DANIEL TSUI LECTUREBEIJING 2005

SANKAR DAS SARMAUNIVERSITY OF MARYLAND

CONDENSED MATTER THEORY CENTERWWW.PHYSICS.UMD.EDU/CMTC

“My connection to Dan”1976

1987-881995-2005

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!

TIDBITS ABOUT QUBITSSankar Das Sarma

• QUBITS = TWO-LEVEL QUANTUM SYSTEM

• LINEAR SUPERPOSITION • QUANTUM ENTANGLEMENT• QUANTUM PARALLELISM

TOPOLOGICAL QUANTUM COMPUTATION www.physics.umd.edu/cmtc

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: ??

Spin Quantum Computation in

Semiconductor Nanostructures

Localized 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

SPINTRONICS• SPIN MATERIALS Diluted magnetic semiconductors (DMS): ferromagnetic

• SPIN DEVICESActive control of (nonequilibrium) spin AND charge

• SPIN QUBITSScalable solid state spin quantum computation

SPINTRONICSSPIN + ELECTRONICS

“Killer” app. : SPIN QUANTUM COMPUTATION!

QUANTUM COMPUTERS HOW TO BUILD A QC

PHYSICS OF QC ARCHITECTURE

• SCALABLE and ROBUST• FAULT TOLERANT• 100-10,000 COUPLED QUBITS

• Qubit dynamics• Qubit coupling, entanglement• Qubit decoherence

What can a QC do?Why build a QC?

• Prime factorizationShor algorithmExponential speedup• Database searchGrover algorithmAlgebraic speedup• Quantum simulationFeynman’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 RequirementsQubits: 2-level quantum systems

Initialization of qubitsControl and manipulation of qubits

Quantum coupling of 2-qubits1- and 2-qubit gates

Quantum error correctionHigh fidelity

Qubit specific measurementLong quantum coherence

Scalability

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

Electron/nuclear spin: An ideal qubit?

Quantum algorithms: Factoring, searching...

Output

tMeasuremen

nn

Input

N naUU 1100011

Quantum gates:

1-qubit: Spin rotation

2-qubit: Exchange interaction

22

Quantum computing with spins

21 SS

1

0

Spin relaxation and manipulation of localized states in semiconductors:

Considerations for solid state quantum computer architectures

Quantum DotQC Architecture

Si Donor Nuclear SpinQC Architecture

Semiconductor implementations

D. Loss and D.P. DiVincenzo, PRA 1998

GaAs quantum dots

Silicon donors (P)

B. Kane, Nature 1998

Fault tolerant if coherence time gateMT 410

R. Vrijen et al., PRA 2000

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 DotsExact Diagonalization TheoryGoing beyond perturbative/Heitler-London exchange gate calculations in coupled dotQC 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

-30 0 30-30

0

30 B=9T

-40 0 40

-40

0

40

Y [n

m]

X [nm]

B=0 T

HOMOPOLAR BINDING IN AN ARTIFICIAL MOLECULE

Schematic Parameter SpaceCyclotron energy Parabolic confinement

Dot separationModified magnetic length

10

1

(magnetic field)

VortexVortexMixingMixing

Level Level CrossingsCrossings

Spin Spin HamiltonianHamiltonian

Small ExchangeSmall Exchange

Three electrons-Three Dots

B=5TR=20nmħ0=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 FermionsArtificial Atom to Artificial Molecule

Two Spins in Two Quantum Dots:Quantum Gates

Single spin qubits

Qubit #1 Qubit #2

•Heisenberg Hamiltonian:

•Quantum gates:

•Heisenberg interaction + local magnetic field gives universal set of quantum gates

B S1 S2

Validity of Heisenberg Exchange HamiltonianFor Spin-Based Quantum Dot Quantum Computers

Our system

Exchange splitting

Energy spectrum

Validity of Heisenberg Exchange Hamiltonian For Six-Electron Double

Quantum Dot

Exchange splitting

Six electron double dot Energy spectrum

Adiabatic Condition• When the system Hamiltonian is changed adiabatically, the system wavefunction can be expanded on the instantaneous eigenstates:

iii tutCt ),()()(

• System evolution is governed by the Schroedinger equation:

• Instantaneous eigenvalues and eigenstates are needed to integrate the Schroedinger equation.

t

ik

N

ki ik

ik dEEiitHk

EEc

tc )(exp

Loss due to non-adiabaticityIn an exchange gate for a double dot

P donors in Si

Exchange in silicon-based quantum computer architectureMOTIVATION

Kane’s proposal for a silicon-based quantum computer

From

the

web

site

of S

NF

at

the

Uni

vers

ity o

f New

Sou

th

Wal

ws

Sydn

ey, A

ustr

alia

B.E.Kane, Nature (1998)

Concern with donor positioning: Each 31P in the array must be

exactly under the A-gate.

BUILDING BLOCKS OF KANE’S PROPOSAL• qubits are the 31P nuclear spins (I=½)

• Spin interactions in Si:31P

nz

ez

nznn

ezBne ABgBH

1-qubit operations

R =

ez

ez

nz

ez

nz

ez

J(R)

AABHRH21

222

111)()(

EXCHANGE

2-qubit operations

Hydrogenic model for P donors in Si

Si (IV) 14 e –

14 p+

P (V) 15 e –

15

p+

~ +_

Asymptotic exchange coupling of two hydrogen atoms (Herring&Flicker, 1964)

o

003

**22

A30*)/(* , *)/exp()*/1()(

)()()](2/[

mmaaarar

rErrUm d

rerU

2

)(

*)/2(exp)*

(*

)( 252

0 aRaR

aeERJ

~

Electrons in Si ( beyond m* and … )

REAL SPACE:Diamond structure

RECIPROCAL SPACE: Brillouin

zone

CONDUCTIONBAND MINIMUM:Anisotropic and

six-fold degenerate

a

Exchange between 31P donors in Si

rKK

Krk rr ii ecueu )(,)(

Envelope functions:

Bloch wavefunctions:

Ground state

]//)[(

2

222221)( bzayxz e

baF

r

)()(6

1)(6

1

rrr

F

Heitler-London triplet-singlet splitting

RkkR

R

KK

RKKKK

)cos()(

)(

22

, ',

)'(2

'

2

i

st

ecc

EEJ

Exchange calculated for two donors along [100]

*

Exchange versus donor displacements

within the Si unit cell

*

PRL 88, 027903 (2002).

3rd neigh.

2nd neigh.(12)

(4)

(6)

*1st neigh.

six-fold degeneracy of six-fold degeneracy of the the Si Si

conduction band conduction band minimum.minimum.Dipolar spin coupling ? Dipolar

gates?

The extreme sensitivity of the exchange coupling to the

relative positioning of the substitutional

donor pair in Si is entirely due to the

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

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?

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

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

Si Dipolar QC• 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

Bound orbital states T1 ~ 1ms (GaAs Quantum dot) (B=1T, T<<1K) 10 s (Si:P)

Electron spin coherence in semiconductor QC’s

Decoherence is dominated by spin-spin interactions:SPECTRAL DIFFUSION

Electron’s Zeeman frequency

fluctuates due to nuclear dipolar flip-flops

RESULTS:T2 ~ 50 s GaAs-QD

>1000 s Si:P

B

Bloch’s equation

*2T

• 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.

MT 1T

||12

11 MT

MT

MBgdtMd

B

Spectral diffusion of a Si:P spin

B

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.

Nuclear pairs are described by Poisson random variables;

Flip-flop rates are calculated using the method of moments, a high temperature expansion.

Theory

The Hamiltonian

nnn ISISA

2

Dependency with 29Si density, sample orientation

TM increases very fast when we remove 29Si !

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

0 10 20 30 40 50 60 70 80 90

20

30

40

50

60200300400500600700

Natural [4.67% 29Si] Theory/2.5 Experiment

Experiment: E. Abe, K.M. Itoh, J. Isoya, S. Yamasaki, cond-mat/0402152.

[011][111]

T M

[s]

[degrees][100]

Theory: R. de Sousa, S. Das Sarma, PRB 68, 115322 (2003)

Enriched [99.22% 29Si] Theory/2.5 Experiment

GaAs quantum dots

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

DYNAMIC NUCLEAR POLARIZATION ?

Quantum theory of spectral diffusion: Cluster expansion results

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0(a)

=600

H

ahn

echo

[ms]

0.1 1

0.1

1

10(b)

stochastic (=600)

=100

=600

=00

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

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 !

Toy paper airplane to 747 jumbo aircrafts

10-15 YEARS FOR <100 QUBITS RESEARCH QCPERHAPS 50 YEARS FOR A ‘COMMERCIAL’ QCBASED ON LINEAR EXTRAPOLATION

Making a quantum computerWhat is the right analogy?

Aviation?Manhattan project?Controlled fusion?Integrated circuits (“chips”)?

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)

Quantum theory of spectral diffusion: Two possible series expansions

/ † †1( ) n BH k Tv Tr U U e U UM

Exact expression:

4 62 4( ) 1 4 nm nm nm nmn m

v c b O c b

Expansion in :

converges whe 1 4

n n mnm

nm

A Acb

Expansion in :

powers of 1 1 !4 nm

nm n m

bc A A

iHU e 1 3

2 n nz nm n m nz mzn n m

H A I b I I I I

with

Essential condit Max 1ion: nmb

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

Quantum theory of spectral diffusion: non-perturbative cluster expansion

00 1

2 | |

( ) 1 ( ) ( )k

kD

k D k

v v O

( ) ( ) ( )D D SS D

v v v

Define “set D” contribution recursively:

Additive version of cluster exp.:

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

iD Si

v v

Examples of |D|=10:

Need 10-site exact solution Need 2-site solution only!

Lowest order cluster expansion: Product of pairs

( ) 1 ( ) exp ( )nm nmn mn m

v v v

101 1091 1074 1073 1061 1034 10

configurations

2

422

2

( ) 1 ( )

41 sin ( )1

1

nm nm

nmnm

nm

nm nm nm

v v

c

c

b c

Exact solution for pair nm

42( ) 1 41 nmmn m

n nmv c bc

42

| |4( ) 1 sin14

1 n m

nnm m nm

Avc

Ac

Cluster expansion interpolates between and expansions at the lowest order!

Spin-orbit coupling in semiconductor heterojunctions

Spin-orbit coupling

Rashba:

Dresselhaus:

)(21),( 22

0* yxmyxV

)(zV0 zeEz

0 z

Spin-flip + phonon

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

ħZ Coupling energy

Spin relaxation in III-V quantum dots

0.01 0.1 1 1010-1

100

101

102

103

104 GaSb

GaAs

InAs

Spi

n-fli

p ra

te 1

/T1 [s

-1]

Longitudinal Magnetic field [Tesla]

InSb

Electrical control of g-factor

Electron penetration into AlGaAs barrier

D.D. Awschalom, Nature 2001

E. Yablonovitch, PRB 2001

)()( 22223/420 BOEEgg

But even without barrier penetration:

1~/)( 00 ggg for E~105 Volt/cm !

Dresselhaus! Rashba!

Electrical manipulation of g-factor in GaAs

20 40 60 80 100 120 140-0.4-0.20.00.20.40.60.81.0 GaAs

Quantum dot radius l0 [nm]

gg

0

0 2 4 6 8 10 12 140.5

0.6

0.7

0.8

0.9

1.0

GaAs

Longitudinal magnetic field [Tesla]

gg

0

Dresselhaus dominated!E=104 V/cm

105 V/cm

Electrical manipulation of g-factor in InAs

10 15 20 25 30 35 400.80.91.01.11.21.31.41.51.61.7

InAs

Quantum dot radius l0 [nm]

gg

0

0 1 2 3 40.8

1.0

1.2

1.4

1.6

1.8InAs

Longitudinal magnetic field [Tesla]

gg

0

Rashba dominated!

E=104 V/cm

g-factor control T1 control !!

0 50 100 150 200100

101

102

103

104

105

106

1 X 104 Volt/cm 5 X 105

7 X 105

S

pin-

flip

rate

1/T

1 [s-1]

Quantum dot radius l0 [nm]