Transcript Numerical Methods

Part 2. Quantum Computer
Qubit :  0   1
G. Feve et al., Science 2007
One Electron Makes Current Flow
Nature 2007
Quantum Computing at 16 Qubits
Quantum Dot Molecule Model
Liu, Chen, and Voskoboynikov, CPC 2006
1. Introduction
2. The Current Spin DFT for the Model System
*. Total Energy Functional
*. Hamiltonian 3D2D
3. Numerical Methods and Algorithms
4. Numerical Results
5. Conclusions
Introduction : motivation
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Quantum computation
*. use quantum mechanical phenomena
( such as superposition and entanglement )
to perform operations on data
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Electronic excitations of coupled QDs
*. artificial molecule (QDM)
*. Förster-Dexter Resonant energy transfer
Introduction : model
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Three vertically aligned InAs/GaAs QDs
A self-consistent algorithm
 Schrödinger-Poisson system
 Cubic eigenvalue problems
 Jacobi-Davidson method and GMRES
 Kohn-Sham orbitals and energies of six electrons
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CSDFT
Density functional theory
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Hohenberg and Kohn, 1964
Kohn, 1998, Nobel Prize in Chemistry
Electronic structure of many-body systems
Many-body electronic wave function (3N variables)
 electronic density ( 3 variables )
The binding energy of molecules in chemistry
The band structure of solids in physics
CSDFT
Current spin density functional theory
 Vignale and Rasolt, 1987
 Electronic structure of quantum dots in
magnetic fields
 Couple to spin and orbital
Total energy functional
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Number of electrons : N
Total spin : S
Spin-up and spin-down :
Total density :
Constraint :
Total energy functional :
kinetic energy
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the electron momentum operator :
is the vector potential induced
by an external magnetic field
KS orbitals and eigenvalues :
Total energy functional :
effective mass
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Energy and position dependent electron effective
mass approximation :
derived from eight-band Kane Hamiltonian
Energy-band gap :
Spin-orbit splitting in the valence band :
Momentum matrix element :
Total energy functional :
hard-wall confinement potential
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induced by a discontinuity of conduction-band
edge of the system components
Total energy functional :
Hartree potential
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Coulomb’s law
Electron-electron interaction
where : Permittivity of vacuum :
Dielectric constant :
Total energy functional :
energy of magnetic field
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Landé factor :
Bohr magneton :
Paramagnetic current density :
Total energy functional :
xc energy
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xc energy per particle depends on the magnetic
field
the external field changes the internal structure
of the wave function
Vorticity :
KS Hamiltonian
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To minimize the total energy of the system,
a functional derivative of
is taken with respect to
under the constraint of the
orbitals
being normalized.
KS Hamiltonian
where
: the orientation of the electron spin along the z axis
KS Hamiltonian : xc energy
where
KS Hamiltonian : xc energy
Perdew and Wang, 1992
Spin polarization :
Wigner-Seitz radius :
Numerical Methods :
2D problem
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Principal quantum number :
Quantum number of the projection of
angular momentum onto the z-axis :
Numerical Methods :
2D problem
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KS equations are then reduced to a 2D problem :
where
Numerical Methods :
2D problem
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Interface conditions :
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Boundary conditions :
Numerical Methods :
Hartree potential
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(3D) is solved by Poisson equation
Numerical Methods :
Hartree potential
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By cylindrical symmetry :
where
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Separating variables :

Numerical Methods :
Hartree potential
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Interface conditions :
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Boundary conditions :
Numerical Methods :
cubic eigenvalue problem
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The standard central finite difference method
 InAs, GaAs, Interface, Boundary
Since the effective mass and the
Landé factor are energy dependent :
Poisson equation :
Numerical Algorithm :
self-consistent
(1)
Set k = 0.
At B=0, first three lowest energies :
we therefore must solve (3.20) six times.
At B=15, first three lowest energies :
we thus solve (3.20) two times.
Numerical Algorithm :
self-consistent
(2) Evaluate
If converges then stop.
Otherwise set
(3) Solve (3.21) for the Hartree potential
by using GMRES.
(4)
Numerical Algorithm :
JD method
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interior, nonsymmetric, degenerate
Instead of using deflation scheme in JD solver,
we compute several eigenpairs simultaneously
and several corrections are incorporated in
search subspace at each iteration.
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M. Crouzeix, etc., The Davidson method, (1994).
G.L.G. Sleijpen, etc., Jacobi-Davidson type methods for
generalized eigenproblems and polynomial eigenproblems, (1996).
Numerical Algorithm :
JD method
Numerical Algorithm :
Numerical Algorithm :
Numerical Results
Numerical Results
Numerical Results
Numerical Results
Numerical Results
Conclusions
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New Model :
nonparabolicity + realistic hard-wall
finite confinement potential + magnetic
field + CSDFT + advanced xc energy
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QDM + Magnetic Field + Electric Field
Control many-electron states
Better Approximation
Block Jacobi-Davidson method