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Part 2. Quantum Computer

Part 2. Quantum Computer. 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

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Part 2. Quantum Computer

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  1. Part 2. Quantum Computer

  2. G. Feve et al., Science 2007 One Electron Makes Current Flow Nature 2007 Quantum Computing at 16 Qubits

  3. 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

  4. Introduction :motivation • Quantum computation*. use quantum mechanical phenomena ( such as superposition and entanglement ) to perform operations on data • Electronic excitations of coupled QDs*. artificial molecule (QDM)*. Förster-Dexter Resonant energy transfer

  5. Introduction :model • 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

  6. CSDFT Density functional theory • Hohenberg and Kohn, 1964Kohn, 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 chemistryThe band structure of solids in physics

  7. CSDFT Current spin density functional theory • Vignale and Rasolt, 1987 • Electronic structure of quantum dots in magnetic fields • Couple to spin and orbital

  8. Total energy functional • Number of electrons : N • Total spin : S • Spin-up and spin-down : • Total density : • Constraint :

  9. Total energy functional :kinetic energy • the electron momentum operator : • is the vector potential induced by an external magnetic field • KS orbitals and eigenvalues :

  10. Total energy functional :effective mass • 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 :

  11. Total energy functional :hard-wall confinement potential • induced by a discontinuity of conduction-band edge of the system components

  12. Total energy functional :Hartree potential • Coulomb’s law • Electron-electron interaction where : Permittivity of vacuum : Dielectric constant :

  13. Total energy functional :energy of magnetic field • Landé factor : • Bohr magneton : • Paramagnetic current density :

  14. Total energy functional :xc energy • xc energy per particle depends on the magnetic field • the external field changes the internal structure of the wave function • Vorticity :

  15. KS Hamiltonian • 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.

  16. KS Hamiltonian where : the orientation of the electron spin along the z axis

  17. KS Hamiltonian : xc energy where

  18. KS Hamiltonian : xc energy Spin polarization : Wigner-Seitz radius : Perdew and Wang, 1992

  19. Numerical Methods :2D problem • Principal quantum number : • Quantum number of the projection of angular momentum onto the z-axis :

  20. Numerical Methods :2D problem • KS equations are then reduced to a 2D problem : where

  21. Numerical Methods :2D problem • Interface conditions : • Boundary conditions :

  22. Numerical Methods :Hartree potential • (3D) is solved by Poisson equation

  23. Numerical Methods :Hartree potential • By cylindrical symmetry : where • Separating variables : 

  24. Numerical Methods :Hartree potential • Interface conditions : • Boundary conditions :

  25. Numerical Methods :cubic eigenvalue problem • The standard central finite difference method InAs, GaAs, Interface, Boundary • Since the effective mass and the Landé factor are energy dependent : • Poisson equation :

  26. 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.

  27. Numerical Algorithm :self-consistent (2) Evaluate If converges then stop. Otherwise set (3) Solve (3.21) for the Hartree potential by using GMRES. (4)

  28. Numerical Algorithm :JD method • interior, nonsymmetric, degenerate • Instead of using deflation scheme in JD solver, we compute several eigenpairs simultaneouslyandseveral corrections are incorporated in search subspace at each iteration. • M. Crouzeix, etc., The Davidson method, (1994).G.L.G. Sleijpen, etc., Jacobi-Davidson type methods forgeneralized eigenproblems and polynomial eigenproblems, (1996).

  29. Numerical Algorithm :JD method

  30. Numerical Algorithm :

  31. Numerical Algorithm :

  32. Numerical Results

  33. Numerical Results

  34. Numerical Results

  35. Numerical Results

  36. Numerical Results

  37. Conclusions • New Model : nonparabolicity + realistic hard-wall finite confinement potential + magnetic field + CSDFT + advanced xc energy • QDM + Magnetic Field + Electric Field • Control many-electron states • Better Approximation • Block Jacobi-Davidson method

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