic elimination correspond to the conditions for which the linear polarizability model of the atoms is valid. Then inserting Eq. (15) into Eqs. (12) and (13), the equations of motion for a dense collection of two-level atoms embedded in a linearly polarizable host are recovered, where l = 1 + (${\omega }_p^2$/3)/(${\omega }_b^2$ - ω ² - ${\omega }_p^2$/3 - iωγ). Now we have the local field enhancement factor, and also the linear dielectric function, in terms of the physical parameters of the harmonic oscillator model. In this case, the imaginary part of the local field enhancement factor, and thus the intrinsic cooperative decay of the two-level atoms, can be seen to be a consequence of the normal decay processes of the linear oscillators.

5. Summary

The presence of a optically linear material component affects the dynamics of dense two- level atoms as a consequence of the Lorentz local-field correction. These effects are local- field enhancement of field and the near dipole-dipole interaction, local cooperative decays, and coherence exchange processes.