Four-Wave Mixing in Asymmetric Double Quantum Dot Molecule–Metal Nanoparticle Assemblies

: In this study, the four-wave mixing (FWM) spectrum of a strongly pumped hybrid structure is theoretically examined. The hybrid structure consists of an asymmetric double semiconductor quantum dot (SQD) molecule and a spherical metal nanoparticle (MNP), which are coupled to-gether via long-range Coulomb interaction. Having as a starting point the Hamiltonian of the system, in the dipole and the rotating-wave approximations, we derive a set of nonlinear density matrix equations, which are numerically solved, in the steady-state limit, and then the FWM coefficient is calculated within a range of values of the pump–probe field detuning. The spectral response of the FWM coefficient is investigated, for different values of the pump-field detuning, the electron-tun-neling rate, and the energy gap between the upper states in the energy-level scheme of the double SQD molecule, while the interparticle distance between the two components of the structure is mod-ified.


Introduction
The coupling between the excitonic and the plasmonic nanoparticles produces collective optical properties, which are quite different from the properties of the individual components, such as emission, dispersion, and absorption. During the last years, these interesting optical effects have attracted scientific interest, both on an experimental and a theoretical level, in hybrid nanostructures that are composed of SQDs and MNPs [1][2][3][4][5]. The study of the Λ-type system that describes the asymmetric double-SQD system has also attracted the scientific interest of several scientists who investigated the pump-probe response and the FWM [6,7], and the Autler-Townes splitting and the tunneling-induced transparency [8,9]. In this work, we are interested in the study of a complex structure in which an asymmetric double-SQD molecule is coupled to an MNP. We start with the Hamiltonian of the system, in the dipole and the rotating-wave approximations, and derive a set of nonlinear density matrix equations, which are numerically solved, in the steady-state limit. Then, the FWM coefficient is calculated within a range of values of the pump-probe field detuning. The spectral response of the FWM coefficient is investigated, for different values of the pump-field detuning, the electron-tunneling rate, and the energy gap between the upper states in the energy-level scheme of the double SQD molecule for different values of the distance between the SQD and MNP.

Methods
We consider a hybrid molecule consisted of a spherical MNP of radius a and a couple of spherical SQDs of radii , in the presence of a polarized external field that is composed of a strong pump field of amplitude a E and angular frequency a ω and a weak probe field of amplitude b a E E << and angular frequency b ω . In the energy level configuration presented in Figure 1, the double SQD molecule is modeled as a three-level Λ-type system, while the basic excitations in the MNP are the surface plasmons, which present a continuous spectrum. The incident field excites an electron occupying the ground state 0 , thus creating an electron-hole pair that is bound to the same SQD (direct exciton state 1 ). Due to the transition of this electron to the second SQD, via tunneling, a hole is induced in the first SQD (indirect exciton state 2 ).  Having as a starting point the Hamiltonian, we can show that, under the rotatingwave approximation, the slowly varying density matrix elements obey a set of five differential equations. We proceed to the expansion of the density matrix elements,

01
ρ , being calculated numerically, in the steady state.

Parameters and Results
In this section, we examine the FWM spectra of a strongly pumped asymmetric tunneling-controlled double SQD-MNP system. The incident field is assumed to have a polarization direction that is parallel to the interparticle axis, and hence, we take

Discussion
In the case of exact resonance Figure 2a-c, the FWM spectra are symmetric, with respect to the vertical axis at In all these cases, the increase of the interparticle distance leads to a transposition of the resonances, toward higher values of the absolute pump-probe field detuning.
In the off-resonance case Figure 2d-f, the spectral symmetry is broken as long as the electron-tunneling coefficient is not negligible, as seen in Figure 2e . In this case, since the dark-state condition 12 ω = Δ   is accomplished, the six-peaked FWM spectrum is turned into a four-peaked spectrum. The rest spectral trends are similar to the ones presented in the previous case.

Summary
To summarize, we studied the FWM spectrum of a strongly pumped hybrid system, which is composed of an asymmetric double SQD molecule that is coupled to a spherical MNP, via long-range Coulomb interaction. In the case of a nonresonant pump field, with a pump-field detuning equal to the energy gap between the upper states of the Λ-type system, the spectra are approximately symmetric and exhibit four resonances, for average values of the electron-tunneling rate, since the dark-state condition is accomplished. However, in the case of exact resonance, the spectra are approximately asymmetric and exhibit a six-resonance profile. In any case, the increase of the center-to-center distance leads to a transposition of the resonances away from the spectral center.