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Journal of Nonlinear Optical Physics & Materials

Study of the absorptive and dispersive responses of molecular systems under stochastic considerations

José Luis Paz, A. Mastrodoménico, Jaime Fernando Cárdenas-García, Luis G. Rodríguez, César Costa Vera

Published Sep 1, 2014• 3 citations
Publisher

Abstract

Absorptive and dispersive nonlinear optical properties have been studied considering a two-level molecular system with permanent dipole moments immersed in a thermal reservoir interacting with a classical electromagnetic field. Our model is based in a stochastic description, where the system–solvent interaction induces random shifts in the Bohr frequency (ω0) that transforms ω0 into a stochastic variable ξ(t). Our results show that it is necessary to neglect the rotating-wave approximation in order to measure the effect of the permanent dipole moments. The two-photon transitions over those with one-photon are favored when the permanent dipole moments are considered.

DOI
10.1142/s0218863514500374

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Absorptive and dispersive nonlinear optical properties have been studied considering a two-level molecular system with permanent dipole moments immersed in a thermal reservoir interacting with a classical electromagnetic field. Our model is based in a stochastic description, where the system–solvent interaction induces random shifts in the Bohr frequency (ω0) that transforms ω0 into a stochastic variable ξ(t). Our results show that it is necessary to neglect the rotating-wave approximation in order to measure the effect of the permanent dipole moments. The two-photon transitions over those with one-photon are favored when the permanent dipole moments are considered.
The solvent effects over optical properties of a two level system in presence of an electromagnetic field was modeled in this work. The system-field interaction was considered by using the dipolar-electric approximation, assuming negligible the molecular permanent dipole moments. The solvent effects were included in the formalism in a stochastic form, through the induced shifts over the Bohr molecular frequency. The coherence was calculated from the optical stochastic Bloch equations (OSBE), and later, it was averaged over all realizations of the stochastic molecular variable. To do that, it was applied the convolution theorem, which was resolved using direct and inverse Fourier transforms.
We calculate the non-linear absorptive and dispersive properties in four wave mixing (FWM) signal for a homogeneously broadened molecular two level system with dipole permanent moments interacting with two classical electromagnetic field,, using (omega) 1 equals (omega) 2, considering two sources of noise, the collective effect of the thermal bath, that is modeled as a random Bohr frequency and the effect of the electromagnetic field random amplitude. This assumptions transforms the optical Bloch equations to a system of multiplicative stochastic differential equations whose solution is found though an systematic treatment of cumulant expansion which lead directly to reduce OBE.
The inclusion of the permanent dipole moments and the solvent on the optical conventional Bloch equations (OCBE) allowed us to obtain analytical expressions for the optical properties of a two-level molecular system. We employed the methodology developed by Colmenares et al.1, in which they model the collisional effect of the solvent through a stochastical function, ξ(t) = ω0 + σ(t), so the OCBE become a set of coupled integro-differential stochastical equations that we solved, up to third order in the incident field, employing the perturbation theory. Once obtained the analytical expressions for the density matrix elements, macroscopic polarization and effective susceptibility of the system, we studied the optical properties derived in the frequency space, inside and outside the rotating wave approximation.
The study of a two-level molecular system with permanent dipole moments (PDM) immersed in a thermal bath and interacting with an electromagnetic field depends on the inclusion of the rotating wave approximation (RWA). A methodology based on cumulant expansions to obtain the average in the Fourier components associated with the coherences and populations, calculated by the optical stochastic Bloch equations, is employed. With these equations, we have modeled the optical responses for the four-wave mixing signal and we have demonstrated that inside the RWA the PDM cannot be measured.
In this work, the thermal reservoir effects over the absorptive and dispersive nonlinear optical properties of a two-level molecular system in presence of classical electromagnetic fields, were modeled. The collective effects proper of the thermal reservoir are modeled as a time dependent frequency, whose manifestation is the broadening of the upper level according to a prescribed random function. Using the stochastic optical Bloch equations, analytical expression for the nonlinear induced susceptibilities and absorptive and dispersive optical properties, were obtained. Numerical calculations were carried out to construct surfaces corresponding to these optical properties as a function of the pump-probe frequency detuning, relationships between the longitudinal and transversal relaxation times, and molecular concentration of solute. Finally, we see an attenuation of these optical responses by the solvent effects and the high pump-intensity.
We have considered the stochastic effect of a thermal reservoir on a two-level system interacting with a classical electromagnetic field. The collective effect of the bath is modelled in this work as a random Bohr frequency whose manifestation is the broadening of the upper level according to a prescribed random function. This assumption transforms the optical Bloch equations to a system of multiplicative stochastic differential equations whose solution is found. After obtaining analytical expressions for the linear absorption coefficient and linear refraction index, numerical results are presented. This is done in the form of tridimensional plots of the optical properties as a function of the pump-frequency detuning factor and the strength or relaxation time of the noise source.