PHYSICAL REVIEW A 94, 043819 (2016) Quantum analysis of plasmonic coupling between quantum dots and nanoparticles SalmanOgli Ahmad Faculty of Electrical and Computer Engineering, Tabriz University, 51666, Tabriz, Iran and Faculty of Chemical Engineering, Hacettepe University, 06800, Ankara, Turkey (Received 18 June 2016; published 13 October 2016) In this study, interaction between core-shells nanoparticles and quantum dots is discussed via the full-quantum- theory method. The electromagnetic field of the nanoparticles is derived by the quasistatic approximation method and the results for different regions of the nanoparticles are quantized from the time-harmonic to the wave equation. Utilizing the optical field quantization, the nanoparticles’ and quantum dots’ deriving amplitudes contributing to the excitation waves are determined. In the current model, two counterpropagating waves with two different frequencies are applied. We derived the Maxwell-Bloch equations from the Heisenberg-Langevin equations; thus the nanoparticles–quantum dots interaction is perused. Moreover, by full quantum analyzing of the analytical expression, the quantum-plasmonic coupling relation and the Purcell factor are achieved. We show that the spontaneous emission of quantum dots can be dramatically manipulated by engineering the plasmon-plasmon interaction in the core-shells nanoparticles. This issue is a very attractive point for designing a wide variety of quantum-plasmonic sensors. Through the investigation of the nanoparticle plasmonic interaction effects on absorbed power, the results show that the nanoparticles’ and quantum dots’ absorption saturation state can be switched to each other just by manipulation of their deriving amplitudes. In fact, we manage the interference between the two waves’ deriving amplitudes just by the plasmonic interactions effect. DOI: 10.1103/PhysRevA.94.043819 I. INTRODUCTION Over the last decade, designs of highly sensitive sensors have led researchers to focus on the optical field and quantum interaction. This interaction involves one atom with a few energy levels and one or more optical resonant modes of the quantized fields. One such system is the well-known Jaynes-Cummings model which is used to analyze an im- portant application such as the transparency induced by an electromagnetic field [1–4]. Some reports are referenced here of such a system dealing with semiclassical analysis, in which the atoms are treated as the quantized form, and with the construction of a few energy levels. Furthermore, the optical field is considered using classical mean-field theory [1,3]. However, there are a number of quantum effects that have no analogs in semiclassical theory; some of them include the collapse and revival of the Rabi oscillation of the atomic inversion, the atomic dipole moment, and the creation of novel state radiation [5]. For these reasons, full quantum analyzing is considered to investigate the quantum-plasmonic interaction in which all subsystem interactions can be fully studied. Indeed, all aspects of the system’s static and dynamic state can be investigated by extra details. It is noteworthy to mention that the type of system analyzing approach becomes important when we want to study a novel and effective phenomenon such as the plasmonic and its interactions. In this system, a plasmonic nanoparticle (NP) is placed close to a quantum dot (QD) which can be used in different applications such as Raman signal enhancement and fluorescence enhancing [6–8]. Plasmonic NPs show a strong localized field close to themselves that significantly enhances the interaction with any QDs around it. Actually, the plasmonic provides a unique ability to manipulate the light through the confinement of the electromagnetic field to the region below the diffraction limit [9]. Also, this phenomenon can be easily manipulated by the NPs’ morphology engineering such as core-shells NPs. In such case, the plasmon-plasmon interaction (the interaction of the core plasmonic and the outer shell plasmonic resonance) provides a new plasmonic resonance peak with a strong amplitude [10–12]. By considering the above statements, in the present work the quantum-plasmonic interaction is analyzed with the full quantum theory in which both the QDs and the NP plasmonic field are quantized [13–18]. In this study, we designed Au/SiO 2 /Au NPs and by engineering the NP mor- phology, such as the manipulation of the silica layer thickness and so on, the plasmonic resonance peak is shifted around 810 nm. The core-shells NPs’ radii are 10, 14, and 18 nm and, as we supposed, the QDs (erbium with radius 3–4 nm) have four energy levels. Due to a special application of this work, two counterpropagating waves at 808 nm (incidence wave) and 1616 nm (pumping wave) are assumed. Indeed, we analyze the effect of two counterpropagating waves on the quantum- plasmonic interactions. After this short Introduction, the article is organized as follows: In Sec. II we initially calculate the field distribution for the designed core-shells NPs in which we used the quasistatic approximation [19–21]. In Sec. III we quantize the NP plasmonic field and derive the system Hamiltonian. In Sec. IV the Heisenberg-Langevin equations are derived, and by these, the important parameters such as Purcell factor and quantum-plasmonic coupling strength are analytically calculated. In Sec. V the Maxwell-Bloch equations are derived from the Heisenberg-Langevin equations and other important parameters such as the power absorbed by the NPs and the QDs are studied. II. QUASISTATIC APPROXIMATION FOR CALCULATION OF CORE-SHELLS NPs’ ELECTRIC FIELD DISTRIBUTION In this section, the core-shells NPs’ electrostatic potential and electric field distribution are examined. We assume 2R < 50 nm, where R is the NP radius; by considering this condition, information about changing in the electronic and optical properties is due to the respective alteration of ε(ω, r ), 2469-9926/2016/94(4)/043819(12) 043819-1 ©2016 American Physical Society