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Journal of Nonlinear Optical Physics & Materials

Effects of spectral diffusion on the nonlinear optical properties in two- and three-state quantum systems in a four-wave mixing signal

Fernando Moncada, José Luis Paz, Luis Lascano, César Costa Vera

Published Dec 1, 2018• 4 citations
Publisher

Abstract

Effects of spectral diffusion on the third-order nonlinear susceptibility for two- and three-level quantum systems, immersed in a thermal reservoir, are evaluated with a four-wave mixing (FWM) signal through the density matrix formalism. For this, inhomogeneously broadened two- and three-level quantum systems are used. In these models, the distribution of natural frequencies follows a Lorentzian function. The absorptive and dispersive nonlinear optical responses are determined for the considered quantum models. The results presented in this work show potential applications in the development of optical switches and provide a potential method to measure spectral diffusion relaxation times in three-level quantum systems.

DOI
10.1142/s0218863518500388

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Effects of spectral diffusion on the third-order nonlinear susceptibility for two- and three-level quantum systems, immersed in a thermal reservoir, are evaluated with a four-wave mixing (FWM) signal through the density matrix formalism. For this, inhomogeneously broadened two- and three-level quantum systems are used. In these models, the distribution of natural frequencies follows a Lorentzian function. The absorptive and dispersive nonlinear optical responses are determined for the considered quantum models. The results presented in this work show potential applications in the development of optical switches and provide a potential method to measure spectral diffusion relaxation times in three-level quantum systems.
We have incorporated spectral diffusion in a system of three quantum states immersed in a thermal bath for the study of optical responses in four-wave mixing spectroscopy, using the density matrix formalism and the development of quantum relaxation theory. The primary objective of this work is to establish a generalized theoretical framework that evaluates how internal relaxation within all three quantum states—including a non-resonant third state—modifies the macroscopic nonlinear polarization of the medium. By introducing specific scaling parameters, we aim to isolate and quantify the influence of spectral diffusion in the third state, a feature typically omitted in existing literature. The results found allow giving explanations about the parametric amplification, as well as the possibility of generation of sensors in regions close to the degeneracy of optical frequencies. In our study we have been able to recognize previously published results for the optical susceptibility responses of absorption and scattering. Our model, unlike those already presented in this topic, assumes all states of the quantum system with internal relaxations. We found that the inclusion of spectral diffusion in the third state significantly impacts the system’s kinetic behavior, specifically eliminating “hole burning” effects in dispersion and parametric amplification in nonlinear absorption. These findings suggest that interband and intraband kinetic constants must be considered simultaneously to accurately model the ultra-fast relaxation times in complex molecular systems like organic dyes. The development of the problem is subject to rotating wave considerations in the selection of the Fourier component terms associated with both the coherences induced in the system and those coming from the canonical populations of the states. In addition, it involves the development of perturbation theory in the amplitudes of the incident electromagnetic fields associated with this nonlinear FWM type technique. Our treatments have been performed for frequency differences in the absorption range of organic dyes, without any restriction in the cases of near-zero tunings. Our proposal for the third state with spectral diffusion effects completely eliminates both the “hole burning” created in the dispersion and the parametric amplification generated in nonlinear absorption, when compared with results with the state outside resonance treated without spectral diffusion effects within the band. This highlights the importance of interband kinetic constants in comparison with those within the band.
We study the nonlinear absorptive and dispersive optical properties of molecular systems immersed in a thermal reservoir interacting with a four-wave mixing (FWM) signal. Residual spin-orbit Hamiltonians are considered in order to take into account the internal structure of the molecule. As system parameters in the dissipation processes, transverse and longitudinal relaxation times are considered for stochastic solute–solvent interaction processes. The intramolecular coupling effects on the optical responses are studied using a molecule model consisting of two coupled harmonic curves of electronic energies with displaced minima in nuclear energies and positions. In this study, the complete frequency space is considered through the pump–probe detuning, without restricting the derivations to only maximums of population oscillations. This approach opens the possibility of studying the behavior of optical responses, which is very useful in experimental design. Our results indicate the sensitivity of the optical responses to parameters of the molecular structure as well as to those derived from the photonic process of FWM signal generation.
The effect of cross relaxation on the third-order nonlinear susceptibility, χ(3) (−ω2, ω1, −ω1, ω2), describing polarization spectroscopy experiments is calculated with a density matrix formalism. By using a model of inhomogeneously broadened two-level systems with a reservoir state and allowing for cross relaxation within the inhomogeneous line, we find that both the ratio of the maximum signal at ω1 = ω2 to the background and the broad component resembling the absorption band depend on the cross-relaxation rate. Also, one additional component with a characteristic width of the order of the spectral-diffusion rate Γ3 is expected to appear, depending on the value of this parameter. Furthermore, we estimate from published data [ Phys. Rev. A17, 1439 ( 1978)] the product of the transverse-relaxation time times the cross-relaxation time T2T3 to be 0.014 psec2 for cyanine dye and 0.0009 psec2 for malachite green oxalate.
A study of the effects of relaxation times on the Four-Wave Mixing (FWM) signal response in frequency space for a two-level system is presented where the collisional effects of the solvent and the presence of the electromagnetic field are considered. The model is a generalization of the conventional two-level approach in that it includes a simplified description of the molecular structure associated with the non-zero permanent dipole moments. Significant changes in both the FWM-intensity and nonlinear optical properties are observed, due mainly to changes in the ratio between the longitudinal and transversal relaxation times.
We report the results of a theoretical study of line shapes obtained in frequency-domain four-wave mixing in systems inhomogeneously broadened due to random fields in the presence of spectral diffusion. The results are based on a solution of the modified optical Bloch equations solved perturbatively to third order in the applied optical fields, assuming a strong redistribution model for the spectral diffusion. The results show that the redistribution of excitation can be directly observed in the line-shape function and can also lead to the presence of interference effects in the line shape.
Using a perturbation theory based on the density-matrix formulation, we study the nonlinear optical responses of a noninteracting three-level model system to consecutive coherent pulsed excitations, as realized in several recent experiments involving excitonic transitions in quantum wells. The terahertz emission, which is a second-order response, and the four-wave mixing signal, a third-order response, are calculated within the rotating-wave approximation, in the presence of detuning, dephasing, and inhomogeneous broadening. We study the quantum beats in the photon echo from transient four-wave mixing experiments of a three-level system. We find that the temporal profile of the photon echo signal in the four-wave mixing experiments depends very strongly on the amount of inhomogeneous broadening of the energy levels involved. Both the position and the intensity of the peaks exhibit a smooth evolution from strong quantum beats to a conventional photon echo as the inhomogeneous broadening increases. These features from our noninteracting model system are compared with a recent experiment, and found to account for a number of experimental observations.