18. March 2009 Mitglied der Helmholtz-Gemeinschaft Quantum Computing with Quantum Dots IFF Spring...

18. March 2009

Mit

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d d

er

Helm

holt

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sch

aft

Quantum Computing with Quantum DotsIFF Spring School

| Carola Meyer

18. March 2009 IFF Spring School Folie 2

Why a quantum computer?


Quantum computing

calculationpreparation read-outtime

classical bit

1 ON 3.2 – 5.5 V

0 OFF -0.5 – 0.8 V

exponentially faster for Fourier transformation (Shor algorithm)

quantum-bit (qubit)

0 1

a10 + a21 =a1a2

H H-1U |A|

time

decoherence


„DiVincenzo“ Criteria

DiVincenzo: Fortschr. Phys. 48 (2000) 9-11, pp. 771-783

A scalable system with well characterized qubits

A qubit-specific measurement capability A („read-out“)

The ability to initialize the state of the qubits to a simple fiducial state, e.g. |00...0>

A „universal“ set of quantum gates U

Long relevant decoherence times, much longer than the gate operation time


Outline

Part I Brief introduction to quantum dots and transport

How can this be used to build a quantum computer?

Measurement of spin states Fast charge measurement Spin to charge conversion

Part II Manipulation of single qubits

SWAP: implementation of a two-qubit gate

Relaxation


Quantum dots

single molecule

metallic (superconducting) nanoparticle

self-assembled QD nanotube

nanowire

1 nm 10 nm 1m

vertical QD

lateral QD

100 nm


Confining Electrons in a Semiconductor


From 3D to 0D

E

D(E)

EFEF

D(E)

E

3D

2D

1D

0D


Gate fabrication


Real Quantum Dot structures

• Ohmic contacts by RTA of Ni/Au/Ge (diffusion from surface to 2DEG)

• Electrical control of dot potential and tunnel barriers

• Electron spins can be polarized with large B and low T

Tel = 150 mK, B = 7 T, g = 0.44| P = 99.9%


transport measurements

source drain

Vg

source drain

Vg

source drain

Vg

Kouwenhoven et al., Science 278, 1788 (`97)


Loss & DiVincenzo Proposal

Loss & DiVincenzo, Phys. Rev. A 57, 120 (1998)

2DEG

gates

• Quantum dots defined in 2DEG by gates

• Coulomb blockade used to fix number of electrons at one per dot

e e e e

• Electron spin used as Qubit


Loss & DiVincenzo: Qubit Manipulation

2DEG

gates

e e e e

high-g layer

B

• Addressing of single qubits by manipulation of g-factor

Loss & DiVincenzo, Phys. Rev. A 57, 120 (1998)

back gates

• Qubit manipulation using spin resonance

Bac

• 2 Qubit operations using J coupling

J-gates

A-gates









( )


Read-Out of Electron Spin

Requirements

• Read-out has to be fast enough→ Shorter than T1 (spin energy relaxation)

• Charges are measured→ Spin to charge conversion

• Back-action on qubit system should be small→ decouple read-out from qubit system


QPC as charge detector

IQPC

DRAIN

SOURCE

RE

SE

RV

OIR

200 nm M R

Q

T

P

• Define QPC by negative voltage on R and Q

• Tune S-D conductance to last plateau at working point

• Change number of electrons in dot: make VM more negative

Working point: max. sensitivity to electrostatic environment



IQPC

DRAIN

SOURCE

RE

SE

RV

OIR

200 nm M R

Q

T

P

N

N-1N-2

Reduce number of electrons in dot:

Change in charge lifts the electrostatic

potential at the QPC constriction,

resulting in a step-like feature in IQPC

Enhance sensitivity



IQPC

DRAIN

SOURCE

RE

SE

RV

OIR

200 nm M R

Q

T

P

Measure differential conductance in IQPC

Coulomb oscillations in dot can be detected by QPC

highly sensitive charge detector (1/8 e)

allows to study QD even when isolated from reservoirs (s. QuBits)



Requirements





How fast is the charge detection?

IQPC

DRAIN

SOURCE

RE

SE

RV

OIR

200 nm M P R

Q

T

• VSD = 1 mV

• IQPC ~ 30 nA• ∆IQPC ~ 0.3 nA

• Observation of singel tunneling events

• Spontaneous back and forth tunneling between dot and reservoir

(a) electron predominantly in reservoir (b) electron predominantly in dot

(a)

(b)

• shortest steps ~ 8 µs


Pulsed-induced tunneling

responseto pulse

IQ

PC (

nA)

Time(ms)

0 0.5 1.0 1.5

responseto electrontunneling

0.0

0.4

0.8

-0.4Real time single electron tunneling


Histograms tunnel time

~ (60 s)-1

0.0 0.5 1.0 1.5-1

0

1

2

3

IQ

PC (

a.u

.)

Time (ms)

~ (230 s)-1

Increase tunnelbarrier

0.0 0.5 1.0 1.5-1

0

1

2

3

Time (ms)


Spin read out principle:

N = 1

SPIN UP

time

charge

0

-e

N = 1 N = 1N = 0

SPIN DOWN

time

charge

0

-e

-1

convert spin to charge


Initialization

Energy selective tunneling

• spin-up will stay in dot

• spin down will tunnel

• wait a few tunneling processes (high polarization in state)

• fast initialization process

|



Requirements





Spin read-out procedure

inject & waitempty QD

Vp

uls

e

read-out spinempty QD

IQ

PC

Nature 430, 431(2004)


Spin read-out results

inject & waitempty QD

Vp

uls

e

read-out spinempty QD

IQ

PC

“SPIN DOWN”

Time (ms)

0 1.00.5 1.5

“SPIN UP”

Time (ms)

0 1.00.5

IQ

PC (

nA)

0

1

2

1.5

Elzerman et al., Nature 430, 431, 2004


More spin down traces

Time (ms)

0 1.5

IQ

PC (

nA)

0

1

2

1.00.5

treadtwait

thold

Thresholdvalue

thold : time the electron spends in the dot

tdetect : 1/1 tunneling time


Verification spin read-out

Waiting time (ms)

Spi

n do

wn

frac

tion

1

exp wait

T

tC

Spin flip


Measurement of T1

B = 8 TT1 ~ 0.85 ms

B = 10 TT1 ~ 0.55 ms

B = 14 TT1 ~ 0.12 ms

• Surprisingly long T1

• T1 goes up at low B

Elzerman et al., Nature 430, 431, 2004



Requirements












( )


quantum measurement

Any more questions about this point?


Drawbacks of read-out

So far:energy-selective read-out(E-RO)

Drawbacks:

(1) energy splitting must be larger than thermal energy

(2) very sensitive to fluctuations in electrostatic potential

(3) high-frequency noise can spoil E-RO (photo-assisted

tunneling)


(3) t = : with

high PR that electron was in state ES

low PR that electron was in state GS

Alternative read-out scheme

Now:

tunnel-rate-selective read-out (TR-RO)

(1) t = 0 : position both levels above chemical potential

(2) electron will tunnel regardless of spin state

ES GS >>

-1 -1GS ES >> >>


Alternative read-out scheme

Now:

tunnel-rate-selective read-out (TR-RO)

ES GS

Advantage:

(1) does NOT rely on large energy splitting

(2) robust against background charge fluctuations

(cause small variation of tunneling rate)

(3) photon-assisted tunneling not important

>>


Singlet-triplet read-out

Experimental conditions:

(1) can be achieved in Quantum Hall regime, where high spin-selectivity is induced by spatial separation of spin-resolved edge channels

(2) can be used for read-out of two-electron dot with electrons in

(a) spin singlet ground state (b) spin triplet state

| S |T


Single-shot read-out

T S/ 20

S 2.5 kHz

T 50 kHz

20 kHz low pass filter


On chip generation of oscillating magnetic fields

Minimum field Bac = 5 mT

fRabi ~ 30 MHz

Single Qubit gate operation

1/2fRabi ~ 15 ns

On-chip design

dissipation: 10 W at 1 mT

250 W at 5 mT

thermal “budget” dilution fridge:

300 W at 100 mK

Compare to spin coherence time


Basics of electron spin resonance

m agne tic fie ld

ab

sorp

tion

m agne tic fie ld

fieldm

odulation

E = h= giµBB0 = 30 µeV für GHz

mS = 1/2

mS = -1/2

B0ener

gy

magnetic field


Detection of continuous wave ESR

| Ground state

AC field lifts Coulomb blockade

Simple concept: BUT hard to prove that signal in current is due to single spin rotation

Engel & Loss, PRL 86, 4648 (01)


Photon-assisted tunneling

0 - hf

N electrons N+1 electrons

0 + hf

- Electron in dot absorbs photon (N+1) → N- Electron in lead absorbs photon N → (N+1)Two side-peaks arise

Electric field couples to charge for < f:


Spin manipulation and detection

Initialization Pull dot levels far below Fermi level to avoid PAT

Switch on hf to change the spin state

Single shot read-out

Pulse spin down level in bias window


Spin manipulation and detection

Initialization Pull dot levels far below Fermi level to avoid PAT

Switch on hf to change the spin state

Single shot read-out

S(0,2)

T(0,2)

by spin blockade

Double quantum dot with one electron in the right dot

Pulse spin down level in bias window

Read-out by lifted spin blockade


Coherent Rabi oscillations


Coherent Rabi oscillations

Idot large

Idot small


SWAP gate implementation in a Double Quantum Dot

Few electron double quantum dot

• Fully tunable 2Qubit system

• Quantum point contact (QPC)as charge detector

• Measure dIQPC/dVL : change of total electron number in double dot

• VL controls number of electrons in left dot

• VPR controls number of electrons in right dot


source

Vleft

drain

Vright

Vtgl Vtgm Vtgr

Current in a double quantum dot

(0,0)

(1,0)

(0,1) (0,2)

(1,1)

(2,0)

(1,2)

Vright

(2,1) (2,2)

Vle

ft


source

Vleft

drain

Vright

Vtgl Vtgm Vtgr

Current in a double quantum dot

(0,0)

(1,0)

(0,1) (0,2)

(1,1)

(0,2)

(1,2)

(2,1) (2,2)

e

h

01234 024681012

Vright

Vle

ft


VL VR

Two electron double quantum dot

• QPC can detect all charge transitions• 2 electron double quantum dot• Tuned between (1,1) and (0,2) state


Spin configurations in a DQD

Spin-Singlet

S = 0

antisymmetric

Spin-Triplet

S = 1; ms = +1, 0, -1

symmetric


Hyperfine coupling in a DQD

• Ga and Ar have a nuclear spin:

about 106 nuclear spins in a quantum dot

• Electrons feel a magnetic field due to hyperfine interaction with these nuclei

• Nuclear spins are not fully polarized

fluctuations lead to a field

• Singlet and Triplet states become mixed

“Overhauser field”

• In an external magnetic field in <z>, |S and |T0 become mixed


Harvard scheme

Singlet ground state

Tilt potential:new charge ground state(1, 1)

B > 0:(1,1) S and (1,1) To

mixing

t = s:transfer to (0,2) ground state

spin selection rules:

• (1,1) S can tunnel to (0,2) S

• (1,1) T to (0,2) S transition is blocked

If charge does NOT return to (0,2) state, spin dephasing

during time s


Harvard scheme

Interdot tunneling:

• hybridization (0,2) – (1,1)

• exchange splitting J()

B = 100 mT perp. field

Strength of J()

controlled by gates


The logical Qubit

1. prepare singlet (0,2) S2. rapid pulse (1 ns) : slow compared to tunnel splitting

separated singlet3. separation time s: rapid back projection into (0,2) S state

1 2 3

How long can the electrons

be separated spatially before

they loose phase coherence?

T2* ~ 8 ns


Spin swap and Rabi oscillations

2oS T

Slow detuning:

rotate intoS

for J 0



S

Read-out

0T



2oS T 2oS T

turn on J()


Spin SWAP and Rabi oscillations


• CNOT can be composed from single qubit rotations and √SWAP

A universal set of quantum gates

Single qubit rotations and the CNOT gate form a universal set

• Single qubit rotations

I dot (

fA)

100

Rotation of spin 1Rotation of spin 2









( )


Entanglement and decoherence


Singlet-triplet spin echo

• refocus separated singlet to undo inhomogeneous dephasing

• apply pulse by pulsed J()

( ) / , 3 , 5EJ



Singlet probability as a functionof detuning and E.

singlet recovery


Spin-spin relaxation times

Spin dephasing time: ~ 8 ns

Spin coherence time: ~ 1.2 s

Time for √SWAP: ~ 180 ps

about 7000 √SWAP operations can be performed during T2

However









( )

Why can’t we already buy a quantum computer ?Why can’t we already buy a quantum computer ?

( )


Spin energy relaxation

nuclei: T1 ~ hours – dayselectrons: T1 ~ ms

spin system is in excited state

relaxation to ground state due to spin-phonon interaction

read-out within T1

dMz

dt= (Mx(t)By My(t)Bx)

Mz M0

T1


Origin of spin-phonon coupling

Spin-orbit interaction is the most important contribution

HSO cannot couple different spin states of the same orbital

New eigenstates can couple to the electric field

Lattice vibrations lead to fluctuations of the electric field

Spin relaxation


Different contributionsnew eigenstates

Only acoustic phonons are relevant → linear dispersion relation

Matrix element:

Piezoelectric phonons dominate

Phonon wavelength much larger than dot size


Breaking time reversal symmetry

All contributions would cancel out without magnetic field applied

Follow one period of lattice vibration (harmonic oscillator)

“van Vleck” cancellation

SO

SO

B0


Magnetic field dependence

All contributions add up to: EZee5


Decoherence due to dephasing spins

magnetization in x,y-plane(superposition)

superposition decays because of dephasing

Slow fluctuations can be refocused

Time ensemble is needed for presented Hahn-echo

However:

From one Hahn-Echo sequence to the next nuclear field takes a new, random and unknown value


Magnetic field fluctuations

Unknown magnetic field

electron spin evolves in an unknown way

Gaussian distribution with standard deviation

T2* = 10 ns

In experiment:

=̂ BN = 2.3 mT

Reduce dephasing

Find a way to decrease of magnetic field

BN


Summary

Proposal for quantum computing with quantum dots

electron spin as qubit

exchange interaction as qubit coupling

Single spin read-out

spin to charge conversion

quantum point contact as charge detector

spin-energy relaxation time (T1) measurement

Quantum gates

single spin rotation

SWAP operation between two qubits

spin-phase relaxation time (T2) measurement

Origin of spin relaxation

spin orbit coupling (T1)

nuclear hyperfine field (T2)


Outlook

• All necessary components not yet implemented in the same device

• Gate implementation still too slow

• Scaling to ~1000 qubits not straight forward

• Improve T2 : Polarize nuclei to >99%

Find materials without nuclear spins and SO coupling

→ carbon based (graphene, carbon nanotubes)

→ silicon (2DEG charge carrier mobility too low)

Any solutions possible?Any solutions possible?

Why can’t we already buy a quantum computer ?Why can’t we already buy a quantum computer ?


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