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Physical review. B, Condensed matter

Cooperative radiative decay in the nonlinear optical response of excitonic nanostructures

Vladimir Chernyak, Shaul Mukamel

Published Jul 15, 1993• 34 citations
Publisher

Abstract

The nonlinear susceptibility ${\mathrm{\ensuremath{\chi}}}^{(3)}$ and the four-wave mixing signal from assemblies of molecules with nonoverlapping charge distributions and an arbitrary geometry are calculated using Green's-function techniques. At low molecular density and under off-resonant excitation we recover the local-field approximation. The role of retarded interactions and cooperative spontaneous emission in the electrodynamics of confined excitons in molecular quantum wells is discussed.

DOI
10.1103/physrevb.48.2470

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The nonlinear susceptibility \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3) and the four-wave mixing signal from assemblies of molecules with nonoverlapping charge distributions and an arbitrary geometry are calculated using Green's-function techniques. At low molecular density and under off-resonant excitation we recover the local-field approximation. The role of retarded interactions and cooperative spontaneous emission in the electrodynamics of confined excitons in molecular quantum wells is discussed.
The stationary nonlinear reflection and the time resolved four wave mixing signal from a molecular monolayer are calculated using Green function techniques. Cooperative resonant nonlinear response found in small aggregates suggests the existence of coherence size of order of optical wavelength. A new peak in the nonlinear reflection spectrum is predicted, which is missed by the local field approximation. For an infinite two dimensional molecular monolayer with transition dipole moments in the lattice plane, the momentum-dependent two exciton decay rate is found to be larger than the sum of the single exciton radiative decay rates, as predicted by the local field approximation.
We study the nonlinear optical response of molecular assemblies consisting of coupled two-level molecules with arbitrary geometry including inhomogeneous broadening and exciton dephasing. An analytical expression for the third-order optical susceptibility is derived based on a certain closure of the equation-of-motion hierarchy. An exact expression for the third-order response of small crystalline aggregates with exciton population relaxation is derived, and used to show that for low relaxation rate \ensuremath{\Gamma} the cooperative part of the nonlinear response scales as ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}1/2}$ in d=1, and up to logarithmic corrections, as ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}1}$ in d=2 and 3. We show that cooperative enhancement is caused by exciton scattering. This is to be contrasted to other cooperative effects such as superradiance, which are related to the size of molecular coherence domains and therefore scale as ${\mathrm{\ensuremath{\Gamma}}}^{\mathrm{\ensuremath{-}}\mathit{d}/2}$. In a numerical study for dipole-dipole interactions in d=1, 2, and 3 dimensions we find that the local-field approximation is valid only in selected cases, and in general fails to reproduce the magnitude and the resonance structure of ${\mathrm{\ensuremath{\chi}}}^{(3)}$.
We derive a rigorous theory of the interaction between photons and spatially extended excitons confined in quantum dots in inhomogeneous photonic materials. We show that beyond the dipole approximation, the radiative decay rate is proportional to a nonlocal interaction function, which describes the interaction between light and spatially extended excitons. In this regime, light and matter degrees of freedom cannot be separated and a complex interplay between the nanostructured optical environment and the exciton envelope function emerges. We illustrate this by specific examples and derive a series of important analytical relations, which are useful for applying the formalism to practical problems. In the dipole limit, the decay rate is proportional to the projected local density of optical states, and we obtain the strong and weak confinement regimes as special cases.
The performance of approximate theories for the collective nonlinear optical response of assemblies of interacting molecules is investigated by comparing their predictions to exact calculations. To this end, the nonlinear absorption coefficient of linear molecular aggregates with energetic disorder is studied with the aid of Monte Carlo simulations. It is shown that the well-known local-field approximation (LFA) cannot describe the near-resonance nonlinear absorption spectrum for disorder values up to the nearest-neighbor intermolecular interactions; surprisingly, it is found that the validity of the LFA does not at all improve with growing disorder. By contrast, the ``excitonic two-level system'' approximation, in which each collective one-photon transition (Frenkel exciton) of the assembly is treated as an independent two-level system, turns out to describe the collective nonlinear response of the aggregates in a much better way and, moreover, rapidly improves for growing disorder values. It is shown that both models correctly describe the linear optical response and that, within the framework of the LFA, a close connection exists between the rotating-wave approximation and the Heitler-London approximation.
In nanostructures with a high excitonic coherence, the interplay between the spatial structure of the excitonic wavefunction and that of the radiation field plays an essential role, causing drastic size effects in the optical response, such as size-resonant enhancement of the radiative decay rate and the nonlinear response and the interchange of quantized levels. This article discusses such peculiar optical effects based on the theoretical treatment beyond the long-wavelength approximation and demonstrates the possibility of designing novel photofunctions with the explicit use of the spatial structures of the radiation field.
The optical response of radiatively coupled semiconductor multiple-quantum-well structures is investigated theoretically. It is shown that the transverse optical field leads to a coupling of exciton states within each well which causes a radiative decay and a mixing of excitonic resonances. Simultaneously, the field-induced long-ranged coupling between different wells leads to collective effects that are very sensitive to the detailed geometry of the structure. For a quantum-well spacing equal to an integer multiple of half the optical wavelength inside the medium, it is predicted that the collective effects cause a stimulated decay of electronic excitations that should be observable in either transmission or reflection geometry. On the other hand, in a quarter wavelength structure, the light-induced coupling causes an interwell energy transport and a splitting of the excitonic resonances, that should be observable as quantum beats in the time-resolved transmitted or reflected signal.