Splitting of the Effective Rabi Frequencies for the Coherent Plasmonic Fields in the Semiconductor Quantum Dot–Metal Nanospheres Hybrids

Splitting of the effective Rabi frequencies for plasmonic fields when the interaction occurs between semiconductor quantum dot and three metallic nanospheres in the presence of three electromagnetic fields is examined. We study theoretically the role of the dipole–dipole interactions in creating many Multipoles. Then, we split the effective Rabi frequencies into three parts according to the quantitative Multipoles of the plasmonic fields. The density matrix equations are derived for the description of the optical properties of the SQD-MNPs nanosystem. We investigate the influence of each part of the effective Rabi frequencies and compare them with each other individually. The three parts of the effective Rabi frequency for the probe field are affected by changing the angles, the strong probe field and other parameters for the hybrid system.


Introduction
There are a lot of efforts have been made to discuss the semiconductor quantum dot (SQD), metallic nanoparticles (MNP), and the effects of their interaction with optical fields, where it is useful to many contemporary scientific disciplines. It can form a hybrid quantum system which is to exhibit a variety of novel optical properties that are required in the field of optoelectronics [1,2]. It is shown that the quantum coherence effects in a QD-MNP system can form a barrier (quantum cage) that spatially confines the spatial extent of the coherently normalized plasmonic field of the MNP [3]. Artuso et al. [4] studied how MNP geometry can be used to tailor local fields, coupling, and dynamics of the hybrid structures and found that the response of the system could be tailored by engineering a metal nanoparticle shape and the placement of SQDs on the MNP to control the individual local near fields that couple the MNPs and SQDs. The population dynamics and the absorption properties of a hybrid nanosystem are investigated, to demonstrate how it depends on the collective molecular states of the QD-MNP system [5]. Anomalous Dipole-Dipole Interaction in an Ensemble of Quantum Emitters and Metallic Nanoparticle Hybrids is discovered by [6]. The coherent transfer of excitonic populations in a semiconductor quantum dot (SQD) is modulated by the surface plasmon of a metallic nanoparticle (MNP) which is investigated in [7]. The potential for controlled population inversion in a coupled system comprised of a semiconductor quantum dot and a metal nanoparticle are analyzed [8]. A theoretical study of the two-photon Rabi oscillations of a heterodimer comprising a semiconductor quantum dot and a metal nanosphere is discovered by [9]. A theory for the photoluminescence of dimer nanohybrids and trimer nanohybrids using the density matrix method is developed [10]. A theory for photoluminescence quenching and plasmonic properties in hybrid nanosystems made from three nanosystems such as quantum emitters, metallic nanoparticles, and graphene is developed [11][12][13]. Effect of dipole-dipole interactions on the one-photon and two-photon photoluminescence in an ensemble of quantum dots doped in a polymer matrix is discussed in [14]. The study of the exciton-plasmons system offers many parameters that can be used to control the optical response of the system [15]. It is found that the multipole treatment of the interaction is crucial for the understanding of strongly interacting exciton-plasmon nanosystems [16]. The preparation of quantum states with a defined spin in a hybrid system consisting of a p-doped semiconductor quantum dot (QD) coupled to a metallic nanoparticle is analyzed by [17]. The transport properties of a single plasmon interacting with a hybrid system composed of a semiconductor quantum dot (SQD) and a metal nanoparticle (MNP) coupled to a one-dimensional surface plasmonic waveguide are investigated [18]. Anomalous Photoluminescence Quenching is produced [19]. Jiang et al. [20] demonstrated theoretically a novel double-hole structure to improve the single-emitter emission rate, and the numerical investigations show that the structure possesses a large local electric field. The nonlinear refractive index of a sample including hybrid molecules composed of AuNPs and SiQDs was investigated theoretically and experimentally [21]. Dipole-Dipole Interaction in Two-Photon Spectroscopy of Metallic Nanohybrids are discussed in [22]. A theory of the quantum yield for plasmonic nanowaveguide where the cladding layer is made of an ensemble of quantum dots and the core layer consists of an ensemble of metallic nanoparticles is developed [23]. Thin bilayer films containing Au and CuS nanoparticles, with surface plasmon resonances, exhibited enhanced third harmonic generation over the individual films as a result of dipole-dipole incoherent interactions between the nanoparticles [24]. The optical properties of a hybrid structure consisting of a metal nanoparticle (MNP) and an asymmetric double semiconductor quantum dot (SQD) molecule, which are coupled together, via long-range Coulomb interaction are studied theoretically [25]. In [26] the influence of the strength of the plasmonexciton dipole interaction for probe field and control field for different parameters of the hybrid SQD-MNPs nanosystem is studied.
In this paper, the present scheme is based on a coupled semiconductor quantum dot (SQD) and metal nanospheres (MNSs) nanosystem in the presence of the three electromagnetic fields. The SQD is taken as a four-level V-type system in which distinct excitonic transitions occur. We derive the polarization of the three electromagnetic fields which induce dipole moments in the three metal nanospheres and semiconductor quantum dot, where these structures interact with each other via the Multipoles interaction. The main focus and motivation of this paper is splitting the effective Rabi frequencies for coherent plasmonic fields into three parts, studying the influence of each part individually, which has a significant effect and comparing the parts with each other. This work is organized as follows: in the "Theoretical Model and Formalism" section, we describe the SQD-MNSs nanosystem, derive the density matrix equations describing the dynamics of the system and obtain the form of the three parts of the effective Rabi frequencies for the hybrid nanosystem. In the "Numerical Results and Discussion" section, we discuss our numerical results. Finally, we present our conclusions in the "Conclusion" section.

Theoretical Model and Formalism
We consider a hybrid nanosystem that consists of a single semiconductor quantum dot (SQD) and three metallic nanospheres (denoted by MNS m ) of the same radii R. The semiconductor quantum dot (SQD) and the environment surrounding the system have a dielectric constant s and b respectively. We suppose that the SQD is characterized by a four-level V − type atomic system designated by �1⟩, �2⟩, �3⟩ , and �4⟩ , where the state �1⟩ is the ground state. The center-to-center distance between the SQD and the three metallic nanospheres have the same distance (denoted by r m ), while the center-to-center distance between the three MNS m is d nm ( n, m = 1, 2, 3 , n ≠ m ), respectively. All these centers are being in two dimensions (2D), plane (ZOX), where the center of the SQD is situated at the origin O. m ( m = 1, 2, 3 ) is the angle confined between the Z-axis and r m for the three MNS m respectively, as illustrated in Fig. 1. The excitonic transitions for the SQD | 1⟩ ⟺ | 2⟩ , | 1⟩ ⟺ | 3⟩ and | 3⟩ ⟺ | 4⟩ are characterized by the transition frequencies 21 , 31 , and 43 , where ij = i − j , i ≠ j = 1, 2, 3, 4 . The system interacts with three electromagnetic fields n (t) = n e i n t with frequency n ( n = 1, 2, 3 ), respectively. These fields create excitons in the semiconductor quantum dot and surface plasmon polaritons (SPP) in the three metallic nanospheres. The excitons and SPP produce dipole electric fields which interact with each other via dipole-dipole interaction (DDI). In this problem, we suppose the three metallic nanospheres are identical. Each MNS m is treated as a classical particle with a dielectric function denoted by m ( ) , according to the generalized Drude theory, it can be written as follows [27]: m ( ) = (1 − where the operator ij = �i⟩⟨j� , ( i, j = 1, 2, 3, 4 ) and ij is the dipole moment of the SQD associated with atomic transition ⟨i� ⟷ ⟨j� . n SQD (for n = 1, 2, 3 ) represent the fields that are falling on the SQD due to the contributions of the system components induced by the electromagnetic fields n . So n SQD can be written as: The fields n,m SQD are produced by three MNS m on SQD under the effect of applied field n and it can be calculated as follows [28]: n,m SQD = The vector dipoles n,m originate from the charge induced on the surface of the three MNS m and direct in the Z-axis where is given by , and the unit vector along the vector m is given ( n,k n,m ) is due to the dipole-dipole interaction between the metallic nanospheres MNS l ( MNS k ) and MNS m under the electromagnetic field n . It is given by: n,l n,m = lm is the unit vector along the vector directed from center of MNS l to center of MNS m as illustrated in Fig. 1. I.e., we can get: , replace l with k in the above equations. We can obtain an expression for n SQD with doing recurrence of the above steps to taking many Multipoles, up to minus of tenth order (d m ) −10 . Then the total Hamiltonian of the SQD is expressed as: where the direction of the electromagnetic fields ( E n ) is along the X-axis. We notice that the effect of the term, which has very small amount of the effective Rabi frequencies, does not appear, when we take the all terms of the effective Rabi frequencies. So, we can put the effective Rabi frequency ( Ω , so we have: and Ω Ω n e f f s r 3 Fig. 1 A schematic diagram of the SQD and three MNS m (hybrid nanosystem). The SQD has four-level V-type configuration coupling with three electromagnetic fields 1 3 Fig. 2 The spectra of the parts of Im Ω Under the electric dipole approximation and the rotatingwave approximation [29], we define the equation of motion of density matrix elements (the master equation) of the SQD coupled to the three MNS m , as follows: and Im Ω Where the density matrix elements have the identity property 3 . W h e r e 1 , 2 and 3 represent the radiative decay rates of the excitation states �2⟩ , �3⟩ and �4⟩ due to spontaneous emission (13) d dt 32  and Im Ω e f f c 23 respectively at respectively. Δ 1 = 21 − 1 , Δ 2 = 31 − 2 and Δ 3 = 43 − 3 are the frequency detuning for the three fields.

Numerical Results and Discussion
In this section, we discuss the role of the three parts of the effective Rabi frequency (Ω e f f 2 ) and its influence in the hybrid nanosystem. The three MNS m are silver (Ag) with plasma frequency and relaxation damping 9.02eV and 0.026eV, respectively [30][31][32]. The probe field E 2 is taken as the strong field to find out its effect on the hybrid nanosystem. The parameters of hybrid nanosystem are taken as R = 6 nm, 12 = 13 = 34 = 1.2 e nm,  Im Ω e f f 23 respectively at 1 = ∕3 and Fig. 2b 1 , b 2 , and b 3

are taken for
Im Ω Im Ω e f f 23 respectively. We notice, when 1 = ∕3 , the distances between the three nanospheres d 1 , d 2 , and d 3 are different. But at 1 = ∕6 , the distances between the three nanospheres d 1 , d 2 , and d 3 are equal. Then when r increases, d 1 , d 2 , and d 3 increase (showing from numerical results by using Matlab). We conclude that when the angle between r i and r j ( i ≠ j, i, j = 1, 2, 3 ) equal 2 ∕3 , the distances d 1 , d 2 , and d 3 are equal. Then the interaction between the Multipoles is strongly effective when equal the distance between the three MNS m and SQD, also the distance between each of the three MNS m . So, the splitting of the  Figure 3a 1 , a 2 and a 3 show two small peaks on the two sides and a sharp peak in Im Ω e f f 2 respectively the middle of Δ 2 when Ω 2 = 1 , while at Ω 2 = 4 , the sharp peak decrease rapidly. Figure 3b 1 , b 2 , and b 3 appear broad window having two holes at Ω 2 = 1 , when increasing Ω 2 ( = 4 ) the two holes disappear from the window and the spectrum shows two peaks on the two sides of Δ 2 . Figure 4 illustrates the spectra of the parts of Im Ω e f f 2 for different values of 1 ( ∕9 , ∕5 , ∕4) . Figure 4a 1 , a 2  Rabi frequency is taken completely, without splitting (as in Fig. 5a 4 ), it exhibits like Fig. 5a Fig. 6, we present three-dimensional plots of the three parts of the Ω e f f 2 as a function of the detuning Δ 2 and the dielectric constant s at 1 = ∕3 . Figure 6a 1 , a 2    Im Ω

Conclusion
We have investigated the exciton-plasmon interaction of the hybrid nanostructure composed of semiconductor quantum dot and three metallic nanospheres via three electromagnetic fields which led to many Multipoles. We noticed when the effective Rabi frequency is taken completely (all terms), the effect of the term which has very small amount of the effective Rabi frequencies does not appear, so the effective Rabi frequencies for coherent plasmonic fields have been split into three parts according to the quantitative of Multipoles of the plasmonic fields and we studied each part separately to show its importance. It is observed that (FPERF) proportional to ( changing the angles and the strong probe field, other parameters like the Rabi frequency Ω 3 and dielectric constants ( b , s ) have also strong effect on these spectra. Lastly, we wish the results of splitting of the effective Rabi frequencies for coherent plasmonic fields can be benefitted and used in analyzing optical experiments of hybrid nanosystems.
Author Contribution All authors contributed to the study conception, design and preparation. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Data Availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.