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Results in Physics

Spectral diffusion in a three-state quantum system: studies of nonlinear optical susceptibilities

José Luis Paz, Edgar Márquez, Cristhian Dávila, Marcos Loroño, Lenin A. González-Paz, José R. Mora, Ysaías J. Alvarado

Published May 29, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1016/j.rinp.2026.108696

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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.
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 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.
The steady-state response of a nonlinear medium to several simultaneously applied monochromatic electromagnetic fields is described by the density matrix method. A Fourier series expansion in terms of ascending powers of the amplitudes of the applied fields is especially useful to describe the parametric response in spectral regions, where the absorption is small. As the resonances of the material system are approached, the general formalism exhibits the mixture and interference between parametric processes and single- and multiple-photon absorption and emission processes. previously discussed examples of the two-level and three-level system are generalized. The reaction of the general nonlinear medium on the electromagnetic fields is incorporated.
The purpose of this research is to study the relaxation of optically generated excitons and carriers in semiconductor microstructures using four wave mixing (FWM) spectroscopy. The systems studied include CdSSe microcrystallite doped glasses and GaAs/AlGaAs multiple quantum well structures (MQWS). Exposing a semiconductor to optical radiation near the fundamental band gap results in the creation of populations of elementary excitations including electrons, holes, and excitons, and also results in the creation of a superposition state between the ground and excited state of the solid. The work in this dissertation examines the relaxation of the populations of elementary excitations and the relaxation of the coherence between the ground and excited state using four wave mixing spectroscopy. First, the nonlinear optical response of simple two level systems is examined in order to provide insight into the types of line shapes expected from semiconductors. It is shown that the line shape is strongly dependent on how the system is coupled to the reservoir and the consequences of coupling to a reservoir are examined in a FWM measurement made in atomic sodium. The first semiconductor system studied is CdSSe microcrystallite doped glass. This system is shown to have a very slow component to the nonlinear response which has a optical intensity dependence and temperature dependence which suggests that the FWM response in these materials is trap mediated. Room temperature FWM measurements in GaAs MQWS enables the measurement of the carrier recombination time and the ambipolar diffusion coefficient. Using the technique of correlated optical fields, a slow component to the nonlinear response was measured showing an interference profile which suggests a possible shift of the exciton resonance due to the optically generated carriers. At low temperatures (2-20 K), measurements of the exciton line shape and relaxation time were made and evidence for exciton spectral diffusion was found. The low temperature line shapes can be qualitatively reproduced using Modified Optical Bloch equations which include the effects of spectral diffusion.
This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process. In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed. In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile. Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given. Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.