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DANIEL TSUI LECTURE BEIJING 2005 SANKAR DAS SARMA UNIVERSITY OF MARYLAND CONDENSED MATTER THEORY CENTER WWW.PHYSICS.UMD.EDU/CMTC PowerPoint PPT Presentation


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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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DANIEL TSUI LECTURE BEIJING 2005 SANKAR DAS SARMA UNIVERSITY OF MARYLAND CONDENSED MATTER THEORY CENTER WWW.PHYSICS.UMD.EDU/CMTC

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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


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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!


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TIDBITS ABOUT QUBITSSankar Das Sarma

  • QUBITS = TWO-LEVEL QUANTUM SYSTEM

  • LINEAR SUPERPOSITION

  • QUANTUM ENTANGLEMENT

  • QUANTUM PARALLELISM

TOPOLOGICAL QUANTUM COMPUTATION

www.physics.umd.edu/cmtc


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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: ??


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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


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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


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SPINTRONICS

SPIN + ELECTRONICS

“Killer” app. : SPIN QUANTUM COMPUTATION!


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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


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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?


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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


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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


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Quantum computing with spins

Electron/nuclear spin: An ideal qubit?

Quantum algorithms: Factoring, searching...

  • 1-qubit: Spin rotation

  • 2-qubit: Exchange interaction

Quantum gates:


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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


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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


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Experiments

GaAs

  • Neighboring quantum dots

  • Single electron in each dot

  • Does a model of this system reproduce the Heisenberg

  • model?


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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?


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Model


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Electron Mitosis

HOMOPOLAR BINDING IN AN

ARTIFICIAL MOLECULE


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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)


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Three electrons-Three Dots

B=5T

R=20nm

ħw0=3meV


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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


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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


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Validity of Heisenberg Exchange Hamiltonian

For Spin-Based Quantum Dot Quantum Computers

Our system

Energy spectrum

Exchange splitting


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Six electron double dot

Energy spectrum

Exchange splitting

Validity of Heisenberg Exchange Hamiltonian For Six-Electron Double Quantum Dot


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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.


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Loss due to non-adiabaticity

In an exchange gate for a double dot


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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


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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


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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)


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a

Electrons in Si(beyond m* and …)

CONDUCTION

BAND MINIMUM:

Anisotropic and

six-fold degenerate

REAL SPACE:

Diamond

structure

RECIPROCAL SPACE: Brillouin zone


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Ground state

Envelope functions:

Bloch wave

functions:

Heitler-London triplet-singlet splitting

Exchange between 31P donors in Si


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*

Exchange calculated for two donors along [100]


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2nd neigh.

(12)

*

1stneigh.

PRL 88, 027903 (2002).

(4)

3rd neigh.

(6)

*

Exchange versus donor displacements

within the Si unit cell


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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?


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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


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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:


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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


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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


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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


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  • 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.


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Spectral diffusion of a Si:P spin

B


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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.


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The Hamiltonian


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Dependency with 29Si density, sample orientation

TM increases very fast when we remove 29Si !


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


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GaAs quantum dots

Spectral diffusion is very important: Ga and As do not have I=0 isotopes !


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DYNAMIC NUCLEAR POLARIZATION ?


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Quantum theory of spectral diffusion: Cluster expansion results

[W.M. Witzel, R. de Sousa, S. Das Sarma, cond-mat/0501503]


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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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Making a quantum computerWhat is the right analogy?

Aviation?

Manhattan project?

Controlled fusion?

Integrated circuits (“chips”)?

Toy paper airplane to 747 jumbo aircrafts

10-15 YEARS FOR <100 QUBITS RESEARCH QC

PERHAPS 50 YEARS FOR A ‘COMMERCIAL’ QC

BASED ON LINEAR EXTRAPOLATION


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Solid state spin quantum computation in semiconductors

CMTC/UMD Spin Quantum Computation Group

  • Sankar Das Sarma

  • Vito Scarola

  • Kwon Park

  • Belita Koiller

  • Xuedong Hu

  • Rogerio De Sousa, Wayne Witzel

  • Juan Delgado, Magdalena Constantin

    Supported by NSA, LPS, ARO, ARDA, UMD

SDS, Michael Freedman, Chetan Nayak

cond-mat/0412343 (PRL 2005)


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with

Expansion in t:

Expansion in l:

Quantum theory of spectral diffusion: Two possible series expansions

[W.M. Witzel, R. de Sousa, S. Das Sarma, cond-mat/0501503]

Exact expression:


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Large sets D are mainly composed of disconnected clusters Si; If clusters are far enough, neglect inter-cluster coupling to get

Quantum theory of spectral diffusion: non-perturbative cluster expansion

Define “set D” contribution recursively:

Additive version of cluster exp.:

Examples of |D|=10:

Need 10-site exact solution

Need 2-site solution only!


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 configurations

Lowest order cluster expansion: Product of pairs

Exact solution for pair nm

Cluster expansion interpolates between t and l expansions at the lowest order!


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Spin-orbit coupling in semiconductor heterojunctions

Spin-orbit coupling

Dresselhaus:

Rashba:


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Spin-flip + phonon

ħZ

Coupling energy

T1~ 10 ms for GaAs (=30 nm, B=1 T)


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Spin relaxation in III-V quantum dots


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Electrical control of g-factor

D.D. Awschalom, Nature 2001

E. Yablonovitch, PRB 2001

Electron penetration into AlGaAs barrier

But even without barrier penetration:

Dresselhaus!

Rashba!

for E~105 Volt/cm !


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Electrical manipulation of g-factor in GaAs

E=104 V/cm

Dresselhaus dominated!

105 V/cm


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Electrical manipulation of g-factor in InAs

Rashba dominated!

E=104 V/cm


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g-factor control  T1 control !!


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