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Journal of Optics

Optical properties of core/shell quantum dot in model of two-level and five-level atom

Ljiljana Stevanović, M Perić, V Pavlović

Published Jul 1, 2025• 2 citations
Publisher

Abstract

Abstract The optical properties of the CdSe/ZnS spherical core/shell quantum dot (CSQD), related to the laser-driven transitions 1 s → 1 p − 1 and 1 s → 2 p − 1 , are the subject of this study. We investigate in detail how transition energies, oscillator strengths, imaginary and real parts of the total, linear and third-order susceptibilities change with the variation of the model parameters, such as CSQD radius, core radius, relaxation rates and laser intensity. The energy levels and wave functions, needed for the calculations of the model parameters, were obtained by solving the radial Schrödinger equation in the effective mass approximation with the corresponding boundary conditions. The interaction of CSQD with laser light was studied in a density matrix approach using a two-level model and, for the first time, a five-level model. Optical Bloch equations were derived under dipole and rotating wave approximations and solved in the steady-state regime to obtain an analytic expression for susceptibility. Special attention was paid to the cases in which nonlinear susceptibility prevailed over linear susceptibility, leading to the appearance of the window in the absorption peak. It was found that susceptibility is more sensitive to the change in the core radius than to the change in the CSQD radius and that a greater laser intensity is needed to change the nature of the absorption and dispersion curves for 1 s → 2 p − 1 then for 1 s → 1 p − 1 transition.

DOI
10.1088/2040-8986/aded9e

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Abstract The optical properties of the CdSe/ZnS spherical core/shell quantum dot (CSQD), related to the laser-driven transitions 1 s → 1 p − 1 and 1 s → 2 p − 1 , are the subject of this study. We investigate in detail how transition energies, oscillator strengths, imaginary and real parts of the total, linear and third-order susceptibilities change with the variation of the model parameters, such as CSQD radius, core radius, relaxation rates and laser intensity. The energy levels and wave functions, needed for the calculations of the model parameters, were obtained by solving the radial Schrödinger equation in the effective mass approximation with the corresponding boundary conditions. The interaction of CSQD with laser light was studied in a density matrix approach using a two-level model and, for the first time, a five-level model. Optical Bloch equations were derived under dipole and rotating wave approximations and solved in the steady-state regime to obtain an analytic expression for susceptibility. Special attention was paid to the cases in which nonlinear susceptibility prevailed over linear susceptibility, leading to the appearance of the window in the absorption peak. It was found that susceptibility is more sensitive to the change in the core radius than to the change in the CSQD radius and that a greater laser intensity is needed to change the nature of the absorption and dispersion curves for 1 s → 2 p − 1 then for 1 s → 1 p − 1 transition.
The energy eigenvalues and eigenfunctions of a core–shell CdSe/ZnS quantum dot structure have been obtained under an effective-mass approximation. The electric transition dipole moment induced by the 1s (inside the well)–1s (outside the well) transition has been calculated. The optical nutation signal induced by the transition has been calculated numerically based on the optical Bloch equations. In particular, the influence of the size variation of the core's radius and the shell's thickness has been studied. It has been shown from the results that the optical nutation signal is sensitive to the size and structure, and that there is an optimal structure and size for the optical nutation phenomenon. The quantum size dependent Rabi frequency and intensity of the optical nutation are also discussed.
Optical nutation in GaAs/Al0.3Gao.7As core-shell quantum dot (CSQD) shined by femtosecond laser has been analyzed theoretically. The nutation signal intensity, arising due to photo-induced electronic transitions between ground state and first excitonic state, has been calculated using time dependent perturbation theory and rotational wave approximation. The dependency of the properties of optical nutation signal viz. its maximum amplitude and the oscillation frequency of the nutation signal has been studied at various CSQD's core radius and shell thickness.
I recent years, significant attention has been given to nonlinear optical phenomena, such as nonlinear optical absorption and dispersion, derived from a driven quantum transition of semiconductor nanostructures, such as semiconductor quantum wells, quantum disks and mainly quantum dots. An important issue that is widely studied is the effect of probe electromagnetic field intensity on the total (nonlinear) absorption coefficient and on the total (nonlinear) index of refraction. Some of the phenomena that are commonly found for large electromagnetic field intensities are the creation of a strong dip in the absorption spectrum near or at resonance (the so-called bleaching effect), negative absorption near or at resonance that leads to optical gain and the change of the slope of the total index of refraction near or at resonance from negative to positive that changes the behavior of the system from fast to slow light. Unfortunately, these phenomena do not exist and are artifacts of a common abuse of the theoretical methodology for the study of the total absorption coefficient and the total index of refraction. The main error is that the total absorption coefficient and the total index of refraction was initially derived from density matrix equations using perturbation theory and includes first-order (linear) as well as third-order (nonlinear) terms. This method gives correct results for small electromagnetic field intensities and it leads to saturation of absorption and dispersion in this regime. However, for larger intensities it gives the misleading results mentioned above due to either failure of the order of perturbation theory considered in the derivation of the total absorption coefficient and the total index of refraction or even failure of perturbation theory at all in the treatment of the problem. In this work, we revisit the problem of the total absorption coefficient and the total index of refraction in a symmetric semiconductor quantum dot structure (we choose the example of a core-shell quantum dot) under a strong probe field excitation. We use the two-level model, solve the relevant density matrix equations under the rotating wave approximation and under steady state conditions, and obtain the correct form of the nonlinear optical susceptibility that is then used for the derivation of the formulae of the total absorption coefficient and the total index of refraction under the interaction with a strong probe field. Actually, the commonly used perturbation theory formulae can be obtained by our formulae using a series expansion and keeping terms up to second order in electric field amplitude. We also derive the electronic structure of the quantum dot system by solving numerically the time-independent Schrodinger equation, under the effective mass approximation, using the potential morphing method and obtain the energy difference and the electric dipole matrix element for the relevant transitions of the quantum dot system. Then, for the specific quantum dot system we compare the results of the total absorption coefficient and the total index of refraction for the two methodologies (the perturbation result and our result) for different electromagnetic field intensities.
The energies of 1s states inside and outside the quantum well of a CdSe/ZnS quantum-dot quantum well (QDQW) and the transition dipole moment between these two states have been calculated by effective mass approximation. By using the calculation results, the longitudinal and transversal relaxation times of CdSe/ZnS have been approximated. We obtain a three-pulse photon echo (TPE) signal on the basis of the two states using the optical Bloch equations. We find that there should be four echoes in TPE from our theoretical calculation. In particular, the effects of the variation in core radius and shell thickness have been studied. It has been shown from the numerical calculation that the TPE signal is strongly dependent on the size and structure of the QDQW, and is dominated by relaxation time, which is also dependent on the size and structure.
In recent years,the subject of optical nonlinearity in semiconductor quantum confined structures such as quantum wires and quantum dots has attracted a considerable attention because of their relevance for studying practical applications and as a probe for the electronic structure of media.It has been shown that the low dimensional quantum systems present very large nonlinear optical susceptibilities .Theoretically,most of the studies on quantum dots are focused on spherical quantum dots for simplicity.The shape of quantum dot depends on growth conditions.In fact,quantum dots are better described as thin disk or cylinder.In the present work third order nonlinear optical susceptibilities in disk shaped quantum dots due to resonant intersubband transition are analyzed by a compact density matrix approach.It has rarely been discussed in literatures.Firstly,we consider an electron moving in a quantum disk whose radius R is very large compared to its thickness d .For simplicity,we assume that the isotropic effective mass approximation is valid within the disk and that the inner potential is null while the outer confining potential is approximately infinite.Using the adiabatic approximation we solve the three dimensional Schrdinger equation and obtain the analytical expressions of wave functions and energy levels in cylindrical coordinates.Secondly,we consider the system is excited by an electromagnetic field with frequency ω. The electromagnetic field is incident with a polarization vector normal to the disk.We treat H′=ezE(t) as a perturbation term.By using the compact density matrix method,we derive a general formula for third order nonlinear optical susceptibilites employing a four level model in quantum disk.Finally,numerical results are illustrated for a typical GaAs quantum disk.We calculate the third order nonlinear optical susceptibilities as a function of the disk thickness.It is shown that the third order nonlinear optical susceptibilities in quantum disk are 10～100 times larger than that in quantum well for various disk thickness.Furthermore,if we choose an optimized thickness of quantum disk,we could obtain a quite large refractive index with a comparatively small absorption coefficient.Therefore,we suggest that the quantum dots would be an excellent system in the field of optical communications and optical switching.
The optical absorption of an isolated Λ-type three-level semiconductor quantum dot under continuous-wave excitation was investigated theoretically using a semiclassical density-matrix approach. The equations of motion were derived from the master equation within the rotating-wave approximation and solved in the steady-state regime to obtain the optical susceptibility and absorption spectrum. The roles of the transition dipole moment and relaxation rate were examined systematically in both weak- and strong-field regimes. In the weak-field regime, the spectrum was dominated by the ∣1⟩↔∣3⟩ transition, while a weaker secondary feature associated with the ∣2⟩↔∣3⟩ transition emerged through relaxation-assisted population redistribution. Increasing the relaxation rate broadened and suppressed the dominant peak, whereas increasing the transition dipole moment mainly broadened the weaker secondary feature. In the strong-field regime, the two absorption channels merged into a broadened composite profile due to the combined effects of stronger coherent driving, power broadening, and relaxation. These results clarify how transition strength and dissipation govern the absorption line shape of an elementary multilevel semiconductor quantum-dot system.