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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

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!

- QUBITS = TWO-LEVEL QUANTUM SYSTEM
- LINEAR SUPERPOSITION
- QUANTUM ENTANGLEMENT
- QUANTUM PARALLELISM

TOPOLOGICAL QUANTUM COMPUTATION

www.physics.umd.edu/cmtc

- 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: ??

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

- SPIN MATERIALS
Diluted magnetic semiconductors (DMS): ferromagnetic

- SPIN DEVICES
Active control of (nonequilibrium) spin AND charge

- SPIN QUBITS
Scalable solid state spin quantum computation

SPINTRONICS

SPIN + ELECTRONICS

“Killer” app. : SPIN QUANTUM COMPUTATION!

- SCALABLE and ROBUST
- FAULT TOLERANT
- 100-10,000 COUPLED QUBITS
- Qubit dynamics
- Qubit coupling, entanglement
- Qubit decoherence

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

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

- 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

Electron/nuclear spin: An ideal qubit?

Quantum algorithms: Factoring, searching...

- 1-qubit: Spin rotation
- 2-qubit: Exchange interaction

Quantum gates:

Si Donor Nuclear Spin

QC Architecture

Quantum Dot

QC Architecture

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

Experiments

GaAs

- Neighboring quantum dots
- Single electron in each dot
- Does a model of this system reproduce the Heisenberg
- model?

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?

Model

Electron Mitosis

HOMOPOLAR BINDING IN AN

ARTIFICIAL MOLECULE

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)

Three electrons-Three Dots

B=5T

R=20nm

ħw0=3meV

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

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

Validity of Heisenberg Exchange Hamiltonian

For Spin-Based Quantum Dot Quantum Computers

Our system

Energy spectrum

Exchange splitting

Six electron double dot

Energy spectrum

Exchange splitting

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

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.

Loss due to non-adiabaticity

In an exchange gate for a double dot

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

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

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)

a

Electrons in Si(beyond m* and …)

CONDUCTION

BAND MINIMUM:

Anisotropic and

six-fold degenerate

REAL SPACE:

Diamond

structure

RECIPROCAL SPACE: Brillouin zone

Ground state

Envelope functions:

Bloch wave

functions:

Heitler-London triplet-singlet splitting

Exchange between 31P donors in Si

*

Exchange calculated for two donors along [100]

2nd neigh.

(12)

*

1stneigh.

PRL 88, 027903 (2002).

(4)

3rd neigh.

(6)

*

Exchange versus donor displacements

within the Si unit cell

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?

B

R. de Sousa et al., cond-mat/031140, PRA 70, 052304 (2003)

Si:P SPIN DIPOLAR GATE QC ARCHITECTURE

- 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?

- J0 leads to error of the order of (J/D)2; Hence the criterium for gate error to be within p is:

- 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

- Long range couplings are corrected with no overhead in gate time (ability to -pulse within 5s 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

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

- Spin-orbit + phonons
- Hyperfine + phonons
- Spin-orbit + photons

- 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.

B

- 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.

TM increases very fast when we remove 29Si !

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

DYNAMIC NUCLEAR POLARIZATION ?

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

- 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 !

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

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)

with

Expansion in t:

Expansion in l:

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

Exact expression:

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

Define “set D” contribution recursively:

Additive version of cluster exp.:

Examples of |D|=10:

Need 10-site exact solution

Need 2-site solution only!

configurations

Exact solution for pair nm

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

Spin-orbit coupling

Dresselhaus:

Rashba:

ħZ

Coupling energy

T1~ 10 ms for GaAs (=30 nm, B=1 T)

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 !

E=104 V/cm

Dresselhaus dominated!

105 V/cm

Rashba dominated!

E=104 V/cm