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“My connection to Dan” 1976 1987-88 1995-2005 2004

DANIEL TSUI LECTURE BEIJING 2005 SANKAR DAS SARMA UNIVERSITY OF MARYLAND CONDENSED MATTER THEORY CENTER WWW.PHYSICS.UMD.EDU/CMTC. “My connection to Dan” 1976 1987-88 1995-2005 2004. My connection to China Lai WY 1983-85 Beijing Xie XC 1983-87; 88-91 USTC Zhang FC 1984-86 Fudan

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“My connection to Dan” 1976 1987-88 1995-2005 2004

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  1. DANIEL TSUI LECTUREBEIJING 2005SANKAR DAS SARMAUNIVERSITY OF MARYLANDCONDENSED MATTER THEORY CENTERWWW.PHYSICS.UMD.EDU/CMTC “My connection to Dan” 1976 1987-88 1995-2005 2004

  2. My connection to China Lai WY 1983-85 Beijing Xie XC 1983-87; 88-91 USTC Zhang FC 1984-86 Fudan He S 1988-92 USTC Li Q 1989-93 USTC Lai ZW 1990-92 USTC(?);Chicago Liu DZ 1990-94 USTC Zheng L (1995-98); Hu J (1997-99) Indiana Hu XD 1998-2003 Beijing;Michigan Zhang Y 2002- USTC; Yale Wang DW Taiwan 1996-2002 Tse GW Hong Kong 2004- Also B.Y.K Hu; K.E. Khor, … SC Zhang (Stanford); R. Zia (Virginia Tech.); DC Tsui (Princeton)…. More than 100 publications with these collaborators! I was in China (Beijing, Shanghai) in 1986 as a guest of the Chinese Academy of Sciences with the Institute of Physics being my host!

  3. TIDBITS ABOUT QUBITSSankar Das Sarma • QUBITS = TWO-LEVEL QUANTUM SYSTEM • LINEAR SUPERPOSITION • QUANTUM ENTANGLEMENT • QUANTUM PARALLELISM TOPOLOGICAL QUANTUM COMPUTATION www.physics.umd.edu/cmtc

  4. A (VERY) BRIEF HISTORY OF COMPUTATION • UNARY: 10,000 YEARS AGO • BINARY: 1,000 YEARS AGO; BITS • ANALOG COMPUTERS: ~ 1000 years • BOOLEAN ALGEBRA: BITS • DIGITAL COMPUTERS: ~ 100 years • QUANTUM MECH.: 100 YEARS AGO • QUBITS: NOW (PERHAPS) • QUANTUM COMPUTERS: ??

  5. Spin Quantum Computation in Semiconductor NanostructuresLocalized Spin 1\2 qubits in Semiconductor Nanostructures(Heisenberg Coupling)X. Hu;R. de Sousa;B. Koiller;V. Scarola; W.Witzel ARDA, ARO, UMD, LPS, NSA

  6. SPINTRONICS • SPIN MATERIALS Diluted magnetic semiconductors (DMS): ferromagnetic • SPIN DEVICES Active control of (nonequilibrium) spin AND charge • SPIN QUBITS Scalable solid state spin quantum computation

  7. SPINTRONICS SPIN + ELECTRONICS “Killer” app. : SPIN QUANTUM COMPUTATION!

  8. QUANTUM COMPUTERSHOW TO BUILD A QCPHYSICS OF QC ARCHITECTURE • SCALABLE and ROBUST • FAULT TOLERANT • 100-10,000 COUPLED QUBITS • Qubit dynamics • Qubit coupling, entanglement • Qubit decoherence

  9. Prime factorization Shor algorithm Exponential speedup Database search Grover algorithm Algebraic speedup Quantum simulation Feynman’s dream Quantum parallelism Entanglement Universal one and two-qubit gates Quantum error correction Boolean vs. Quantum P/NP some day?? Topological QC What can a QC do?Why build a QC?

  10. Minimal QC Requirements Qubits: 2-level quantum systems Initialization of qubits Control and manipulation of qubits Quantum coupling of 2-qubits 1- and 2-qubit gates Quantum error correction High fidelity Qubit specific measurement Long quantum coherence Scalability

  11. PROPOSED QC ARCHITECTURES (far too many) • ION TRAPS • LIQUID STATE NMR • NEUTRAL ATOM OPTICAL LATTICE • CAVITY QED • SQUIDS, JOSEPHSON JUNCTIONS • COOPER PAIR BOXES • ELECTRON SPINS IN SOLIDS (GaAs, Si) • SOLID STATE NMR • ELECTRON STATES ON HE-4 SURFACE • QUANTUM HALL STATES

  12. Quantum computing with spins Electron/nuclear spin: An ideal qubit? Quantum algorithms: Factoring, searching... • 1-qubit: Spin rotation • 2-qubit: Exchange interaction Quantum gates:

  13. Spin relaxation and manipulation of localized states in semiconductors:Considerations for solid state quantum computer architectures Si Donor Nuclear Spin QC Architecture Quantum Dot QC Architecture

  14. Semiconductor implementations GaAs quantum dots D. Loss and D.P. DiVincenzo, PRA 1998 Silicon donors (P) B. Kane, Nature 1998 R. Vrijen et al., PRA 2000 Fault tolerant if coherence time

  15. Experiments GaAs • Neighboring quantum dots • Single electron in each dot • Does a model of this system reproduce the Heisenberg • model?

  16. Spin Transitions in Few Electron Quantum Dots Exact Diagonalization Theory Going beyond perturbative/Heitler-London exchange gate calculations in coupled dot QC architectures ATOM to MOLECULE Vito Scarola WHEN IS THE 2-ELECTRON QUNTUM DOT A ‘MOLECULE’ WITH TUNABLE EXCHANGE COUPLING? WHEN IS IT JUST AN ARTIFICIAL 2-ATOM SYSTEM?

  17. Model

  18. Electron Mitosis HOMOPOLAR BINDING IN AN ARTIFICIAL MOLECULE

  19. Schematic Parameter Space Dot separation Modified magnetic length 1 0 Cyclotron energy Parabolic confinement Small Exchange Spin Hamiltonian Vortex Mixing Level Crossings 1 (magnetic field)

  20. Three electrons-Three Dots B=5T R=20nm ħw0=3meV

  21. Conclusion • Exact diagonalization allows accurate • treatment of strongly interacting regime • Exchange splitting (J) oscillates with • magnetic field • Trial state analysis implies singlet-triplet • transitions of Composite Fermions • Artificial Atom to Artificial Molecule

  22. Two Spins in Two Quantum Dots:Quantum Gates S1 S2 B Single spin qubits Qubit #1 Qubit #2 • Heisenberg • Hamiltonian: • Quantum gates: • Heisenberg interaction + local magnetic field gives • universal set of quantum gates

  23. Validity of Heisenberg Exchange Hamiltonian For Spin-Based Quantum Dot Quantum Computers Our system Energy spectrum Exchange splitting

  24. Six electron double dot Energy spectrum Exchange splitting Validity of Heisenberg Exchange Hamiltonian For Six-Electron Double Quantum Dot

  25. Adiabatic Condition • When the system Hamiltonian is changed adiabatically, the system wavefunction can be expanded on the instantaneous eigenstates: • System evolution is governed by the • Schroedinger equation: • Instantaneous eigenvalues and eigenstates are needed to integrate the Schroedinger equation.

  26. Loss due to non-adiabaticity In an exchange gate for a double dot

  27. Exchange in silicon-based quantum computer architecture MOTIVATION Kane’s proposal for a silicon-based quantum computer B.E.Kane, Nature (1998) P donors in Si Concern with donor positioning: Each31P in the array must be exactly under the A-gate. From the website of SNF at the University of New South Walws Sydney, Australia

  28. 1-qubit operations 2-qubit operations R = EXCHANGE • BUILDING BLOCKS OF KANE’S PROPOSAL • qubits are the 31P nuclear spins (I=½) • Spin interactions in Si:31P

  29. Hydrogenic model for P donors in Si ~ + _ P (V) 15 e – 15 p+ Si (IV) 14 e – 14 p+ Asymptotic exchange coupling of two hydrogen atoms (Herring&Flicker, 1964)

  30. a Electrons in Si(beyond m* and …) CONDUCTION BAND MINIMUM: Anisotropic and six-fold degenerate REAL SPACE: Diamond structure RECIPROCAL SPACE: Brillouin zone

  31. Ground state Envelope functions: Bloch wave functions: Heitler-London triplet-singlet splitting Exchange between 31P donors in Si

  32. * Exchange calculated for two donors along [100]

  33. 2nd neigh. (12) * 1stneigh. PRL 88, 027903 (2002). (4) 3rd neigh. (6) * Exchange versus donor displacements within the Si unit cell

  34. The extreme sensitivity of the exchange coupling to the relative positioning of the substitutional donor pair in Si is entirely due to the six-fold degeneracy of the Si conduction band minimum. Dipolar spin coupling ? Dipolar gates?

  35. Qubits are dipolar coupled single electron spins B R. de Sousa et al., cond-mat/031140, PRA 70, 052304 (2003) Si:P SPIN DIPOLAR GATE QC ARCHITECTURE

  36. Gate imperfection in the presence of exchange • Long-range dipolar ~1/R3 is much stronger than short-range exchange for large inter-donor separation; How large should be the separation so that J can be neglected? • J0 leads to error of the order of (J/D)2; Hence the criterium for gate error to be within p is:

  37. Gate times and donor separation • Separations of the order of 300 Å allows easier lithography; • Gates are 106 times slower than exchange coupling; however there is no need for exchange control and donor positioning with atomic precision. Using 28Si we expect T2~T1~ seconds for B~1T

  38. Si Dipolar QC • Long range couplings are corrected with no overhead in gate time (ability to -pulse within 5s is required). • Dipolar implementation is reliable, its advantages/disadvantages should be compared with other proposals without exchange (for example, Skinner, Davenport, Kane, PRL 2003, which requires electron shuttling between donors); • Dipolar coupling insensitive to electronic structure: No inter-valley interference, interstitial defects are also good qubits; • “Top-down” construction schemes based on ion implantation can be used even though they lack atomic precision in donor positioning. • Can be scaled up

  39. Electron spin coherence in semiconductor QC’s Bound orbital states T1 ~ 1ms (GaAs Quantum dot) (B=1T, T<<1K) 10 s (Si:P) Decoherence is dominated by spin-spin interactions: SPECTRAL DIFFUSION Electron’s Zeeman frequency fluctuates due to nuclear dipolar flip-flops B RESULTS: T2 ~ 50 ms GaAs-QD >1000 ms Si:P

  40. Spin-orbit + phonons • Hyperfine + phonons • Spin-orbit + photons Bloch’s equation • Spectral diffusion(nuclear spins, time dependent magnetic fields) • Dipolar / exchange coupling between “like” spins • Unresolved hyperfine structure • Different g-factors • Inhomogeneous fields • Dipolar / exchange between “unlike” spins.

  41. Spectral diffusion of a Si:P spin B

  42. Nuclear induced spectral diffusion • Nuclear spins flip-flop due to their dipolar interaction; • Electron’s Zeeman frequency fluctuates in time due to nuclear hyperfine field. Theory • Nuclear pairs are described by Poisson random variables; • Flip-flop rates are calculated using the method of moments, a high temperature expansion.

  43. The Hamiltonian

  44. Dependency with 29Si density, sample orientation TM increases very fast when we remove 29Si !

  45. Spin-1/2 theory of nuclear spectral diffusion: Comparison with experiment

  46. GaAs quantum dots Spectral diffusion is very important: Ga and As do not have I=0 isotopes !

  47. DYNAMIC NUCLEAR POLARIZATION ?

  48. Quantum theory of spectral diffusion: Cluster expansion results [W.M. Witzel, R. de Sousa, S. Das Sarma, cond-mat/0501503]

  49. Conclusions • Electrical control of single spin dynamics is promising for III-V quantum dots because of spin-orbit coupling; • The spin of localized states interact weakly with the phonons at low T: Nuclear induced spectral diffusion if the dominant decoherence mechanism; • Isotopically purified Si:P donor spins can be coherent for ~1000 s (B = 0.3 Tesla); 60 ms already measured ! (S.A. Lyon, 2003) • GaAs quantum dots (or donors) coherent for only 1 – 100 s, but TM /J > 106 !

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