single spin qubits, qubit gates and qubit transfer with...

Seigo Tarucha

Dep. of Appl. Phys. The Univ. of Tokyo

International School of Physics "Enrico Fermi”: Quantum Spintronics and Related Phenomena June 22-23, 2012 Varenna, Italy

Single Spin Qubits, Qubit Gates and Qubit Transfer with Quantum Dots

Micro-magnet technique for qubits and two qubit gates

• Two-qubit gate for controlling entanglement

• Fast Z-rotation and CPHASE

Qubit transfer

• Multiple QDs array

• Electron transfer between between distant QDs

… Joint project with Dr. Bauerle and Munier, Grenoble

Outline

Double QD for Qubit Gates

Individual two qubitsM. Pioro-Ladrier et al. Nat. Phys. 2008; T. Obata et a. PRB 2010

Two qubit gate for controlling entanglementR. Brunner et al. PRL 2011

10

|0> = |↑>, |1>=|↓>

1001

Time control of spin exchange coupling→ entangled state

Time control of spin rotation → single qubit

“Universal set of quantum gates”

Qubit Hamiltonian

Hqubit = (z/2)[|0><0| - |1><1|] + (x(t)/2)[|0><1| + |1><0|]

Well-defined two states|0> and |1>

Hybridization of two states

z

|0>…-z/2

|1>…z/2

= (z/2)z + (x/2) x

= +

z Define two eigen states

x Mix up two eigen states

|0>=|↑>|1>=|↓>

S-T0 Qubit Hamiltonian

Hqubit = (-J/2)[|S><S| - |T0><T0|] + gBBnz[|S><T0| + |T0><S|]

= (-J/2)z + gBBznuc x

=|↑↓>

|↑↓>|↓↑>

|S>

|T0>

Bnucz

= |↓↑>

Temporal J manipulation with inter-dot detuning:

J

J >> Bznuc |↑↓> ↔ |↓↑>

J << Bznuc |S> ↔ |T0>

|0>=|S>|1>=|T0>

Charge Qubit with a Tunnel-coupled Double QD

E0 = (EL+ ER)/2 = 0 EL=-/2, ER=/2

EL

ER

e e

H = (-1/2)[(|0><0|-|1><1|)-2t(|0><1|+|1><0|)]

E0

=(-1/2)[z +2tx] where B0 = (,0,) and BAC=(2t,0,0)

=ħB0 2t= ħB1cos(t)

2t

0 1

z

y

x

B0/ħ

BAC2t/ħ

Fictitious magnetic fields for universal rotation

Microwave

Non-adiabatic

B0

BAC

EZeeman=hfAC

Single Spin Qubit with QD

|0>=|↑>|1>=|↓> B = B0z + B1xcost

HESR = -(1/2) ħ0z – (1/2) ħ1xcost

Qubit concept = Electron spin resonance Loss and DiVincenzoPRA 1998

Single spin qubits:(GaAs) Koppens et al. Science 2006

Nowack et al. Science 2007 Pioro-Ladriere, et al. Nat. Phys. 2008 Obata et al. PRB 2010Brunner et al. PRL 2011

(InAs) Nadj-Perge et al. Nature 2010

Apply ESR for single dot with a single electronusing a local AC B1 field

|↑>

|↓>

B1

Qubit Hamiltonian for Spin RotationHqubit = (-z/2)z + (-x/2) x

Time evolution

Larmor precession

Qubit Gates

|↓>

|↑>

xy

zRx() =2fRabit

Ry() = Rx(: t→t-/2) =2fRabit

Rz() = Rx(/2)Ry()Rx(/2)

NOT : Rx()

HADAMARD : Rx(/2)

RxRy

RzROTATION

…Temporal detuning of EZeeman2tfL (EZeeman=hfL)

Our proposal : Micro-magnet technique

0Bx(ｘ)e

Pioro-Ladriere et al. Nat. Phys. 2008

-magnet

Tokura et al. PRL 2006

Oscillation of an electron under a stray field by ac voltage

Local AC B Field Driven by AC Current or AC Voltage

ac B

DC B0 ac I

AC current to on-chip coil

IAC = 1 mA Bac~ 1 mT rotation: ~ 80 ns

Koppens et al. Science 2006

Nowack et al. Science 2007

S

From voltage to B fieldUsig spin-orbit interaction

Bloc=(Exp)

Spin Qubit with a Micro-magnet Technique

[

x [100]

y [010]

110]

Hspin = HZeeman + HSOHZeeman= (1/2)gBBext

HSO= (-pyx + pxy) + (-pxx+pyy)

Rashba Dresselhaus

Local B Field Generation by SO Effect

EacBext

Levitov and Rashba, PRB 2003Golovach et al. PRB 2006

lSO-1 =m*/(+)h for Bext, Eac//[110]

fRabi=(gB|Beff|)/2h ∝ Eac‧Beff

[110]

=m*/(-)h for Bext, Eac//[110]

fRabi ∝Eac‧Bext

Nowack et al. Science 2007

∝Bext

Spin Qubit Using SO Effect : GaAs QD

Rabi oscillations fRabi ∝Eac Eac increase

[110]

Micro-magnet for Spin Qubit with Quantum Dot

magnet

ｚ

ｘ

Stray field

Bext

～T/m

Out-of plane Bx(ｚ)

10 to50 mT/0.1mIn-plane Bz = excess local Zeeman field

B0= Bext +Bz∝ fESR

fac

Bac = B1xcos2fac

75

bSL ~ 0.6 T/m (saturation)

z (m)‐0.5 0.5

‐75

B x(m

T)

0 bSL

0

dot 1 dot 2

B0 M 70 nm

80 nm

90 nm

300 nmMCo = 1.8 T

0.3 m

Simulation

Addressing Two Individual Spin Qubits

Bext= BESR‐Bz1 =BESR‐Bz2Local DC B0+Local AC BAC

To manipulate more spins in a multiple QD:

MMM M

In-plane stray field Bz at each dot depends on the micro-magnet geometry

B0=Bext +Bz

MW

gBBESR=hfMW

Local Zeeman field

ESR at two different Bext

Formation of a triplet state blocks electron transition…Pauli spin blockade

Ono and ST, Science 2002PRL 2004

P-SB is lifted by spin flip…most sensitive spin information detector

ESR/Qubit Readout using Pauli Effect

IQPC

External fieldQ

PC

cur

rent

EZeeman = hf

Two Spin Qubits in Double Quantum Dot

B0R B0L

@ 1T CW EDSRLeft spin Right spin

B0R

B0L

QPC charge sensing to detect to charge change in the double dot

15 mT

Left dotRight dot

T. Obata et al. PRB 2010R. Brunner et al. PRL 2011

-17 dBm, fRabi≈0.85 MHz




Right dotLeft dot

Rabi Frequencies vs. MW FieldRabi frequency ~ PMW

1/2 = EMW

Left spin

Right spin

Obata et al. PRB 2010

Brunner et al. PRL2010

Entanglement Control

Exchange = (-J/4)1·2

Hexc = (-J/4)1·2= -J(1/4)(I + 1·2) + (J/4)I= (-J/2)USWAP + (J/4)I

Temporal Operation of Spin Exchange Coupling

exp[-iHexct/ℏ]

= exp[iJUSWAPt/2ℏ]exp[-iJt/4ℏ]

= Icos(Jt/2ℏ) + iUSWAPsin(Jt/2ℏ)

= iUSWAP for Jt=h/2

U√SWAP for Jt=h/4

Time evolution

How to control exchange coupling?

S(0,2)

(N1,N2)=(0,0)

S(1,1)

(1,0)

(0,1)

S(2,0)

(2,2)(1,2)

(2,1)Detuning

S(1,1)

S(1,1)

S(1,1) S(0,2)

S(2,0)

N1

N2

Detuning

Tokura et al. Springer 2009

S(1,1)

Two-electron States in Double QDSmall detuning : J=4t 2/UH

Large detuning: J

T0(1,1)T-

T+

Energy detuning = E1-E2

S(0,2)-S(1,1) S(2,0)-S(1,1)

T(1,1)

S(0,2)S(2,0)

S(1,1), S(0,2)T+, T-, T0(1,1)

Etriplet=0

S(0,2)

J~√2t

S(0,2)-S(1,1)

Detuning

T(1,1)

S(0,2)

T-

T+

T0

0

Ene

rgy

S(1,1)

Control of Pauli Blockade and Exchange Coupling

Pauli spin blockadeK. Ono et al.

Science 2002

S(0,2)

(N1,N2)=(0,0)

S(1,1)

(1,0)

(0,1)

S(2,0)

(2,2)(1,2)

(2,1)

S(0,2)-S(1,1)

Detuning

J = Etriplet - Esinglet

J~0J>0

J

S(0,2)

J~√2t

S(0,2)-S(1,1)

Detuning

S(0,2)

T-

T+

T0

0

Ene

rgy

S(1,1)

Quantum Gate Operation with Double QD

Pauli spin blockade

(2,1)

J~0J>0

J = Etriplet - Esinglet

Use for initialize and readout

J switch for entanglement control

Spin rotation at J=0

Exchange control at J≠0

Magic basis: Bell statesHill and Wooters PRL 1997

12121212

Concurrence of Two Qubit Entanglement

Preparation

|↑>|↓> with J=0

Exchange operation

Jex = ħex with finite Jex =/2 for √SWAP

= for SWAP

Readout of S0using Pauli effect

Change of chargestate from (1,1) to (0.2)

|ex cos ex| ↑↓ sin ex ↓↑ exp ex

P:(1

,1)→

(0,2

)

0

1

ex/SWAP0 1 3 42

Concurrence=|sin2ex|

Hill and Wooters, PRL1997

|↑>|↑>

Initialization

Two-qubit Gate to Control and Detect Single State

/2 rotation Exchange control

What we measure is P:(1,1) to (0,2)

Controlled Manipulation of the Two Spin Exchange

Single weight: Pbright (ex)

…1/4: partially entangled

…1/2 : simple product

…0 : simple product

| ↑⟩|↑⟩ | ↑⟩ |↓⟩

⨂|↑⟩ | ⟩ | ⟩

exc :SWAP |2> = (1/2)(T+ - T- - √2S0)

√SWAP 1 [(2+i)T+ -iT- +i√2S0)]23/2

|2> =

SWAP2n |2> = T+

R. Brunner et al. PRL 2011

ex/SWAP10 2 3 40

0.5

P:(1

,1) →

(0,2

)=P

(S0)

Concurrence

SWAP fidelity ~ 98%Rabi fidelity ~ 50%

Exchange + Spin rotationProbability of finding the singlet in the output |2>R. Brunner et al. PRL 2011

Calculation of concurrence using parameters derived from experiments

Control-PHASE

Loss & DiVincenzo, PRA 1998CNOT with rotations of 1 spin and SWAP/SWAP1/2

HH|↑↑>|↑↓>|↓↑>|↓↓>

|↑↑>|↑↓>|↓↑>|↓↓>

Input Output

Quantum circuit for CNOT

z z z

Rz(θ) = XRy(θ) X = Y Rx(−θ) Y

Rotation about z

Preparation of 4 logical basis statesX2+SWAP

|↓>

|↑>

xy

z

RxRy

Rz

Multiple qubit system in multiple QDs

Connection between Distant Qubit Systems

Connection between distant quantum systems with qubit state transfer

Photons：Quantum cryptography,communication

a| > + b| >

Multiple quantum dots in an inhomogeneous Zeeman field

50 nm

150 nm

200 nm

-1 m 1 m

13 bits

T. Takakura et al. APL 2011

2DEG depth

Inhomogenous B Field Induced by Micro-magnet

Electron Transfer by Surface Acoustic Wave (SAW)

piezoelectric（GaAs)

IDT

I = efSAW

1 nsec for 3 m<< spin T2

2efSAW

SAW Induced Electron Transport

d d Gate electrode

I = nefEnergy

Single Electron Transfer between Distant QDs

3 m

Detector QDSource QD

“Low” IQPC

“High” IQPC

Electron loading in source QD

Charge states of two QDs probed by two QPC charge sensors

QDQD


“High” IQPC

“High” IQPC

Taking out an electron in SAW



“High” IQPC

“Low” IQPC

Putting an electron in SAW onto detector QD


Before SAW

After SAW

Before SAW

After SAW

Before SAW

After SAW

Before SAW

After SAW

Transfer time < T2* (=25 nsec )

16m-D23

SAW SAW

Single Electron Transfer between Distant QDs> 94 %

McNeil et al. Nature 477, 439 (2011)

Summary

• Developed a micro-magnet technique for making single spin qubits, which is applicable to non-magnetic material systems, i.e. Si, C-based.

• Realized a two-qubit gate of combined spin rotation and exchange control, which enables control and detect the partial entanglement.

• Proposed micro-magnet techniques to implement fast z-rotation and CPHASE.

• Proposed a SAW technique of transferring qubitstate between distant QDs.

single spin qubits, qubit gates and qubit transfer with...

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