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Physical Review A

Nonlinear susceptibilities of molecular aggregates: Enhancement of χ ( 3 ) by size

Francis C. Spano, Shaul Mukamel

Published Nov 1, 1989• 292 citations
Publisher

Abstract

We investigate the third-order nonlinear-optical susceptibility ${\ensuremath{\chi}}^{(3)}$ in molecular aggregates and its scaling with aggregate size. The aggregate is modeled as a collection of N-interacting, homogeneously broadened two-level systems, where the interaction includes both the static dipole-dipole coupling and superradiant coupling. Our expression for ${\ensuremath{\chi}}^{(3)}$, derived using Liouville-space Green-function techniques, contains contributions from excitons as well as biexcitons. The scaling of ${\ensuremath{\chi}}^{(3)}$ with aggregate size and its dependence on the laser detuning and homogeneous and inhomogeneous dephasing rates are analyzed. ${\ensuremath{\chi}}^{(3)}$ contains terms that scale as \ensuremath{\sim}${N}^{2}$ and \ensuremath{\sim}N(N-1). However, under off-resonant conditions these terms interfere destructively, resulting in a linear \ensuremath{\sim}N scaling, which is precisely what is expected for N monomers. Detailed calculations of phase-conjugate degenerate four-wave mixing and pump-probe spectroscopy are performed. In the first case we predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton line shape. Pump-probe spectroscopy shows a series of biexciton absorption lines, some of which are superradiantly broadened.

DOI
10.1103/physreva.40.5783

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We investigate the third-order nonlinear-optical susceptibility \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) in molecular aggregates and its scaling with aggregate size. The aggregate is modeled as a collection of N-interacting, homogeneously broadened two-level systems, where the interaction includes both the static dipole-dipole coupling and superradiant coupling. Our expression for \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3), derived using Liouville-space Green-function techniques, contains contributions from excitons as well as biexcitons. The scaling of \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) with aggregate size and its dependence on the laser detuning and homogeneous and inhomogeneous dephasing rates are analyzed. \ensuremathχ(3){\ensuremath{\chi}}^{(3)}\ensuremathχ(3) contains terms that scale as \ensuremath{\sim}N2{N}^{2}N2 and \ensuremath{\sim}N(N-1). However, under off-resonant conditions these terms interfere destructively, resulting in a linear \ensuremath{\sim}N scaling, which is precisely what is expected for N monomers. Detailed calculations of phase-conjugate degenerate four-wave mixing and pump-probe spectroscopy are performed. In the first case we predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton line shape. Pump-probe spectroscopy shows a series of biexciton absorption lines, some of which are superradiantly broadened.
We investigate phase conjugate degenerate four-wave mixing (DFWM) in molecular aggregates consisting of N interacting, homogeneously broadened two-level systems in a cyclic configuration with a dimension much smaller than an optical wavelength. The interaction includes both the static dipole-dipole coupling and superradiant coupling. We show that, in general the size dependence of χ(3) is determined by the relative magnitudes of the homogeneous dephasing rate and the superradiant decay rate. We predict the existence of a narrow dephasing-induced resonance superposed on the broader superradiant exciton lineshape.
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 ${\mathrm{\ensuremath{\chi}}}^{(3)}$.
Using a reduced equation of motion for the density matrix which accounts for spontaneous emission and superradiance, we analyze the fluorescence and transient grating (TG) decays from a dilute, optically thin distribution of molecular aggregates. We find that the fluorescence is a limited form of superradiance, where cooperativity is restricted to the number N of two level systems which make up the aggregate. The dependence of the linear aggregate decay rate on the amount of inhomogeneous broadening, which is randomly distributed according to Gaussian line shape of width (1/e)2σ, is calculated. It is shown that σ must be comparable in magnitude to the nearest neighbor dipole–dipole coupling V and not the superradiant decay rate, in order to quench the superradiance.
We explain the utility of third-order nonlinear susceptibility, χ (3) (ω 2 ; ±ω 1 , ∓ω1{\mp}\omega_{1}∓ω1​, ω 2 ), for the observation of resonances with homogeneous widths from inhomogeneously broadened distributions in three-level systems related to Raman or two-photon allowed electronic transitions. The line narrowing and high resolution capabilities are demonstrated theoretically using a model for inhomogeneous distributions of transitions in which we regard an absorption spectrum as the composition of some inhomogeneously broadened substructures. The experimental methods to separate a signal having frequency ω 2 from ingoing beam at the same frequency are outlined.
J-aggregates are the subject of great interest due to high values of optical susceptibilities of intensive excitonic resonance, originated from delocalization of electronic excitation in molecular aggregate. In [1,2] resonant enhancement of third-order susceptibiltity of J-aggregates of pseudoisocyanine (PIC) solutions was observed. The dephasing time of excitonic transition of PIC J-aggregate solutions was determined in [3] as 16 ps by low-temperature photon echo, and T2=0.17 ps at room temperature by measuring the dispersion of nonlinear susceptibility in the process of four-wave mixing [2].
The coherent response of excitons in semiconductor nanostructures measured in four-wave mixing (FWM) depends strongly on the inhomogeneous broadening of the exciton transition. We investigate GaAs/Al0.3Ga0.7As single quantum wells (SQW) and AlxGa1-xAs mixed crystals. Additional to the usual phase-space filling nonlinearity, excitation-induced dephasing (EID) and biexciton formation (BIF) are important. EID leads to a strong dependence of the signal on the angle between the linear input polarizations. We find that EID persists in inhomogeneous systems, showing that the mutual density-dependent dephasing rate difference between two subsystems within the inhomogeneous distribution is strongly dependent on their energy difference. BIF is strongly affecting the cross-linear polarized FWM response. The signal for positive delay is dominated by the transitions from the one-exciton state X to the two-exciton states. Here, the third-order polarization at X-XX is not at the same resonance as the first-order polarization at 0-X. Consequently, the rephasing of the microscopic third-order polarization to the macroscopic FWM photon echo is blurred by the non-perfect correlation of X and XX energies, leading to a fast and non-exponential signal decay in delay time. For inhomogeneous broadenings larger than the biexciton binding energy, we find an enhanced biexciton binding energy, and a quenching of the oscillator strength of the unbound biexciton.