# A Review of Many-Body Interactions in Linear and Nonlinear Plasmonic Nanohybrids

## Abstract

**:**

## 1. Introduction

## 2. Surface Plasmon Polaritons

## 3. Dipole-Dipole Interactions

_{ij}is the DDI coupling constant and is found as

_{DDI}has the effect of the long-range DDI (i.e., r

^{−3}). The average in E

_{DDI}has been evaluated in references [30,31,32,33,34,35,36] by using the method of Lorentz [49], The expression of the DDI field is found as and is written as

^{−3}. Putting Equation (18) into Equation (14), we obtained the expression of the DDI as

^{−3}. The higher-order terms r

^{−6}were neglected because they are weak.

## 4. Linear and Nonlinear Plasmonics and Density Matrix Method

_{ij}. To study the Kerr nonlinearity, we applied the probe field between $|\mathbf{1}\rangle $ and $|\mathbf{2}\rangle $. A schematic diagram of the four-level quantum emitter is shown in Figure 2.

_{ij}= |i><j| is the exciton creation operators for $\left|i\rangle \leftrightarrow \right|j\rangle $. Parameter Ω

_{p}is called the Rabi frequency. The first term in ${H}_{int}$ is the exciton-probe field interaction. The second term is exciton-SPP field interaction. The third term is exciton-DDI-MNP interaction. The last term is exciton-DDI-QE field interaction.

_{ij}is the exciton decay rates.

## 5. Exciton Decay Rates Due to Dipole-Dipole Interaction

_{ij}= |i><j| is the exciton creation operator for transition $\left|i\rangle \leftrightarrow \right|j\rangle $ where i and j stand for 1, 2, 3, and 4. Meanwhile, the operators ${p}_{nm,{k}_{z}}^{\u2020}{p}_{nm,{k}_{z}}^{}$ are the photon creation and annihilation operators, respectively. The coupling constant appearing in Equation (43) was found as

_{int}is the interaction term given in Equation (44) and D

_{k}is the DOS which has been calculated in Equation (8).

_{0}is the radiative decay rate when QD is in the vacuum. Please note the following relationship ${\gamma}_{21}={\Gamma}_{P}^{}+{\Gamma}_{SPP}^{}$, ${\gamma}_{23}={\Gamma}_{DDI}^{MNP}$ and ${\gamma}_{34}={\Gamma}_{DDI}^{QE}$. Note that the radiative and nonradiative decay rates can have large values when exciton energies ε

_{12}, ε

_{23,}and ε

_{34}are close to ${\epsilon}_{nm}$. This is an interesting finding of the paper.

## 6. Dressed States: Exciton-DDI Coupling

## 7. Results and Discussion

_{2}. Some of the examples for the energy (frequency) physical quantities are Rabi frequency (Ω

_{P}), exciton frequencies, probe detuning, and decay rates. In our numerical simulations, we used γ

_{23}/γ

_{21}= γ

_{34}/γ

_{21}=1 and Ω

_{P}/γ

_{21}= 1. The probe detuning (δ

_{p}) and DDI detuning (δ

_{d}) were measured with respect to the decay rate γ

_{21}.

_{p}/${\mathsf{\gamma}}_{21}$ = 0), the Kerr intensity has a peak. The zero detuning means that the probe field frequency is in resonance with the exciton frequency ${\omega}_{21}$ (i.e., ${\omega}_{p}={\omega}_{21})$.

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

- 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] - Yannopapas, V.; Paspalakis, E. Giant enhancement of dipole-forbidden transitions via lattices of plasmonic nanoparticles. J. Mod. Opt.
**2015**, 62, 1435–1441. [Google Scholar] [CrossRef] - Tame, M.S.; McEnery, K.R.; Özdemir, Ş.K.; Lee, J.; Maier, S.A.; Kim, M.S. Quantum plasmonics. Nat. Phys.
**2013**, 9, 329–340. [Google Scholar] [CrossRef] [Green Version] - Törmä, P.; Barnes, W.L. Strong coupling between surface plasmon polaritons and emitters: A review. Rep. Prog. Phys.
**2015**, 78, 013901. [Google Scholar] [CrossRef] [PubMed] - Achermann, M. Exciton−Plasmon Interactions in Metal−Semiconductor Nanostructures. J. Phys. Chem. Lett.
**2010**, 1, 2837. [Google Scholar] [CrossRef] - Singh, M.R.; Guo, J.; Fanizza, E.; Dubey, M. Anomalous photoluminescence quenching in metallic nanohybrids. J. Phys. Chem. C
**2019**, 123, 10013–10020. [Google Scholar] [CrossRef] - Balakrishnan, S.; Najiminaini, M.; Singh, M.R.; Carson, J.J.L. A study of angle dependent surface plasmon polaritons in nano-hole array structures. J. Appl. Phys.
**2016**, 120, 034302. [Google Scholar] [CrossRef] - Antón, M.A.; Carreño, F.; Melle, S.; Calderon, O.G.; Granado, E.C.; Singh, M.R. Optical pumping of a single hole spin in ap-doped quantum dot coupled to a metallic nanoparticle. Phys. Rev. B
**2013**, 87, 195303. [Google Scholar] [CrossRef] [Green Version] - Racknor, C.; Singh, M.R.; Zhang, Y.; Birch, D.J.S.; Chen, Y. Energy transfer between a biological labelling dye and gold nanorods. Methods Appl. Fluoresc.
**2013**, 2, 015002. [Google Scholar] [CrossRef] [Green Version] - Singh, M.R.; Black, K. Anomalous dipole-dipole interaction between ensemble of quantum emitters in metallic nanoparticle hybrids. J. Phys. Chem. C
**2018**, 122, 26584–26591. [Google Scholar] [CrossRef] - Yudson, V.I.; Singh, M.R. Lattice-gas model for electron-hole coupling in disordered media. Phys. Rev. B
**1998**, 58, 16202. [Google Scholar] [CrossRef] - Singh, M.R.; Guo, J.; Chen, J. A Theoretical Study of Fluorescence Spectroscopy of Quantum Emitters Coupled with Plasmonic Dimer and Trimer. J. Phys. Chem. C
**2019**, 123, 17483–17490. [Google Scholar] [CrossRef] - Singh, M.R. The effect of the dipole–dipole interaction in electromagnetically induced transparency in polaritonic band gap materials. J. Mod. Opt.
**2007**, 54, 1739–1757. [Google Scholar] [CrossRef] - Singh, M.R. Dipole-Dipole Interaction in Photonic-Band-Gap Materials Doped with Nanoparticles. Phys. Rev. A
**2007**, 75, 043809. [Google Scholar] [CrossRef] - Schmidt, H.; Imamoglu, A. Giant Kerr nonlinearities obtained by electromagnetically induced transparency. Opt. Lett.
**1996**, 21, 1936–1938. [Google Scholar] [CrossRef] [PubMed] - Wang, H.; Goorskey, D.; Xiao, M. Enhanced Kerr Nonlinearity via Atomic Coherence in a Three-Level Atomic System. Phys. Rev. Lett.
**2001**, 87, 073601. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yan, X.-A.; Wang, L.-Q.; Yin, B.-Y.; Jiang, W.-J.; Zheng, H.-B.; Song, J.-P.; Zhang, Y.-P. Effect of spontaneously generated coherence on Kerr nonlinearity in a four-level atomic system. Phys. Lett. A
**2008**, 372, 6456–6460. [Google Scholar] [CrossRef] - Khoa, D.X.; Van Doai, L.; Son, D.H.; Bang, N.H. Enhancement of self-Kerr nonlinearity via electromagnetically induced transparency in a five-level cascade system: An analytical approach. J. Opt. Soc. Am. B
**2014**, 31, 1330–1334. [Google Scholar] [CrossRef] - Ren, J.; Chen, H.; Gu, Y.; Zhao, D.; Zhou, H.; Zhang, J.; Gong, Q. Plasmon-enhanced Kerr nonlinearity via subwavelength-confined anisotropic Purcell factors. Nanotechnology
**2016**, 27, 425205. [Google Scholar] [CrossRef] [PubMed] - Doai, L.V.; Khoa, D.X.; Bang, N.H. EIT enhanced self-Kerr nonlinearity in the three-level lambda system under Doppler broadening. Phys. Scr.
**2015**, 90, 45502. [Google Scholar] [CrossRef] - Sheng, J.; Yang, X.; Wu, H.; Xiao, M. Modified self-Kerr-nonlinearity in a four-level N-type atomic system. Phys. Rev. A
**2011**, 84, 053820. [Google Scholar] [CrossRef] - Singh, M.R. Two-photon absorption in photonic nanowires made from photonic crystals. J. Opt. Soc. Am. B
**2009**, 26, 1801–1807. [Google Scholar] [CrossRef] - Berland, K.; So, P.; Gratton, E. Two-photon fluorescence correlation spectroscopy: Method and application to the intracellular environment. Biophys. J.
**1995**, 68, 694–701. [Google Scholar] [CrossRef] [Green Version] - Jung, J.-M.; Yoo, H.-W.; Stellacci, F.; Jung, H.-T. Two-Photon Excited Fluorescence Enhancement for Ultrasensitive DNA Detection on Large-Area Gold Nanopatterns. Adv. Mater.
**2010**, 22, 2542–2546. [Google Scholar] [CrossRef] - Li, X.; Kao, F.-J.; Chuang, C.-C.; He, S. Enhancing fluorescence of quantum dots by silica-coated gold nanorods under one- and two-photon excitation. Opt. Express
**2010**, 18, 11335–11346. [Google Scholar] [CrossRef] [PubMed] - Gao, D.; Agayan, R.R.; Xu, H.; Philbert, M.A.; Kopelman, R. Nanoparticles for Two-Photon Photodynamic Therapy in Living Cells. Nano Lett.
**2006**, 6, 2383–2386. [Google Scholar] [CrossRef] [Green Version] - Yuan, H.; Khoury, C.G.; Hwang, H.; Wilson, C.M.; A Grant, G.; Vo-Dinh, T. Gold nanostars: Surfactant-free synthesis, 3D modelling, and two-photon photoluminescence imaging. Nanotechnology
**2012**, 23, 075102. [Google Scholar] [CrossRef] [Green Version] - Albota, M.; Beljonne, D.; Brédas, J.L.; Ehrlich, J.E.; Fu, J.Y.; Heikal, A.A.; Hess, S.E.; Kogej, T.; Levin, M.D.; Marder, S.R.; et al. Design of Organic Molecules with Large Two-Photon Absorption Cross Sections. Science
**1998**, 281, 1653–1656. [Google Scholar] [CrossRef] [Green Version] - Singh, M.R.; Persaud, P.D. Dipole−dipole Interaction in two-photon spectroscopy of metallic nanohybrids. J. Phys. Chem. C.
**2020**, 124, 6311–6320. [Google Scholar] [CrossRef] - Singh, M.R.; Persaud, P.D.; Yastrebov, S. A study of two-photon florescence in metallic nanoshells. Nanotechnology
**2020**, 31, 265203. [Google Scholar] [CrossRef] [PubMed] - Fejer, M.M. Nonlinear Optical Frequency Conversion. Phys. Today
**1994**, 47, 25–32. [Google Scholar] [CrossRef] - Cerullo, G.; De Silvestri, S. Ultrafast optical parametric amplifiers. Rev. Sci. Instrum.
**2003**, 74, 1–18. [Google Scholar] [CrossRef] - Sugioka, K. Progress in ultrafast laser processing and future prospects. Nanophotonics
**2017**, 6, 393–413. [Google Scholar] [CrossRef] [Green Version] - Potma, E.O.; De Boeij, W.P.; Wiersma, D.A. Nonlinear coherent four-wave mixing in optical microscopy. J. Opt. Soc. Am. B
**2000**, 17, 1678–1684. [Google Scholar] [CrossRef] [Green Version] - Yannopapas, V.; Paspalakis, E. Optical properties of hybrid spherical nanoclusters containing quantum emitters and metallic nanoparticles. Phys. Rev. B
**2018**, 97, 205433. [Google Scholar] [CrossRef] - Tohari, M.M.; Lyras, A.; AlSalhi, M.S. Giant Self-Kerr Nonlinearity in the Metal Nanoparticles-Graphene Nanodisks-Quantum Dots Hybrid Systems Under Low-Intensity Light Irradiance. Nanomaterials
**2018**, 8, 521. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Terzis, A.; Kosionis, S.; Boviatsis, J.; Paspalakis, E. Nonlinear optical susceptibilities of semiconductor quantum dot—Metal nanoparticle hybrids. J. Mod. Opt.
**2015**, 63, 451–461. [Google Scholar] [CrossRef] - Liu, Q.; He, X.; Zhao, X.; Ren, F.; Xiao, X.; Jiang, C.; Zhou, X.; Lu, L.; Zhou, H.; Qian, S.; et al. Enhancement of third-order nonlinearity in Ag-nanoparticles-contained chalcohalide glasses. J. Nanoparticle Res.
**2011**, 13, 3693–3697. [Google Scholar] [CrossRef] - Kelly, K.L.; Coronado, E.; Zhao, L.L.; Schatz, G.C. The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric Environment. J. Phys. Chem. B
**2003**, 107, 668–677. [Google Scholar] [CrossRef] - Singh, M.R. Enhancement of the second-harmonic generation in a quantum dot–metallic nanoparticle hybrid system. Nanotechnolgy
**2013**, 24, 125701. [Google Scholar] [CrossRef] [PubMed] - Cox, J.D.; Singh, M.R.; Von Bilderling, C.; Bragas, A.V. A Nonlinear Switching Mechanism in Quantum Dot and Metallic Nanoparticle Hybrid Systems. Adv. Opt. Mater.
**2013**, 1, 460–467. [Google Scholar] [CrossRef] - Singh, M.R. Theory of all-optical switching based on the Kerr nonlinearity in metallic nanohybrids. Phys. Rev. A
**2020**, 102, 013708. [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] - Mazenko, G.F. Quantum Statisitcal Mechanics; John Wiley and Sons Inc.: New York, NY, USA, 2000; Section 5.8. [Google Scholar]
- Kittel, C. Introduction to Solid State Physics, 6th ed.; John Wiley and Sons Inc.: New York, NY, USA, 1996; Chapter 13. [Google Scholar]
- Ali Omar, M. Elementary Solid State Physics; Addison-Wesley: New York, NY, USA, 1993; Section 8.11. [Google Scholar]
- Gerstein, J.L.; Smith, F.W. The Physics and Chemistry of Materials; John Wiley and Sons Inc.: New York, NY, USA, 2001; Chapter 15. [Google Scholar]
- Eyring, H. Statistical Mechanics and Dynamics; Dover: New York, NY, USA, 1952. [Google Scholar]
- Lorentz, H. The Theory of Electrons; Dover: New York, NY, USA, 1952. [Google Scholar]
- Novotny, L.; Hecht, B. Principle of Nano-Optics; Cambridge University Press: Cambridge, UK, 2006. [Google Scholar]
- Sarid, D.; Challener, W.A. Modern Introduction to Surface Plasmons: Theory, Mathematica Modeling, and Applications; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Hanamura, E.; Kawabe, Y.; Yamanaka, A. Quantum Nonlinear Optics; Springer: Tokyo, Japan, 2007. [Google Scholar]
- Boyd, R.W. Nonlinear Optics, 3rd ed.; Academic Press: New York, NY, USA, 2008. [Google Scholar]
- Singh, M.R. Electronic, Photonic, Polaritonic and Plasmonic Materials; Wiley Custom: Toronto, ON, Canada, 2014. [Google Scholar]
- Scully, M.O.; Zubairy, M.S. Quantum Optics; Cambridge University Press: London, UK, 1997. [Google Scholar]
- Leoński, W.; Tanaś, R. Possibility of producing the one-photon state in a kicked cavity with a nonlinear Kerr medium. Phys. Rev. A
**1994**, 49, R20–R23. [Google Scholar] [CrossRef] - Imamoḡlu, A.; Schmidt, H.; Woods, G.; Deutsch, M. Strongly Interacting Photons in a Nonlinear Cavity. Phys. Rev. Lett.
**1997**, 79, 1467–1470. [Google Scholar] [CrossRef] - Leoński, W.; Miranowicz, A. Kerr nonlinear coupler and entanglement. J. Opt. B Quantum Semiclass. Opt.
**2004**, 6, S37. [Google Scholar] [CrossRef] [Green Version] - Kalaga, J.K.; Leoński, W.; Szczȩśniak, R. Quantum steering and entanglement in three-mode triangle Bose–Hubbard system. Quantum Inf. Process.
**2017**, 16, 265. [Google Scholar] [CrossRef] - Peřina, J., Jr.; Lukš, A.; Kalaga, J.K.; Leoński, W.; Miranowicz, A. Non-classical light at exceptional points of a quantum PT-symmetric two-mode system. Phys. Rev. A
**2019**, 100, 53820. [Google Scholar] [CrossRef] [Green Version] - Antón, M.; Carreño, F.; Melle, O.S.; Calderón, E.; Cabrera-Granado, M. Singh Transparency in semiconductor-metal nanoparticle hybrid system. Phys. Rev. B
**2013**, 87, 195303. [Google Scholar] [CrossRef] [Green Version] - Singh, M.R. Photon transparency in metallic photonic crystals doped with an ensemble of nanoparticles. Phys. Rev. A
**2009**, 79, 013826. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of a nanohybrid which consists of a noninteracting metallic nanoparticles (MNPs) and interacting quantum emitters (QEs).

**Figure 2.**A schematic diagram of a four-level quantum dots (QDs) is plotted. Energy levels are denoted as |1>, |2>, |3> and |4>. The probe field and SPP field are applied in the transition $\left|1\rangle \leftrightarrow \right|2\rangle $. The DDI-MNP and DDI-QD fields are acting in the transitions $|2\rangle \leftrightarrow |3\rangle $ and $\left|3\rangle \leftrightarrow \right|4\rangle $, respectively.

**Figure 3.**(

**a**) (left): The band structure of the SPPs is plotted as a function of the normalized energy and wavevectors. (

**b**) (right): The DOS of the SPPs is plotted as a function of the normalized energy.

**Figure 4.**(

**a**) (left): The Kerr absorption intensity (${I}_{kerr}/{I}_{0})$ is plotted as a function of the normalized probe detuning ${\delta}_{p}={\delta}_{21}$. The solid, dash, and dash-dotted lines are plotted for ${\Pi}_{SPP}^{}=0.1$, ${\Pi}_{SPP}^{}=0.7$, and ${\Pi}_{SPP}^{}=0.7$, respectively. (

**b**) (right): The Kerr absorption intensity (${I}_{kerr}/{I}_{0})$ is plotted as a function of the SPP coupling Π

_{SPP}(normalized unit) and the probe detuning (normalized unit).

**Figure 5.**(

**a**) (left): The Kerr intensity (${I}_{kerr}/{I}_{0})$ plotted as a function of the probe detuning $\left({\delta}_{p}={\delta}_{21}\right)$. The solid, dash and dash-dotted lines are plotted for the detuning parameter ${\Phi}_{DDI}^{}=0.1$, ${\Phi}_{DDI}^{}=0.5$, and ${\Phi}_{DDI}^{}=2.0,$ respectively. (

**b**) (right): The Kerr intensity (${I}_{kerr}/{I}_{0})$ plotted as a function of the probe detuning $\left({\delta}_{p}={\delta}_{21}\right)$ and the DDI coupling ${\Phi}_{DDI}^{}$.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Singh, M.R.
A Review of Many-Body Interactions in Linear and Nonlinear Plasmonic Nanohybrids. *Symmetry* **2021**, *13*, 445.
https://doi.org/10.3390/sym13030445

**AMA Style**

Singh MR.
A Review of Many-Body Interactions in Linear and Nonlinear Plasmonic Nanohybrids. *Symmetry*. 2021; 13(3):445.
https://doi.org/10.3390/sym13030445

**Chicago/Turabian Style**

Singh, Mahi R.
2021. "A Review of Many-Body Interactions in Linear and Nonlinear Plasmonic Nanohybrids" *Symmetry* 13, no. 3: 445.
https://doi.org/10.3390/sym13030445