Download Read-out characteristics

Mitglied der Helmholtz-Gemeinschaft
Quantum Computing with
Quantum Dots
IFF Spring School
18. March 2009
| Carola Meyer
Why a quantum computer?
18. March 2009 IFF Spring School
Folie 2
Quantum computing
quantum-bit (qubit)
0  
a
a1 0 + a2 1 = a1
2
1  
classical bit
1  ON  3.2 – 5.5 V
0  OFF  -0.5 – 0.8 V
calculation
decoherence
preparation
Y0

H

U
read-out
 -1
H

Y|A|Y
time
time
exponentially faster for Fourier transformation (Shor algorithm)
18. March 2009 IFF Spring School
Folie 3
„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
18. March 2009 IFF Spring School
Folie 4
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
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Quantum dots
single molecule
self-assembled QD
1 nm
metallic
(superconducting)
nanoparticle
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10 nm
lateral QD
100 nm
vertical QD
nanotube
1mm
nanowire
Folie 6
Confining Electrons in a Semiconductor
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Folie 7
From 3D to 0D
3D
2D
0D
D(E)
EF
EF
E
D(E)
E
1D
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Folie 8
Gate fabrication
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Folie 9
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
P|  = 99.9%
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Tel = 150 mK, B = 7 T, g = 0.44
Folie 10
transport measurements
source
drain
Vg
Kouwenhoven et al., Science 278, 1788 (`97)
18. March 2009 IFF Spring School
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Loss & DiVincenzo Proposal
Loss & DiVincenzo, Phys. Rev. A 57, 120 (1998)
gates
e
e
e
e
2DEG
• Quantum dots defined in 2DEG by gates
• Coulomb blockade used to fix number of electrons at one per dot
• Electron spin used as Qubit
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Folie 12
Loss & DiVincenzo: Qubit Manipulation
Loss & DiVincenzo, Phys. Rev. A 57, 120 (1998)
J-gates
gates
B
e
e
e
e
Bac
2DEG
back
gates
A-gates
high-g layer
• Qubit manipulation using spin resonance
• Addressing of single qubits by manipulation of g-factor
• 2 Qubit operations using J coupling
18. March 2009 IFF Spring School
Folie 13
„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
18. March 2009 IFF Spring School
Folie 14
Read-Out of Electron Spin
Requirements
• Charges are measured
→ Spin to charge conversion
• Read-out has to be fast enough
→ Shorter than T1 (spin energy relaxation)
• Back-action on qubit system should be small
→ decouple read-out from qubit system
18. March 2009 IFF Spring School
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QPC as charge detector
T
Working point: max. sensitivity to
electrostatic environment
IQPC
RESERVOIR
200 nm
DRAIN
Q
MP R
SOURCE
• 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
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QPC as charge detector
T
IQPC
RESERVOIR
200 nm
DRAIN
N
Q
MP R
N-1
N-2
SOURCE
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
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Folie 17
QPC as charge detector
T
IQPC
RESERVOIR
200 nm
DRAIN
Q
MP R
SOURCE
Measure differential conductance in IQPC
Coulomb oscillations in dot
can be detected by QPC
highly sensitive charge detector (1/8 e)
18. March 2009 IFF Spring School
allows to study QD even when isolated
from reservoirs (s. QuBits)
Folie 18
Read-Out of Electron Spin
Requirements
• Charges are measured
→ Spin to charge conversion
• Read-out has to be fast enough
→ Shorter than T1 (spin energy relaxation)
• Back-action on qubit system should be small
→ decouple read-out from qubit system
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RESERVOIR
How fast is the charge detection?
T
DRAIN
(a)
IQPC
Q
G
(b)
200 nm
M P R
SOURCE
• VSD = 1 mV
• IQPC ~ 30 nA
• ∆IQPC ~ 0.3 nA
• shortest steps ~ 8 µs
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• Observation of singel tunneling events
• Spontaneous back and forth tunneling
between dot and reservoir
(a) electron predominantly in reservoir
(b) electron predominantly in dot
Folie 20
Pulsed-induced tunneling
response
to electron
tunneling
DIQPC (nA)
0.8
response
to pulse
0.4
0.0
-0.4
0
0.5
1.0
Time (ms)
18. March 2009 IFF Spring School
1.5
Real time single
electron tunneling
Folie 21
Histograms tunnel time
3
3
G ~ (60 ms)-1
2
DIQPC (a.u.)
2
1
1
0
0
-1
0.0
G ~ (230 ms)-1
1.0
0.5
Time (ms)
1.5
-1
0.0
0.5
1.0
Time (ms)
1.5
Increase tunnel
barrier
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Spin read out principle:
convert spin to charge
charge
SPIN UP
0
-e
time
N=1
charge
SPIN DOWN
0
-e
N=1
18. March 2009 IFF Spring School
N=0
N=1
~G-1
time
Folie 23
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
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Read-Out of Electron Spin
Requirements
• Charges are measured
→ Spin to charge conversion
• Read-out has to be fast enough
→ Shorter than T1 (spin energy relaxation)
• Back-action on qubit system should be small
→ decouple read-out from qubit system
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Spin read-out procedure
DIQPC Vpulse
inject & wait
empty QD
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read-out spin
empty QD
Nature 430, 431(2004)
Folie 26
Spin read-out results
Elzerman et al., Nature 430, 431, 2004
read-out spin
empty QD
DIQPC (nA)
DIQPC Vpulse
inject & wait
2
“SPIN UP”
empty QD
“SPIN DOWN”
1
0
0
18. March 2009 IFF Spring School
0.5
1.0
Time (ms)
1.5
0
0.5
1.0
Time (ms)
1.5
Folie 27
More spin down traces
DIQPC (nA)
twait
tread
2
1
Threshold
value
thold
0
0
0.5
1.0
1.5
Time (ms)
thold : time the electron spends in the dot
tdetect : 1/G1 tunneling time
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Spin down fraction
Verification spin read-out
0.3
 twait 
  C exp  

 T1 
0.2
0.1
0.0
0.5
1.0
1.5
12
Waiting time (ms)
Spin flip
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Measurement of T1
B=8T
T1 ~ 0.85 ms
B = 10 T
T1 ~ 0.55 ms
•
•
Surprisingly long T1
T1 goes up at low B
B = 14 T
T1 ~ 0.12 ms
Elzerman et al., Nature 430, 431, 2004
18. March 2009 IFF Spring School
Folie 30
Read-Out of Electron Spin
Requirements
• Charges are measured
→ Spin to charge conversion
• Read-out has to be fast enough
→ Shorter than T1 (spin energy relaxation)
• Back-action on qubit system should be small
→ decouple read-out from qubit system
18. March 2009 IFF Spring School
Folie 31
„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
18. March 2009 IFF Spring School
Folie 32
quantum measurement
Any more questions about this point?
18. March 2009 IFF Spring School
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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)
18. March 2009 IFF Spring School
Folie 34
Alternative read-out scheme
Now:
tunnel-rate-selective
read-out (TR-RO)
G ES
>>
G GS
(1) t = 0 : position both levels above chemical potential
(2) electron will tunnel regardless of spin state
(3) t = t: with
-1 >> t
G GS
>> G -1
ES
high PR that electron was in state ES
low PR that electron was in state GS
18. March 2009 IFF Spring School
Folie 35
Alternative read-out scheme
Now:
tunnel-rate-selective
read-out (TR-RO)
G ES
>>
G 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
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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 | S 
(b) spin triplet state |T 
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Single-shot read-out
G T / G S  20
G S  2.5 kHz
G T  50 kHz
20 kHz low pass filter
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Single-shot read-out
G T / G S  20
G S  2.5 kHz
G T  50 kHz
20 kHz low pass filter
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On chip generation of oscillating magnetic fields
On-chip design
Minimum field Bac = 5 mT
fRabi ~ 30 MHz
Single Qubit gate operation
1/2fRabi ~ 15 ns
Compare to spin
coherence time
dissipation: 10 mW at 1 mT
250 mW at 5 mT
thermal “budget” dilution fridge:
300 mW at 100 mK
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absorption
Basics of electron spin resonance
energy
mS = 1/2
B0
magnetic field
DE = hn = giµBB0 = 30 µeV für n  9 GHz
18. March 2009 IFF Spring School
field
modulation
mS = -1/2
magnetic field
magnetic field
Folie 52
Detection of continuous wave ESR
Ground state |  
Engel & Loss, PRL 86, 4648 (01)
AC field lifts Coulomb blockade
Simple concept: BUT hard to prove that signal in current is due
to single spin rotation
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Photon-assisted tunneling
Electric field couples to charge for G< f:
- Electron in dot absorbs photon (N+1) → N
- Electron in lead absorbs photon N → (N+1)
Two side-peaks arise
e0 - hf
N electrons
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e0 + hf
N+1 electrons
Folie 54
Spin manipulation and detection
Initialization
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Pull dot levels far below
Fermi level to avoid PAT
Pulse spin down
level in bias window
Switch on hf to change
the spin state
Single shot read-out
Folie 56
Spin manipulation and detection
Double quantum dot with one electron in the right dot
T(0,2)
S(0,2)
Initialization
by spin blockade
18. March 2009 IFF Spring School
Pull dot levels far below
Fermi level to avoid PAT
Pulse spin down
level in bias window
Switch on hf to change
the spin state
Single
Read-out
shotbyread-out
lifted
spin blockade
Folie 57
Coherent Rabi oscillations
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Folie 58
Coherent Rabi oscillations
Idot large
Idot small
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Folie 59
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
• VP controls number of electrons in right dot
R
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Folie 60
Current in a double quantum dot
Vtgl
(2,1)
Vtgr
(2,2)
Vleft
(2,0)
Vtgm
source
(1,0)
(1,1)
(1,2)
(0,0)
(0,1)
(0,2)
drain
Vleft
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Vright
Vright
Folie 61
Current in a double quantum dot
412 103 8
(2,1)
41 2
00
Vtgl
Vtgm
Vtgr
(2,2)
Vleft
(0,2)
26
source
(1,0)
(1,1)
drain
(1,2)
h
e
(0,0)
(0,1)
(0,2)
Vleft
18. March 2009 IFF Spring School
Vright
Vright
Folie 62
Two electron double quantum dot
e0
VL
VR
• QPC can detect all charge transitions
• 2 electron double quantum dot
• Tuned between (1,1) and (0,2) state
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Folie 63
Spin configurations in a DQD
Spin-Singlet
Spin-Triplet
S=0
S = 1; ms = +1, 0, -1
antisymmetric
symmetric
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Folie 64
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
“Overhauser field”
• Nuclear spins are not fully polarized
 fluctuations lead to a field
• Singlet and Triplet states become mixed
• In an external magnetic field in <z>, |S and |T0 become mixed
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Folie 65
Harvard scheme
spin selection rules:
Singlet ground state
• (1,1) S can tunnel to (0,2) S
• (1,1) T to (0,2) S transition is blocked
Tilt potential:
new charge ground state
If charge does NOT return to (0,2) state, spin dephasing
(1, 1)
during time ts
B > 0:
(1,1) S and (1,1) To
mixing
t = ts:
transfer to (0,2) ground
state
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Harvard scheme
Interdot tunneling:
• hybridization (0,2) – (1,1)
• exchange splitting J(e)
Strength of J(e)
controlled by gates
B = 100 mT perp. field
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The logical Qubit
T2* ~ 8 ns
How long can the electrons
be separated spatially before
they loose phase coherence?
12
3
1. prepare singlet (0,2) S
2. rapid pulse (1 ns) : slow compared to tunnel splitting
separated singlet
3. separation time ts: rapid back projection into (0,2) S state
18. March 2009 IFF Spring School
Folie 68
Spin swap and Rabi oscillations
Slow detuning:
rotate S into
   S  To
for J

2
0
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Spin swap and Rabi oscillations
Read-out
  S
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  T0
Folie 70
Spin swap and Rabi oscillations
   S  To
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
   S  To
2
turn on J(e)

2
Folie 71
Spin SWAP and Rabi oscillations
18. March 2009 IFF Spring School
p
3p
5p
Folie 72
A universal set of quantum gates
Single qubit rotations and the CNOT gate form a universal set
• Single qubit rotations
Idot (fA)
100
• CNOT can be composed from single qubit rotations and √SWAP
Rotation of spin 2
18. March 2009 IFF Spring School
Rotation of spin 1
Folie 73
„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
18. March 2009 IFF Spring School
Folie 74
Entanglement and decoherence
18. March 2009 IFF Spring School
Folie 75
Singlet-triplet spin echo
• refocus separated singlet to undo inhomogeneous dephasing
• apply p pulse by pulsed J(e)
J (e ) t E /  p , 3p , 5p
18. March 2009 IFF Spring School
Folie 76
Singlet-triplet spin echo
Singlet probability as a function
of detuning and tE.
singlet recovery
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Folie 77
Singlet-triplet spin echo
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Folie 78
Spin-spin relaxation times
Spin dephasing time: ~
8 ns
Spin coherence time: ~ 1.2 ms
Time for √SWAP:
~ 180 ps
 about 7000 √SWAP operations
can be performed during T2
However
18. March 2009 IFF Spring School
Folie 79
„DiVincenzo“ Criteria
DiVincenzo:
Fortschr.
Phys. 48 (2000)
pp. 771-783 computer ?
Why
can’t
we already
buy9-11,
a quantum




()
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
18. March 2009 IFF Spring School
Folie 80
Spin energy relaxation
spin system is in excited state
1
relaxation to ground state due
to spin-phonon interaction
read-out within T1
nuclei:
T1 ~ hours – days
electrons: T1 ~ ms
0
dMz
M  M0
= g (Mx(t)By  My(t)Bx)  z
dt
T1
18. March 2009 IFF Spring School
Folie 81
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
18. March 2009 IFF Spring School
Folie 82
Different contributions
new eigenstates
Only acoustic phonons are relevant → linear dispersion relation

Matrix element:
Piezoelectric phonons dominate

Phonon wavelength much larger than dot size

18. March 2009 IFF Spring School
Folie 83
Breaking time reversal symmetry
All contributions would cancel out without magnetic field applied
 “van Vleck” cancellation
Follow one period of lattice vibration (harmonic oscillator)
SO
SO
B0

18. March 2009 IFF Spring School
Folie 84
Magnetic field dependence
All contributions add up to:
18. March 2009 IFF Spring School
G  DEZee5
Folie 85
Decoherence due to dephasing spins
magnetization in x,y-plane
(superposition)
1
1
superposition decays because
of dephasing
Slow fluctuations can be refocused
However:
Time ensemble is needed for presented Hahn-echo
0
0
From one Hahn-Echo sequence to the next
nuclear field takes a new, random and unknown value
18. March 2009 IFF Spring School
Folie 86
Magnetic field fluctuations
Unknown magnetic field
 electron spin evolves in an unknown way
BN
Gaussian distribution with standard deviation
In experiment:
T2* = 10 ns
^
=
BN = 2.3 mT
Reduce dephasing
 Find a way to decrease s of magnetic field
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Folie 87
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)
18. March 2009 IFF Spring School
Folie 88
Outlook
Why can’t we already buy a quantum computer ?
• All necessary components not yet implemented in the same device
• Gate implementation still too slow
• Scaling to ~1000 qubits not straight forward
Any solutions possible?
• 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)
18. March 2009 IFF Spring School
Folie 89
Dilbert
18. March 2009 IFF Spring School
Folie 90