Single Spin Qubits, Qubit Gates and Qubit Transfer with...
Transcript of 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
-18 dBm, fRabi≈0.75 MHz
-19 dBm, fRabi≈0.70 MHz
-21 dBm, fRabi≈0.525 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
Detector QDSource QD
“High” IQPC
“High” IQPC
Taking out an electron in SAW
Single Electron Transfer between Distant QDs
Detector QDSource QD
“High” IQPC
“Low” IQPC
Putting an electron in SAW onto detector QD
Single Electron Transfer between Distant QDs
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.