# Pump–Probe Optical Response and Four-Wave Mixing in a Zinc–Phthalocyanine–Metal Nanoparticle Hybrid System

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## Abstract

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## 1. Introduction

## 2. Theory

## 3. Results and Discussion

#### 3.1. Zinc–Phthalocyanine Molecular Complex

#### 3.2. Numerical Results

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

B3LYP | Becke, 3-parameter, Lee–Yang–Parr |

DFT | Density Functional Theory |

FWM | Four-Wave Mixing |

MNP | Metallic Nanoparticle |

TD-DFT | Time-Dependent Density Functional Theory |

## References

- Zhang, W.; Govorov, A.O.; Bryant, G.W. Semiconductor-metal nanoparticle molecules: Hybrid excitons and the nonlinear dispersion effect. Phys. Rev. Lett.
**2006**, 97, 146804. [Google Scholar] [CrossRef] - Yan, J.-Y.; Zhang, W.; Duan, S.-Q.; Zhao, X.-G.; Govorov, A.O. Optical properties of coupled metal-semiconductor and metal-molecule nanocrystal complexes: Role of multipole effects. Phys. Rev. B
**2008**, 77, 165301. [Google Scholar] [CrossRef] - Artuso, R.D.; Bryant, G.W. Strongly coupled quantum dot-metal nanoparticle systems: Exciton-induced transparency, discontinuous response, and suppression as driven quantum oscillator effects. Phys. Rev. B
**2010**, 82, 195419. [Google Scholar] [CrossRef] - Singh, M.R.; Schindel, D.G.; Hatef, A. Dipole-dipole interaction in a quantum dot and metallic nanorod hybrid system. Appl. Phys. Lett.
**2011**, 99, 181106. [Google Scholar] [CrossRef] - Kosionis, S.G.; Terzis, A.F.; Yannopapas, V.; Paspalakis, E. Nonlocal effects in energy absorption of coupled quantum dot–metal nanoparticle systems. J. Phys. Chem. C
**2012**, 116, 23663. [Google Scholar] [CrossRef] - Schindel, D.; Singh, M.R. A study of energy absorption rate in a quantum dot and metallic nanosphere hybrid system. J. Phys. Condens. Matter
**2015**, 27, 345301. [Google Scholar] [CrossRef] [PubMed] - Nugroho, B.S.; Malyshev, V.A.; Knoester, J. Tailoring optical response of a hybrid comprising a quantum dimer emitter strongly coupled to a metallic nanoparticle. Phys. Rev. B
**2015**, 92, 165432. [Google Scholar] [CrossRef] - Hapuarachchi, H.; Gunapala, S.D.; Bao, Q.; Stockman, M.I.; Premaratne, M. Exciton behavior under the influence of metal nanoparticle near fields: Significance of nonlocal effects. Phys. Rev. B
**2018**, 98, 115430. [Google Scholar] [CrossRef] - Mohammadzadeh, A.; Miri, M. Optical response of hybrid semiconductor quantum dot-metal nanoparticle system: Beyond the dipole approximation. J. Appl. Phys.
**2018**, 123, 043111. [Google Scholar] [CrossRef] - Evangelou, S. Tailoring second-order nonlinear optical effects in coupled quantum dot-metallic nanosphere structures using the Purcell effect. Microelectron. Eng.
**2019**, 215, 111019. [Google Scholar] [CrossRef] - Thanopulos, I.; Paspalakis, E.; Yannopapas, V. Plasmon-induced enhancement of nonlinear optical rectification in organic materials. Phys. Rev. B
**2012**, 85, 035111. [Google Scholar] [CrossRef] - Carreño, F.; Antón, M.A.; Paspalakis, E. Nonlinear optical rectification and optical bistability in a coupled asymmetric quantum dot-metal nanoparticle hybrid. J. Appl. Phys.
**2018**, 124, 113107. [Google Scholar] [CrossRef] - Domenikou, N.; Thanopulos, I.; Yannopapas, V.; Paspalakis, E. Nonlinear Optical Rectification in an Inversion-Symmetry-Broken Molecule Near a Metallic Nanoparticle. Nanomaterials
**2022**, 12, 1020. [Google Scholar] [CrossRef] [PubMed] - Li, J.-H.; Shen, S.; Ding, C.-L.; Wu, Y. Magnetically induced optical transparency in a plasmon-exciton system. Phys. Rev. A
**2021**, 103, 053706. [Google Scholar] [CrossRef] - Sadeghi, S.M. Gain without inversion in hybrid quantum dot–metallic nanoparticle systems. Nanotechnology
**2010**, 21, 455401. [Google Scholar] [CrossRef] [PubMed] - Kosionis, S.G.; Terzis, A.F.; Sadeghi, S.M.; Paspalakis, E. Optical response of a quantum dot–metal nanoparticle hybrid interacting with a weak probe field. J. Phys. Condens. Matter
**2013**, 25, 045304. [Google Scholar] [CrossRef] [PubMed] - Sadeghi, S.M. Ultrafast plasmonic field oscillations and optics of molecular resonances caused by coherent exciton-plasmon coupling. Phys. Rev. A
**2013**, 88, 013831. [Google Scholar] [CrossRef] - Zhao, D.-X.; Gu, Y.; Wu, J.; Zhang, J.-X.; Zhang, T.-C.; Gerardot, B.D.; Gong, Q.-H. Quantum-dot gain without inversion: Effects of dark plasmon-exciton hybridization. Phys. Rev. B
**2014**, 89, 245433. [Google Scholar] [CrossRef] - Lu, Z.; Zhu, K.-D. Enhancing Kerr nonlinearity of a strongly coupled exciton–plasmon in hybrid nanocrystal molecules. J. Phys. B
**2008**, 41, 185503. [Google Scholar] [CrossRef] - Paspalakis, E.; Evangelou, S.; Kosionis, S.G.; Terzis, A.F. Strongly modified four-wave mixing in a coupled semiconductor quantum dot-metal nanoparticle system. J. Appl. Phys.
**2014**, 115, 083106. [Google Scholar] [CrossRef] - Kosionis, S.G.; Paspalakis, E. Control of self-Kerr nonlinearity in a driven coupled semiconductor quantum dot—Metal nanoparticle structure. J. Phys. Chem. C
**2019**, 123, 7308–7317. [Google Scholar] [CrossRef] - Singh, M.R.; Yastrebov, S. Switching and Sensing Using Kerr Nonlinearity in Quantum Dots Doped in Metallic Nanoshells. J. Phys. Chem. C
**2020**, 124, 12065–12074. [Google Scholar] [CrossRef] - Malyshev, A.V.; Malyshev, V.A. Optical bistability and hysteresis of a hybrid metal-semiconductor nanodimer. Phys. Rev. B
**2011**, 84, 035314. [Google Scholar] [CrossRef] - Nugroho, B.S.; Iskandar, A.A.; Malyshev, V.A.; Knoester, J. Bistable optical response of a nanoparticle heterodimer: Mechanism, phase diagram, and switching time. J. Chem. Phys.
**2013**, 139, 014303. [Google Scholar] [CrossRef] [PubMed] - Solookinejad, G.; Jabbari, M.; Nafar, M.; Ahmadi, E.; Asadpour, S.H. Incoherent control of optical bistability and multistability in a hybrid system: Metallic nanoparticle-quantum dot nanostructure. J. Appl. Phys.
**2018**, 124, 063102. [Google Scholar] [CrossRef] - Lu, Z.; Zhu, K.-D. Slow light in an artificial hybrid nanocrystal complex. J. Phys. B
**2009**, 42, 015502. [Google Scholar] [CrossRef] - Li, J.-B.; Kim, N.-C.; Cheng, M.-T.; Zhou, L.; Hao, Z.-H.; Wang, Q.-Q. Optical bistability and nonlinearity of coherently coupled exciton-plasmon systems. Opt. Express
**2012**, 20, 1856. [Google Scholar] [CrossRef] - Li, J.-B.; He, M.-D.; Chen, L.-Q. Four-wave parametric amplification in semiconductor quantum dot-metallic nanoparticle hybrid molecules. Opt. Express
**2014**, 22, 24734. [Google Scholar] [CrossRef] - Kosionis, S.G.; Paspalakis, E. Pump-probe optical response of semiconductor quantum dot–metal nanoparticle hybrids. J. Appl. Phys.
**2018**, 124, 223104. [Google Scholar] [CrossRef] - Singh, S.K.; Abak, M.K.; Tasgin, M.E. Enhancement of four-wave mixing via interference of multiple plasmonic conversion paths. Phys. Rev. B
**2016**, 93, 035410. [Google Scholar] [CrossRef] - Fashina, A.; Nyokong, T. Nonlinear optical response of tetra and mono substituted zinc phthalocyanine complexes. J. Lumin.
**2015**, 167, 71. [Google Scholar] [CrossRef] - Vladimirova, Y.V.; Zudkov, V.N. Quantum optics in nanostructures. Nanomaterials
**2021**, 11, 1919. [Google Scholar] [CrossRef] [PubMed] - Antón, M.A.; Carreño, F.; Calderón, O.G.; Melle, S.; Cabrera, E. Radiation emission from an asymmetric quantum dot coupled to a plasmonic nanostructure. J. Opt.
**2016**, 18, 025001. [Google Scholar] [CrossRef] - Johnson, P.B.; Christy, R.W. Optical constants of the noble metals. Phys. Rev. B
**1972**, 6, 4370. [Google Scholar] [CrossRef] - Boyd, R.W. Nonlinear Optics, 3rd ed.; Elsevier Academic Press: San Diego, CA, USA, 2003; pp. 277–328. [Google Scholar]
- Boyd, R.W.; Raymer, M.G.; Narum, P.; Harter, D. Four-wave parametric interactions in a strongly driven two-level system. J. Phys. Rev. A
**1981**, 24, 411. [Google Scholar] [CrossRef] - Frisch, M.J.; Trucks, G.W.; Schlegel, H.B.; Scuseria, G.E.; Robb, M.A.; Cheeseman, J.R.; Scalmani, G.; Barone, V.; Petersson, G.A.; Nakatsuji, H.; et al. Gaussian 09, Revision A.02; Gaussian, Inc.: Wallingford, CT, USA, 2016. [Google Scholar]

**Figure 1.**(

**a**) Schematic depiction of a quantum system located at a distance d from the surface of the MNP. (

**b**) Energy level scheme of the dressed states, $|+\rangle $ and $|-\rangle $, and the transitions between them in the case of negative detuning $\Delta =\omega -{\omega}_{21}$. TP, RL, and AC stand for the three-photon resonance process, the stimulated Rayleigh scattering process, and the AC-Stark effect, respectively. ${\mathsf{\Omega}}^{\prime}=\sqrt{{\Delta}^{2}+{|{\mathsf{\Omega}}_{a}|}^{2}}$ denotes the effective Rabi frequency.

**Figure 2.**(

**a**) The field modification factor of the field, f, and (

**b**) the Purcell factor, g, as a function of the distance d between the quantum system and the surface of the Au-MNP for energy equal to 1.9445 eV, which corresponds to the transition frequency ${\omega}_{21}$ of the molecular quantum system.

**Figure 3.**The molecular quantum system used: the zinc–phthalocyanine complex is composed of carbon (gray), hydrogen (white), oxygen (red), nitrogen (blue), and zinc (light blue) atoms. The complex is not planar; however, the phthalocyanine part of the complex is planar, coinciding with the $xz$-plane, as schematically shown.

**Figure 4.**The dispersion (

**a**) and absorption (

**b**) spectra, as a function of the frequency mismatch of the applied fields $\delta $, for different distances d of the quantum system from the surface of the MNP. We assume that $|{\mathsf{\Omega}}_{a}^{f}|=40\phantom{\rule{3.33333pt}{0ex}}\mu {\mathrm{s}}^{-1}$, ${\gamma}_{d}=0$, and $\Delta =0$.

**Figure 5.**The effective Rabi frequency $|{\mathsf{\Omega}}^{\prime}|$, as a function of the distance d, for different values of the detuning $\Delta $. We also take $|{\mathsf{\Omega}}_{a}^{f}|=40\phantom{\rule{3.33333pt}{0ex}}\mu {\mathrm{s}}^{-1}$ and ${\gamma}_{d}=0$.

**Figure 6.**The FWM spectrum, as a function of the frequency mismatch of the applied fields $\delta $, for different distances d of the quantum system from the surface of the MNP. We assume that $|{\mathsf{\Omega}}_{a}^{f}|=40\phantom{\rule{3.33333pt}{0ex}}\mu {\mathrm{s}}^{-1}$, ${\gamma}_{d}=0$, and $\Delta =0$.

**Figure 7.**The dispersion (

**a**) and absorption (

**b**) spectra, as a function of the frequency mismatch of the applied fields $\delta $, for different distances d of the quantum system from the surface of the MNP. We assume that $|{\mathsf{\Omega}}_{a}^{f}|=40\phantom{\rule{3.33333pt}{0ex}}\mu {\mathrm{s}}^{-1}$, ${\gamma}_{d}=0$, and $\Delta =-3{\Gamma}^{f}$.

**Figure 9.**(

**a**) The dispersion and (

**b**) the absorption spectra versus $\delta $ in the presence of the MNP for different $\Delta $ with $|{\mathsf{\Omega}}_{a}^{f}|=40\phantom{\rule{3.33333pt}{0ex}}\mu {s}^{-1}$, ${\gamma}_{d}=0$, and $d=25$ nm.

**Figure 10.**The FWM spectrum as a function of $\delta $ in the presence of the MNP for different $\Delta $ with $|{\mathsf{\Omega}}_{a}^{f}|=40\phantom{\rule{3.33333pt}{0ex}}\mu {\mathrm{s}}^{-1}$, ${\gamma}_{d}=0$, and $d=25$ nm.

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**MDPI and ACS Style**

Domenikou, N.; Kosionis, S.G.; Thanopulos, I.; Yannopapas, V.; Paspalakis, E.
Pump–Probe Optical Response and Four-Wave Mixing in a Zinc–Phthalocyanine–Metal Nanoparticle Hybrid System. *Micromachines* **2023**, *14*, 1735.
https://doi.org/10.3390/mi14091735

**AMA Style**

Domenikou N, Kosionis SG, Thanopulos I, Yannopapas V, Paspalakis E.
Pump–Probe Optical Response and Four-Wave Mixing in a Zinc–Phthalocyanine–Metal Nanoparticle Hybrid System. *Micromachines*. 2023; 14(9):1735.
https://doi.org/10.3390/mi14091735

**Chicago/Turabian Style**

Domenikou, Natalia, Spyridon G. Kosionis, Ioannis Thanopulos, Vassilios Yannopapas, and Emmanuel Paspalakis.
2023. "Pump–Probe Optical Response and Four-Wave Mixing in a Zinc–Phthalocyanine–Metal Nanoparticle Hybrid System" *Micromachines* 14, no. 9: 1735.
https://doi.org/10.3390/mi14091735