Nonlinear Optical Rectification in a Polar Molecule-Plasmonic Nanoparticle Structure ^†

Abstract

We theoretically study the nonlinear optical rectification of a Zinc-phthalocyanine molecular complex, modelled as a polar two-level quantum system, interacting with an optical field near a gold nanoparticle. We use the steady-state solution of the density matrix equations for determining the nonlinear optical rectification coefficient in this case. We further use first-principle electronic structure calculations for determining the energies of the molecular states involved and the corresponding transition and permanent electric dipole moments, as well as first-principle classical electromagnetic calculations for calculating the influence of the metallic nanoparticle on the decay rates of the molecular states due to the Purcell effect and on the external electric fields applied on the molecule. We investigate the nonlinear optical rectification coefficient in the absence and the presence of the plasmonic nanoparticle for various parameters, such as the field polarization and the distance between the molecular complex and the plasmonic nanoparticle. We find that the nonlinear coefficient can be significantly enhanced for specific field polarization and at suitable distance between the molecule and the plasmonic nanoparticle. We also find that this process is highly efficient at weak field intensity, zero pure dephasing rate and for small values of the transition dipole moment.

1. Introduction

The study of nonlinear optical properties of quantum systems, like quantum dots and molecules, near plasmonic nanostructures has attracted significant interest in the past decade. Several nonlinear phenomena have been studied in quantum systems next to plasmonic nanostructures, like second [1] and third harmonic generation [2], Kerr nonlinearity [3,4], four-wave mixing [5], optical bistability [6] and nonlinear optical rectification [6,7]. The latter occurs in asymmetric quantum systems and it can be strongly influenced, enhanced or suppressed, depending on the particular plasmonic nanostructure used.

In this work, we theoretically study the nonlinear optical rectification of a polar two-level quantum system (QS), a Zinc-phthalocyanine molecular complex, interacting with an optical field near a gold nanoparticle. Initially, we use the steady-state solution of the density matrix equations for determining the proper form of the nonlinear optical rectification coefficient. We then use ab initio electronic structure calculations for determining the electronic structure of the molecule under study, i.e., the necessary molecular state energies and the corresponding transition and permanent electric dipole moments (PDMs). We also use first-principle classical electromagnetic calculations for calculating the influence of the metallic nanoparticle (MNP) on the decay rates of the molecule due to the Purcell effect and on the electric field applied in the molecule in the presence of the MNP. We then use the above to investigate the form of the corresponding nonlinear optical rectification (NOR) coefficient in the absence and presence of the plasmonic nanoparticle for various parameters, such as the field polarization and the distance between the molecular complex and the plasmonic nanoparticle. We find that the NOR coefficient can be quite enhanced for specific field polarization and for suitable distance between the molecule and the plasmonic nanoparticle. Also, we find that high efficiency of this process is obtained for weak field intensity, zero pure dephasing rate and for small value of the transition dipole moment.

2. Theoretical Model

We consider a hybrid structure composed of a polar two-level QS and an MNP, as the one depicted in Figure 1. The QS models the Zinc-phthalocyanine molecular complex exhibiting unequal PDMs, µ₁₁ and µ₂₂, due to the violation of the inversion symmetry, which introduce new coupling terms and affect the electromagnetic response of the system. The local dielectric function of the gold nanoparticle is taken from spectroscopic data and the hybrid system is embedded in a loss-free environment with unity relative dielectric constant.

The QS described above interacts with an applied electric field with time-dependent form, $E\left(t\right)={\mathrm{E}}_0{e}^{-i\omega t}+c.c.$, where ${\mathrm{E}}_0$ is the field amplitude as modified by the presence of the MNP. Also, the decay rate of the QS is modified due to the presence of the MNP by the Purcell effect. Here, we consider that the electric field is polarized either radially (z axis) or tangentially (x axis) with respect to the surface of the MNP. In Figure 2a,b, we present the modified electric field factors and the modified decay rates as function of the distance $d$ between the QS and the MNP, correspondingly, calculated with the EM Green’s tensor technique [8]. Significant modification, both enhancement and suppression of the electric field amplitude and the decay rates are shown, depending on the distance $d$ and on the polarization of the field. For the determination of the NOR coefficient, we use the density matrix equations, as derived for coherent field-matter interaction of an asymmetric QS [6,9,10].

3. The Zinc-Phthalocyanine Molecular Complex

We use parameters for the zinc-phthalocyanine molecular complex shown in Figure 1, with carbon (gray), hydrogen (white), oxygen (red), nitrogen (blue), and zinc (light blue) atoms. From ab initio electronic structure calculations [11,12], we obtain that ħ${\mathsf{\omega }}_{12}$ = 1.9445 eV; furthermore, the corresponding values of the PDMs and the transition dipole moments, in every direction with respect to the MNP (Figure 1), are presented in

Table 1. Numerical values of dipole moments of the molecular complex along the z and x direction.

Dipole Moments/D	z Axis	x Axis
µ11	0.6836	−3.8632
µ22	0.7192	−3.9342
µ12	2.9768	0.3594

. Thus, for a transition along the z axis of the molecule, we use ${\mathsf{\Gamma }}_0^z$ ≈ 11.42 MHz, and for a transition along the x axis, we use ${\mathsf{\Gamma }}_0^x$ ≈ 167.25 kHz.

4. Results and Discussion

In this work we focus on the influence of the MNP to the electromagnetic response of an asymmetric QS interacting with an external electric field. It is however important to mention that since the electric fields, which influence the NOR coefficient, correspond to different field intensities I, saturation effects are present.

In Figure 3, we observe that the NOR is highly efficient at weak field intensity. In case of non-zero $I$, the system reaches saturation with small changes of the intensity. Moreover, ${\chi }_0^{\left(2\right)}\left(\delta \right)$ is stronger when the field is polarized along the x axis than the z axis. Here, $\delta$ denotes the detuning, i.e., frequency difference between the applied electric field frequency and ${\mathsf{\omega }}_{12}$.

In Figure 4, the optical response of the system changes and we obtain weak nonlinear rectification for radial polarization and high efficiency process for tangential polarization of the field for zero pure dephasing. We note that we choose for (a) $d=50$ nm and for (b) $d=31.5$ nm since at these values, the minimal value of the decay rate, as shown in Figure 2b, and thus the maximum of ${\chi }_0^{\left(2\right)}\left(\delta \right)$ is attained for $\delta =0$ and $I=0$. Next, in panels (c)–(f) of the same figure, we focus on the influence of the distance $d$ from the QS to the surface of the MNP on the NOR coefficient, by increasing $I$. We observe that ${\chi }_0^{\left(2\right)}\left(d\right)$ for $I=0$ behaves inversely to the modified spontaneous decay rate, shown in Figure 2b. Also, we observe that for tangential polarization of the field, the maximal NOR coefficient value occurs at shorter distance, as $I$ increases. Lastly, in panels (e) and (f), the NOR efficiency is evidently suppressed, when the pure dephasing rate increases, here denoted by ${\gamma }_d$. Moreover, in panel (f), when the external electric field is applied, the maximum value of the NOR coefficient occurs at larger $d$, in comparison to the case shown in panel (d), due to the variation of ${\gamma }_d$.

5. Summary

In summary, here, we theoretically studied the nonlinear optical rectification of a Zinc-phthalocyanine molecular complex, modelled as a polar two-level quantum system, interacting with an optical field near a gold nanoparticle. We used first-principle electronic structure calculations and first-principle classical electromagnetic calculations and investigated the NOR coefficient in the absence and the presence of the plasmonic nanoparticle for various parameters, such as the field polarization and the distance between the molecular complex and the plasmonic nanoparticle. We found that the nonlinear coefficient can be significantly enhanced for specific field polarization and at suitable distance between the molecule and the plasmonic nanoparticle. We also showed that this process is highly efficient at weak field intensity, zero pure dephasing rate and for small values of the transition dipole moment. Our results may have important applications in nanophotonics and current quantum technology.