Array of nanoparticles coupling with quantum-dot: Lattice plasmon quantum features Ahmad Salmanogli a, b, * , H. Selcuk Gecim b a Faculty of Electrical and Computer Engineering, Tabriz University, 51666, Tabriz, Iran b Faculty of Engineering, Çankaya University Electrical and Electronics Engineering Department, Ankara, Turkey ARTICLE INFO Keywords: Quantum theory Plasmonic Lattice plasmon Second-order correlation function Purcell factor Quantum-dot ABSTRACT In this study, we analyze the interaction of lattice plasmon with quantum-dot in order to mainly examine the quantum features of the lattice plasmon containing the photonic/plasmonic properties. Despite optical properties of the localized plasmon, the lattice plasmon severely depends on the array geometry, which may influence its quantum features such as uncertainty and the second-order correlation function. To investigate this interaction, we consider a closed system containing an array of the plasmonic nanoparticles and quantum-dot. We analyze this system with full quantum theory by which the array electric far field is quantized and the strength coupling of the quantum-dot array is analytically calculated. Moreover, the system's dynamics are evaluated and studied via the Heisenberg-Langevin equations to attain the system optical modes. We also analytically examine the Purcell factor, which shows the effect of the lattice plasmon on the quantum-dot spontaneous emission. Finally, the lattice plasmon uncertainty and its time evolution of the second-order correlation function at different spatial points are examined. These parameters are dramatically affected by the retarded field effect of the array nanoparticles. We found a severe quantum fluctuation at points where the lattice plasmon occurs, suggesting that the lattice plasmon photons are correlated. 1. Introduction In the recent years, metal nanoparticles (NPs) with the plasmonic properties are considered as indispensable components in a wide range of the different applications [1–4]. These applications are based on the plasmon response of the nanostructures and enhancement of the local electric fields at their surfaces [4–6]. Plasmon resonance studies usually start with the investigation of the NPs interaction with an incidence wave. This feature will be very powerful by merging their properties such as plasmon-plasmon interaction, which opens a new field with high sensitive applications [5,6]. For plasmon-plasmon interactions, it is better to consider the NPs dimer optical properties. In dimer, when NPs inter-distance is decreased, due to the NPs near-field interaction effect, the high intensity localized plasmonic is generated at the gap between the two NPs. The arisen amplitude, actually, depends on the NPs inter-distance and dimer morphological properties [5,6]. Therefore, by increasing the NPs inter-distance, the introduced plasmon resonance at the gap region is decreased dramatically. More importantly, the extinc- tion and scattering resonance peak is hardly alterable by the NPs inter-distance manipulation. It has to be noted that the manipulation of the NPs morphological and NPs inter-distance cause shifting the plas- monic resonance frequency. The control this shift is a very difficult task. In this regard, for sensitive applications such the single molecule detec- tion, the nanostructure plasmon frequency should be precisely controlled. In recent years, several useful works have been conducted on the nanostructures containing 1-D or 2-D chains of the plasmonic NPs [7–9]. This type of plasmon is a modified version of plasmon in which the NPs near-field plasmon resonance interacts with the photonics modes. In other words, the lattice plasmon is generated due to the interaction of the NPs plasmonic field with the photonic modes which creates a unique mode like a laser. The chain of plasmonic NPs enable the far-field pho- tonic modes easily interact with the NPs plasmonic mode. It is notable that with engineering the NPs size, array structure, NPs inter-distance, and polarization direction, it is possible to manipulate and control the nanostructure plasmon resonance frequency. The optical properties of this nanostructure provide an indispensable key in the more attractive biomedical applications [8]. The significant properties of this type of plasmon were the main motivations of the present study for studying the quantum feature of this novel phenomenon. Several related works have been done for investigation of the surface plasmon wave-particle duality * Corresponding author. Faculty of Engineering, Çankaya University Electrical and Electronics Engineering Department, Ankara, Turkey E-mail address: tirdad.zey@gmail.com (A. Salmanogli). Contents lists available at ScienceDirect Physica E: Low-dimensional Systems and Nanostructures journal homepage: www.elsevier.com/locate/physe https://doi.org/10.1016/j.physe.2018.03.006 Received 27 January 2018; Accepted 6 March 2018 Available online 7 March 2018 1386-9477/© 2018 Elsevier B.V. All rights reserved. Physica E: Low-dimensional Systems and Nanostructures 100 (2018) 54–62