A study of two-dimensional quantum dot helium in a magnetic field
Golam Faruk* and Orion Ciftja,
Department of Electrical Engineering and Department of Physics
Prairie View A&M University, Prairie View, Texas 77446
Introduction
Quantum dots, namely two-dimensionally confined electrons
in a semiconductor interface, appear to be the building blocks
on designing a quantum computer and they can be used to
build switching and memory devices for high speed
applications.
In this work we study a simple quantum dot system consisting
of two interacting electrons trapped in a two-dimensional
parabolic potential. This system is called quantum dot helium
and its properties are fascinating especially in a magnetic
field.
We applied the exact numerical diagonalization technique to
study several properties of quantum dot helium subject to a
perpendicular magnetic field. We obtained the ground level
energy and angular momentum for various combinations of
Coulomb correlation strengths and magnetic field. Our results
show that there are multiple spin singlet-to-triplet transitions
as function of the magnetic field, suggesting a possible use of
this system as a quantum bit (“qubit”) of a quantum computer.
Model
Objective
 The objective of this work is to study ground state and
excited state properties of two-dimensional quantum dot
helium as a function of the magnetic field
for a variety of values of the Coulomb correlation relative
to the confinement energy.
 By adopting the exact numerical diagonalization method
we can determine to a high degree of accuracy the critical
magnetic fields at which a spin singlet to triplet transition
happens. At the same time we want to obtain almost exact
estimates for the ground and excited state energies of this
system.
Graph-1
Quantum dot helium consists of two electrons that are confined
in a two-dimensional parabolic potential and repel each other
through a Coulomb potential. When this system is subjected to a
perpendicular magnetic field, the quantum Hamiltonian can be
written as:
ˆ


p

e
.
A
(
r
)
1
1
Hˆ =
2
2m
m 2 2  p2  e.A(rˆ2 ) m 2 2 e2
 r1 w0 
 r2 w0 
2
2m
2
rˆ1  rˆ2
2
where m is the mass of the electron, –e (e>0) is
the electron’s charge, w is the angular frequency
of the two-dimensional parabolic potential, A(r) is
the vector potential generating a magnetic field, B
in the z direction, and the linear momentum
operator, p and position, r are given in a polar
system of coordinates.
0
Graph-2
Method
The exact numerical diagonalization (ED) method is a standard
technique used to solve numerically the Schrödinger equation
for a quantum system. The key idea of the method is to
diagonalize the Hamiltonian matrix in a suitable chosen basis.
The resulting matrix eigenvalues correspond to the
numerically exact energy eigenvalues of the quantum system.
The only uncertainty associated with ED results comes from
the statistical error that can be reduced as much as we want
to. In this sense the ED method is extremely useful to obtain
very accurate estimates for energies and related quantities of
a given system. Contrary to other methods, the results
obtained from the ED method are unaffected by
approximations or limitations, except the computer power.
The ED method works very effectively for small systems of up
to eight electrons, therefore is perfectly suitable to our study
that includes only two electrons. Most of the calculations do
not require excessive computer power and can be performed
in conventional computers.
Energy spectrum, ε = E/ĥwo in zero magnetic field for
two values of Coulomb correlation parameter,
l=e^2/ĥw0 as a function of angular momentum, |mz|
Ground state energy as a function of magnetic field,
g =wc/w0 for values of l from 0 (bottom) to 10 (top)
Selected References
Graph-3
Results and conclusions
l=3
l=2
l=1
l=0
The relative motion (r=|r1-r2|) ground state wave function for
quantum dot helium in zero magnetic field for several
values of Coulomb parameter, l. The parameter aμ is the
parabolic oscillator’s length.
By using large matrices we numerically diagonalized the
Hamiltonian of quantum dot helium in the basis functions of FockDarwin eigenstates.
 We obtained the ground level energy and angular momentum for
different values of Coulomb correlation and magnetic field.
 Our results show that in absence of magnetic field, or in absence
of Coulomb interaction the ground state always has zero angular
momentum therefore it corresponds to a singlet (spin S=0) state.
However when both magnetic field and Coulomb interaction are
present, we observed multiple spin singlet (S=0) to spin triplet
(S=1) transitions as function of the magnetic field.
As the magnetic field increases, the ground state changes to
nonzero angular momentum values with alternating odd and even
parity, thus suggesting a possible use of this system as a “qubit” of
a quantum computer, with S=0 and S=1 being the two possible “on”
and “off” states.
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Acknowledgements
Part of this work was supported by the office of the VicePresident for Research and Development of Prairie View
A&M University through a 2003-2004 Research
Enhancement Program grant. One of the authors (F. G.)
would like to acknowledge the support given through the
Title III Program of the U.S. Ministry of Education.