# Quantum Mechanical and Optical Properties of Asymmetric Double Wells and two Electron Quantum Dots

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## Date

2012-06

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Addis Ababa University

## Abstract

In this dissertation the energy of the ground and excited state of symmetric and
asymmetric parabolic double quantum wells (DQWs) are investigated by using our
formulated trial wave functions and the Hamiltonian. For symmetric DQWs, calculation
of the frequency corresponding to the ground state energy splitting describes
that tunneling takes place in the duration of short time when the frequency of the
photon is in tera hertz (THz) region. However the tunneling time for excited state
energy splitting is shorter than that of the ground state energy splitting due to the
decrement of barrier width for parabolic DQWs.
The asymmetric DQWs with resonant levels (the ground state energy level in one
well coincides with the first excited state energy in another well) is analyzed. The
splitting of these levels and the tunneling times are calculated. If the typical life
time of the excited state is much smaller than the tunneling time between wells, the
charged particle can radiate as the result of quantum transition from the excited state
to the ground state. In the opposite case, the asymmetric DQWs can be treated as
a metastable excited nano-system regardless of that the dipole transition from the
excited state to the ground state is permitted. The life time of this metastable state
can be considerably reduced by putting it in the resonant cavity. The possibility of
coherent radiation of an ensemble of asymmetric DQWs is discussed.
Moreover the analytical expressions of absorption coefficients and refractive index
changes for parabolic asymmetric DQWs have been determined using the compact
density matrix approach. The calculation shows that at some frequency range the
asymmetric DQWs become left handed media with negative values of total refractive
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index. This behavior of the asymmetric DQWs at some frequency range is of great
importance for metamaterial science.
The low lying energy levels of 3D two electron QDs with parabolic confinement and
rigid confinement (the wave functions vanish at the surface of the QD) are obtained by
the variational method and perturbation techniques. The quantum states of the QD
are divided in to the para- and ortho- states like in the theory of helium atom. Our
numerical calculations show that the energy differences between the ground and first
excited states of the para- dots are practically linear function of the coupling constant
. This allows one to propose the phenomenological formulas for the energy levels
as a function of a typical size of QD. The spin function of the para-dot is singlet
with total spin zero. One of the spin function of the ortho-dot is not factorizable
symmetric combination. The two functions relate to the entanglement states, which
are important for quantum information processing.
The lowest transition energy of one and two electron GaAs QDs are calculated
by solving the corresponding Schr¨odinger equation. For the two electron GaAs QD
we calculated the real and imaginary parts of the dielectric constant, the refractive
index, and the absorption coefficients as functions of the incident photon energy. The
maxima of these quantities show a blue shift comparing with those of the one electron
QD due to the Coulomb and exchange interaction between electrons.
Reducing the size of quantum dots results in blue shift of the maxima of the above
mentioned quantities for both one and two electron QDs. But the blue shift in the
case of two electron QD is larger than that one electron QD due to the energies of
Coulomb and exchange

## Description

## Keywords

Two Electron Quantum Dots