1. Introduction

It is well known that in a system of interacting atoms the local field acting on each atom is different from the macroscopic electric field described by Maxwell’s equation[1, 2]. In its simplest form (valid for harmonic linear systems) the local field caused by the near dipole-dipole interactions between atoms is given by the Lorentz formula[3]

with E(t) being the time dependent macroscopic electric field and P the macroscopic polarization of the system which is determined self consistently from the equation

Here n is the atom concentration and α is a single atom nonlinear polarizability. Eqs.(1) and (2) result in the Clausius-Mossoti (CM) equation for the macroscopic susceptibility χ_DD.

χ_e ≡ 4πnα is the susceptibility of non-interacting atoms. It has been shown [1, 2, 4, 5] that local field corrections in the form of Eq.(1), and hence Eq.(2), are valid for a static as well as a time dependent field E(t). In particular, in gases Eq.(1) corresponds to the account of hard core interactions [4, 6] which is equivalent of excluding the polarization induced electric field inside the Lorentz sphere[1, 5]. The variables P(ω), E(ω), and E_L (ω) in Eq. (2) are the Fourier components of their time dependent analogs.

We will be interested here in nonlinear optical systems with their size less than the resonant wavelength, where the CM relation has been employed recently to treat a number of phenomena such as intrinsic optical bistability [7], linear and nonlinear spectral shifts[8, 9], lasing without inversion and an absorptionless index of refraction[10, 11], electromagnetically induced transparency [12], laser instabilities and chaos in inhomo-geneously broadened media[13], etc. In particular we will concentrate below on widely discussed[11, 12, 14, 15, 16] nonequilibrium three level Λ systems possessing quantum coherence between the lower two levels. It has been suggested[10, 11] to use the atom concentration as a control parameter and shown through the employment of the CM relation that an enhancement of two orders of magnitude would be achieved in the refractive index compared with the noninteracting atom limit.

 

Figure 1: Real (1) and imaginary (2) part of the susceptibility χ_e of non-interacting atoms⁹

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In this paper we go beyond the CM limit by considering a situation where the optically active atoms are frozen in random positions in a crystal or amorphous host. We will show that in such a situation the effect of configurational fluctuations associated with near dipole-dipole interactions leads to the suppression of the refractive index compared to the predictions based on CM equation.

2. Local mean field equations

In order to clarify the approach below note that the polarization P in Eq.(2) can be written as

where n = N/V ₀ $\widehat{\mu }$_i is a single atom dipole moment operator; the angular brackets denote the quantum statistical average at given random atom positions and the overbar denotes the configurational average over the atom positions. In fact, m_i is the classical dipole moment associated with the i-th atom located at point r_i. It is induced by the interaction of the i-th atom with the other atoms and with the applied electric field.

Eqs.(1) and (2) are self-consistent mean field equations for the average polarization P. They describe the dielectric susceptibility of interacting atoms in terms of the single atom polarizability α. However, in disordered systems there are limitations on the applicability of mean field theory due to configurational fluctuations in atom positions resulting in fluctuations of the local field E_Li and the local polarization m_i. This is especially important in systems with the sign-changeable dipole-dipole interaction. In order to estimate qualitatively the effect of configurational fluctuations we will apply the local mean field formalism developed in the theory of spin glasses[17] and disordered ferromagnets and ferroelectrics[18, 20, 21].

The local mean field formalism is based on the physically transparent assumption that the local average value of the dipole moment of each atom is given by

 

Figure 2: Real (1) and imaginary (2) part of the susceptibility χ_DD obtained from Eqs.(11)–(14)

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where E_iL is the local mean field experienced by the i-th atom due to dipole-dipole interactions with its neighbors in the presence of the external electric field. In the retar-dationless approximation (i.e., the distance between the atoms is less than the resonance wavelength) E_iL can be written as

Eqs.(5) and (6) are self-consistent equations for the local polarization. The employed local mean field approximation is equivalent to the Hartree or local density approximations which neglect the effect of quantum correlations. The latter effect, which also modifies Eq.(1), has been discussed for two level atoms interacting with electromagnetic field in Ref.[4, 6]. Note that for a given form of interaction J_ij, Eqs.(5) and (6) can be solved with the use of computer simulation techniques. The analytical solution below invokes an additional approximation of the distribution function of the local fields.

For the dipole-dipole interaction

The effect of retardation on the form of the dipolar field has been discussed, e.g., in papers[22, 6, 23, 7]. In Eqs.(6),(7) we assumed for simplicity that all dipole moments m_i are oriented along the axis x and n _ij = r _ij/r_ij where r _ij = r _i - r _j is the radius-vector separating atoms i and j; E_iL in Eq.(6) is the x-component of the local electric field.

The mean field approximation(1) corresponds to the replacement of E_iL in Eq.(6) by

Indeed, the volume integral in Eq.(8) can be replaced by two surface integrals over the outer surface of the sample giving rise to depolarizing field E_dep, and the inner spherical Lorentz surface giving rise to the Lorentz local field $\frac{4\pi }{3}P$. In Eq.(1) E = E_ex + E_dep.

 

Figure 3: Real (1) and imaginary (2) part of the susceptibility χ_DD given by the Clausius-Mossoti equation⁵

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It is convenient to rewrite Eq.(6) in the form

Eqs.(9) and (10) imply that ∑_i m_i/V ₀ is a self-averaging variable equal to macroscopic polarization P. The polarization P defined by Eq.(2),(5) can be written in terms of the distribution function f(e) = ̅δ(e - e_i) of the local field e; δ(z) is the Dirac delta-function. For small values of E i.e., for linear response with respect to the macroscopic field (but not the local field) one can neglect the dependence of polarizability α on E and P and we obtain

Eq.(11) transforms to the CM equation in the limit f(e) → δ(e). Also κ_e = χ_e in linear systems where the susceptibility does not depend on the applied field. However, as we will show below, taking account of the finite width of f(e) caused by the configurational fluctuations in atom positions is important for properly estimating the effect of the dipole-dipole interaction on the macroscopic susceptibility χ_DD in nonlinear systems.

3. Macroscopic susceptibility in systems with coherence

We consider 3-level Λ systems with quantum coherence between the lower two levels which was a subject of recent study[11]. The susceptibility χ_e of non-interacting atoms has been shown[16] to be proportional to dimensionless parameter C = 2πμ ² n/ħγ which characterizes the strength of the dipole-dipole interaction. γ is the characteristic rate of dissipative processes. The frequency dependence of χ′_e and χ′′_e is presented in Fig. 1 for C=4.6 and the values of other parameter used in Ref. [11]. One can see that at the dimensionless detuning frequency Δ = (ω - ω ₀)/γ ≈ -0.59, where ω is the frequency the incident light and ω ₀ is the resonance frequency, the susceptibility χ′_e reaches its maximum value ≈ 3 and χ′′_e ≈ 0. In this situation the CM equation predicts a significant enhancement[11] of χ_DD compared with χ_e.

In order to take into account the effect of configurational fluctuations of the local field we calculated the function f(e) in the assumption that m′′_i ≈ 0. This assumption is true in the vicinity of Δ ≈ -0.59 where χ′_e peaks and χ′′_e ≈ 0. The calculation of f(e) has been performed in the self-consistent manner[18] based on the modification of statistical theory of local field distributions developed for the analysis of line shapes in inhomogeneously broadened spectra[19].

We obtain

The parameter M satisfies the equation

The nontrivial solution of Eq.(14) determines the value of M which is the order parameter of the nonequilibrium spin glass state characterizing the noncoherent oscillations of the atom dipole moments [24].

The values obtained for the susceptibility χ_DD calculated with the use of Eqs.(11),(13),(14) and the explicit form of single atom polarizability α[16] are presented in Fig. 2. In Fig. 3 we show, for comparison, the values of χ_DD given by the CM equation. One can see that effect of configurational disorder results in significant suppression of the susceptibility compared with the predictions based on the CM equation. Note, however, that for the more exact estimation of χ_DD at frequencies of external field ω ≠ ω ₀ one need to calculate the single atom polarizability α(ω) being a subject of strong oscillating local field at frequency ω ₀. The results of such calculations will be reported elsewhere.

4. Conclusion

We have shown that the effect of configurational fluctuations of the local field in disordered nonlinear optical systems leads to the suppression of the system susceptibility compared with the predictions based on the mean field theory. Explicit results were obtained for the susceptibility of the three level Λ system with quantum coherence between the two lower levels, which are randomly distributed in crystals and amorphous hosts and show the spin glass phenomena. We expect that the similar suppression of the susceptibility would take place in gases. However, in the latter case the correlation effects between the translational and electronic degrees of freedom might prevent the realization of the optical spin glass state.