12) Solid-State Systems12.1 Solid state NMR/EPR

A-Gates J-Gates

12.1 Solid state NMR/EPR12.2 Superconducting systems12.1 Solid state NMR/EPR12.2 Superconducting systems12.3 Semiconductor qubits

|0>

|1>

Spins insolids

Solid-State NMR / EPR12.1

12.1.1 Scaling behavior of NMR quantuminformation processors

12.1.2 31P in silicon12.1.3 N@C6012.1.4 Other proposals12.1.5 Single-spin readout

Why Solids ?

Liquid state NMR is an excel-lent system for small quantumregisters.

For > 10 qubits, problems arise:- addressability- decoherence

Solids provide possible solutions:- many qubits- local addressing- low temperature

Scaling Behavior

1000

0.001

1

0 10 20# qubits

Sign

al

Signal loss for pps preparation

N

N

CC

C

CC

C

ND3+

O

O-

H

D

H H

H

H

1

A B

3 52

4

6

79

8

1011

12-qubit system: (Mahesh)

L-Histidine

Execution time / ms

Scaling Behavior

2 4 6 8 10 12# Qubits

2 4 6 8 10 12

# Qubits

ln (signal)

2 4 6 8 10 12

Fidelity (simulated)

# Qubits

Solid-State NMR

23Na : I = 3/2 = 2 qubit

mz= -3/2

+3/2

+1/2

-1/2

HQ

-120000-60000060000120000

-5

5

15

25

35

45

-120000-60000060000120000

3

2

1

2

12

−

3

2−

EH. Kampermann, and W.S. Veeman, ’Quan-tum Computing Using Quadrupolar Spinsin Solid State NMR’, Quantum InformationProcessing 1, 327 (2002).

-360000

-260000

-160000

-60000

40000

140000

240000

340000

-120000-60000060000120000

-4,00E+01

-3,00E+01

-2,00E+01

-1,00E+01

0,00E+00

1,00E+01

2,00E+01

3,00E+01

4,00E+01

-120000-60000060000120000

-520000

-420000

-320000

-220000

-120000

-20000

80000

180000

-120000-60000060000120000-1,20E+02

-1,00E+02

-8,00E+01

-6,00E+01

-4,00E+01

-2,00E+01

0,00E+00

2,00E+01

4,00E+01

6,00E+01

-120000-60000060000120000

-400000

-350000

-300000

-250000

-200000

-150000

-100000

-50000

0

50000

-120000-60000060000120000

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

-120000-60000060000120000

32

1

2

1

2−

32

−

E

32

1

2

1

2−

32

−

E

32

1

2

1

2−

32

−

E

Electron-Nucleus Systems

Electron-nucleus entanglingin solid malonic acid radical

M. Mehring, J. Mende, and W. Scherer,Phys. Rev. Lett. 90, 153001 (2003).

Pseudopure state preparation “S-Bus”Electron spin mediates

nucleus-nucleus couplings

Ce:CaF2

A-Gates J-Gates

31P in Solicon12.1.2 31P in silicon

The Qubitsnuclear spin = qubitelectron spin = control

31P = shallow donor

H = -ωI Iz - ωS Sz - a I . S

Relevant Interactions

nuclearZeeman

electronZeeman

hyperfine

Qubit

|0>|1>

Transition frequency ω0 = ωI + a/2(high field approximation)

Control

Why 31P in Si ?

Long decoherence times:

Electron spin T2 ~ 60 ms in 28Si @ 7KNuclear spin T1 > 10 h

Excellent technology base

28Si has nuclear spin 0Natural abundance: 4.6% 29Si

Spin-orbit coupling small

Initialization

000000000

Quantumregister

DiVincenzo’s rule 2:Initialization into a well defined state.

|0> =

ground state:

n↑-n↓n↑+n↓

≈ 1electrons:

Boltzmann factor @ 100 mK, 2T:

n↑-n↓n↑+n↓

≈ 5.10-3nuclei:

Initialize nuclear spin qubitby microwave pulse

Initialization

RelaxationDissipation required

Optical spin injection throughSiGe superlattices or quantum dots

Alternatives:

Electrical spin injection

Readout

Modify Frequency : A-Gatesa ~ |ΨΨel(rn)|2H = -ωI Iz - ωS Sz - a I . S

Barrier

Si31P qubit

large

ν0 ~ 90 MHz

|ΨΨel(rn)|

Barrier

A Gate

Si31P qubit

+

small

ν0 ~ 50 MHz (for U ~ 0.7 V)

|ΨΨel(rn)|

Electronic Frequency Tuning

A-Gate Voltage / VNuc

lear

spin

reso

nanc

efr

eque

ncy

/MH

z

Barrier

A Gate

Si31P qubit

+

Barrier

J Gate

Si

+

31P qubit

Modify Couplings : J-Gates

J gate draws electrons into overlap region

coupling operator: HJ = J I1. I2

effective qubit-coupling: 75 kHz depends on B-field, gates

Tuning of Couplings

Donor Separation / nm0 10 20 30

Exc

hang

eFr

eque

ncy

/Hz

109109

1010

1011

1012

2µBB/h = 56 GHz

Donor Placement

How to put exactly one atom at the right position ?

31P+

31P Implantation

Si substrate

AFM Tiphole≥ 4 nm

Focused ion beam

Depositing 31PPhys. Rev. B, 64:161401 (2001).

STM of H-terminated Si surface

Chemisorbed 31P

Phys. Rev. B, 64:161401 (2001).

Hydrogen desorbed

After PH3 dosing

Qubit ReadoutDiVincenzo’s rule 5: Qubit-selective readout.

1) Transfer qubit to electron spin

Gate

31P qubit readout donor

Coupling J

Ene

rgy

|↑↑>

|↑↓+↓↑>

|↑↓-↓↑>

|↓↓>electron spins

2) Transfer to readout donorScan converts |0> toelectron singlet state

nuclear spin

|0> =|1> =

3) Detectionof singlet state

singlet pair can form D-

electrostatic detection of D-

Current State

Xiao et al., Nature, 430, 435 (2004).

Single spin ESR

00

0

0

0

1

2

3

100 200 300 400 500τ (ns)

Q(1

05 e)

31P in Si B0 = 350.3 mT

PMW= 251 W

126 W

63 W

32 W

τ

Stegner et al., Nature Physics 2, 835–838 (2006).

Coherent excitation

GeSi0.23Ge0.77 barrierSi0.15Ge0.85

Si0.4Ge0.6

Si0.23Ge0.77 barriern-Si0.4Ge0.6 ground planeSi-Ge buffer layer

Si substrate

R. Vrijen et al, Phys. Rev. A 62, 012306 (2000)

SiGe Spin-Transistor

Modification of Kane proposal:- Use SiGe heterostructure- Use electron spin instead of nuclear- Only one type of gates needed

g-factor Tuning

g D=2

.00

g T=0

.82

gav=2.00

gate off

Con

duct

ion

Ban

dE

nerg

y/m

eV

z

<111> Substrate

gav~1.5

biased gate

1-qubit gates

also changes Bohr radiuscan be used for 2-qubit gates

M. Friesen et al., PRB 67, 121301 (2003).

Quantum-Dot Qubits

Electrostatically confinedquantum dots in Si - SiGe QW 4 qubits

2 qubits

Gated Exchange

HJ = J S1. S2

J = 0

J ≠ 0 gate

Tuning of exchange in 2-qubit system

4-qubit system

J = 0

J23 = 100 MHz

Endohedral Fullerenes

N@C60, P@C60

0 100 200 3000,1

1

10

100

1000

Temperature / K

Rel

axat

ion

Tim

eT

1/m

s

Decoherence Time

Why N@C60 ?

5

based on dipole-dipolecoupling @ 1 nm

Switching Time

may be ~ 10 ns

vs.

# gates ~108

effusion cell

ion source

ProductionImplantation into empty cages

HMI Berlin

Purification by HPLC

660640620600580560540

C60

N@C60

C60

N@C60

HPLC enrichmentStep (N@C60/C60)

1x (0.05%)6x (~10%)7x (~40%)8x (~65%)

10x (~90%)

Retention time / s

HPLC Chromatograms

338033603340332033003280

HPLC Step2x3x4x5x

10x

Magnetic field / G

EPR Spectra

Nano-PositioningUniversity of Nottingham

C59N on Si

4.6nm

4.6nm

3.8nm

Accuracy of positioning determined by surface lattice constant.

Manipulation process does not induce additional defects on underlying surface

before after

spaceseparate leads

bit 1 bit 2 bit 3

Solid-State Computer

Addressing Qubits

1H 13Cqbit 1

qbit 2

NMR in Liquids

ωω1ωω2

monochromatic radiationaffects only one type of spins

Addressing N@C60

Wires(not to scale)

Current pulses through µm-scale wirescould shift frequencies by multi-MHz

Phys. Rev. A 65, 052309 (2002).

-0.4

-0.2

-0.1

0.1

0.2field / T

-0.4 0 0.2 0.4

position x / µm

for I = 1A, ∆ = 1 µm

Magnetic Field∆

-0.2-0.4 0.2 0.4

0.8

0.6

0.4

0.2

0

dBG/dxT/µm

position x / µm

Field gradient

20

10

∆νMHz

Nearest neighbourfrequency difference

Si

Frequency Selection

Implement 1-qubit gates by frequency- or field switching

PositionResonance frequency

JJ

What about 2-qubit gates?

2-Qubit GatesRequired: Interaction

rθθ

Dipole-dipole coupling

Edd =µ0

4π (1-3cos2θ)γ1 γ2

r3

Edd

h ~ 50 MHz

N@C60, 1.1 nm distance

Switch on / off ?

Switchable Coupling

electron spin

nuclear spin:15N, 31P

+ (Nuclei) : long decoherence timesno coupling

+ (Electrons) : fast gatesstrong coupling

hyperfinecoupling

Use which as Qubits ?

coupling off

passive state:QuInfo in nucleiactivate:SWAP QuInfo

|i> |i+1>SWAP

coupling on

15

+ (Electrons) : fast gatesstrong coupling

• Magnetic ResonanceForce Microscope

• Single Molecule Transistor•Optical Detection viaDiamond N/V center

• Scanning Tunneling MicroscopeElectron Spin Resonance

Single-Spin Readout

Voltage

Currenttip

Magnetic field

Graphite surface

BDPA clusterSpinprecession

STM EPR

First demonstration: Manassen et al., PRL 62, 2531 (1989).

Detect modulation of tunnel current at Larmor frequency

STM-ESR spectra of BDPAon HOPG (red and black) andbare HOPG (blue).

534 536 538 540Frequency (MHz)

STM EPR

C. Durkan and M. E. Welland, Electronic spin detection in molecules using scanning-tunneling microscopy-assisted electron-spin resonance. Appl. Phys. Lett., 80, 458 (2002).

10 nm

200 nm x 200 nm area

485 525 575

TEMPOConventional EPR

Frequency / MHz

STM EPR of TEMPOTEMPO on graphite : S=1/2, I=1 : hyperfine splitting

Single-Spin Detectionby optically detected magnetic resonance

Las

er

Fluorescence

RFRadio frequency / MHz

Flu

ores

cenc

e

J. Wrachtrup, A. Gruber, L. Fleury, and C.v.Borczyskowski, Chem. Phys. Lett. 267, 179 (1997).

single 1H in pentacene

Single-Spin Detectionby optically detected magnetic resonance

Las

er

Fluorescence

RF F. Jelezko, C. Tietz, A. Gruber, I. Popa, A.Nizovtsev, S. Kilin, and J. Wrachtrup, Sin-gle Mol. 2, 255 (2001).

Diamond N/V center

Single Center : Photon Antibunching

τ / µs0.0 0.2 0.4

1

2

0

g(2)(τ) = <I(t) I(t+τ)><I(t)>2

t

EM image

Fluorescence

Single Center : SpectroscopyNV - containing nanocrystals

2700 2800 2900 3000

MW

Z

X,Y

Flu

ores

cenc

e

MW frequency / MHz

N-V

z

x,y

Ene

rgy

N/V

Detecting N@C60 Spins

use N/V center as field probe

2700 2800 2900 3000

Flu

ores

cenc

e

MW frequency / MHz

N

N/V + N@C60

2700 2800 2900 3000

Coupling to N@C60

Control, 99,9 % pure C60

MW frequency / MHz

Single Spin Force Detection

Crystal Lattice Quantum ComputerF. Yamaguchi, and Y. Yamamoto, ‘Crystal latticequantum computer‘, Appl. Phys. A 68, 1 (1999).

Detection e.g. by vibrating cantilever

Superconducting Systems12.2

12.2.1 Charge Qubits12.2.2 Flux Qubits12.2.3 Gate Operations12.2.4 Readout

Superconducting QubitsMacroscopic quantum stateEasily coupled to each otherCan be integrated

Josephson junction

Must operate at < 50 mKNon-dissipative:

Only superconducting elementsElectron temperature < 50 mK

Energy scales:

log

E

0

Relevant ObservablesFlux = phase differenceCharge Q

kT

Harmonic Oscillator

C L

Classical d2Qdt2

QLC+ = 0

Quantum

[ ,Q] = i h

= h0(n+ )12

~Position ~Momentum

2

2L +H = Q2

2C

Typical values:

Dimension ~ 10 µmL ~ 0.1 nHC ~ 1 pF0/2π ~ 16 GHz

Main challenges:Coupling to environmentVariability of engineered circuits

Cooper Pair BoxCoulomb energy of a box:

q2

2CEC =q(2ne)2

2C=2n2e2

C=

H = EC(N-Ng)2

# Cooper pairson island

tunable byoffset voltage

BoxPhase differenceacross junction

tunable by fluxthrough split junction

H = EC(N-Ng)2 - EJ cos

Hamiltonian including tunnel junction

Gate voltage

Ene

rgy

n n+1

Charge Qubit

Box

ReservoirTunnel junction

Control electrode

Qubit Hamiltonian:Qubit states: |n>, |n+1>

Gate voltage

Ene

rgy

Gate Operations

Tuning thez-“field”

Ej

Ej’(’)

Tuning thex-“field”

ΦΦEj

Flux Qubit

Flux Φ

Ene

rgy

ΦΦΦΦ’

Josephson Qubit

box

reservoir

pulse gate1 µm

probedc gate

: tunnel junction: capacitor

box

φ

Rb,Cb

EJ,CJ

Vg

Vp

Vb

Cp

Cg

Flux Qubit

Rabi Oscillations

Resonant Excitation

Rabi pulse duration / nsOcc

upat

ion

prob

abili

tyof

|1>

Current Device

DiC

arlo

etal

.,N

atur

e,46

0,24

0(2

009)

.

Grover Search

Semiconductor Qubits12.3

12.3.1 Materials12.3.2 Excitons in quantum dots12.3.3 Electron spin qubits

Coupled Quantum Dots

Qubits

Dot 1 Dot 2

|00>

Dot 1 Dot 2

|01>

Dot 1 Dot 2

|10>

Dot 1 Dot 2

|11>

Physicalstate

Logicalstate

M. Friesen et al. , Phys. Rev. Lett. 92, 037901 (2004).

QDot Readout

|0>|1>

Single-step readoutby microwave excitation

Irradiation at ν12 creates charge oscillation if qubit is in |1> state

Can also be used for initialization

Asymmetric QW

Spin-Filter Readout

R. Hanson, L.H.W.v. Beveren, I.T. Vink, J.M. Elzerman, W.J.M. Naber, F.H.L. Koppens,L.P. Kouwenhoven, and L.M.K. Vandersypen, ‘Single-Shot Readout of Electron Spin Statesin a Quantum Dot Using Spin-Dependent Tunnel Rates’, Phys. Rev. Lett. 94, 196802 (2005).

∆I Q

PC

(a.u

.)V

P(a

.u.)

time

time

b

c

N =1

waiting time readout

N =1 2

N =2 1

initializationpreparation &

twait

→

→

frac

tion

of’T

’

e

waiting time (ms)twait

0 84

1

0.8

0.6

0.4

0.2

0

α(1

--

)α

β3 Γ

T

3 ΓT

Γ S+

20

∆I Q

PC

(nA

)

40

30

20

10

0

0 1.5time (ms)

d

’T’

’S’

’T’

’S’

’S’

S

T

S

T

S

TS

T

B =0 .02T

S

T1/T1

- /2.58twait0.15+0.80 e

a

200 nm

T

QΓ

rese

rvoi

rIQPC

L P M

Qubits in Quantum DotsElectrostatically controlledpair of quantum dots

Occupation numbers

Qubit

[1] J.M. Elzerman, R. Hanson, J.S. Greidanus, L.H.W.v. Beveren, S.D. Franceschi, L.M.K.Vandersypen, S. Tarucha, and L.P. Kouwenhoven, 'Few-electron quantum dot circuit with integratedcharge read out', PRB 67, 161308 (2003).