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Hideaki Takayanagi NTT Basic Research Laboratories, NTT Corporation, Japan
NTTNTT 物性科学基礎研究所物性科学基礎研究所
髙 柳 英 明髙 柳 英 明
Superconducting Flux Qubit as a Macroscopic Artificial Atom
Outline
1. Quantum Information Research at NTT2. Fux Qubit3. Single-Shot Measurement4. Multi-Photon Absorption5. Rabi Oscillation6. Conclusions
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Head: H. Takayanagi
About 20 researchers participate to the projectwhich consists of five sub-projects.
Four qubit-research projects and a quantum cryptography one.
QIT Project in NTT Basic Research Laboratories
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SQUID
Coupled QDs(artificial molecule) Exciton in QDs
Quantum gate operationRabi oscillationSingle-shot measurementMulti-photon absorptionRabi oscillation
Four Kinds of Qubit
Atom Chip
cooled atom
Solid-State Qubits
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Quantum cryptography with a single photon
電気光学変調器
AmpGene-rator
時間間隔解析器
時間間隔解析器
AliceBob
HeliumCryostat
Quantumdot
lens
Pin-holeLens
Single-modefiber Grating
Space filter
BeamSplitter
Half-wavelength ¼ wavelength
Splitter
50%-50%BeamSplitter
Detector 1
Detector 2
Detector 3
Detector 4
waveguide
Counter Photon 0
Mirror Attenuator
Titanium-Sapphire Laser
Lens
BeamSplitter
Testing
Nature, 420 (2002) 762
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0 2.5 5 7.5 10 12.5 15 17.5
0
2.5
5
7.5
10
12.5
15
17.5
Josephson persistent current QubitJosephson persistent current Qubit
Josephson Energy : cos( 2 a -+= EJU ) )cos (a- --1 cos- 2 2 f 1
2
Phase difference
+ + 2 f = 1 2
3q =
)( 2
1 p
1 2
)(2
1 m 1
2
qubit = f 0
EJ
1
2
p
m
=0.6
mp
U
=0.8
U
mp
=1.0
m p
U
=0.6
mp
U
=0.8
U
mp
=1.0
m p
U
=0.6
mp
U
=0.8
U
mp
=1.0
m p
U
=0.6
mp
U
=0.8
U
mp
=0.8
U
p
m
J. E. Mooij et al.,Science 285, 1036 (1999).
f = qubit / 0
f = qubit / 0 = 0.5
B
EJ
EJ
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Schematic qubit energy spectrum Schematic qubit energy spectrum
0.49 0.50 0.51
-10
-5
0
5
10
15
Ene
rgy
(GH
z)qubit /
0.4 0.5 0.6-100
0
100
Ene
rgy
(GH
z)
qubit/
)(
)(
2
1
f
f
5.00qubit f
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Three-Josephson-junction Loop:Description
)2cos(coscos2 2121 fE
U
J
Josephson Energy:f 2321
• Flux quantization:
)cos1( JJ EU
• Josephson Energy (1 junction):
• Coupling energy (1 junction):
EC = e2/ 2C EJ
C
EJ ; C
ext= f 0
3
1 2 EJ
C
J.E. Mooij 、 et al (1999)
<1.0
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Three-Josephson-junction Loop:Energy Diagram
)(2
1
)(2
1
21
21
m
p
2 minima in each unit cell.
m
p
p
U
m
Top View
f=0.5
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Three-Josephson-junction Loop: Dependence of the Potential
=0.6 =0.8
=1.0
pm
U
m m pp
UU
If increases, the barrier height :• increases between the two minima of one unit cell• decreases between the minima of adjacent cells
f=0.5
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Three-Josephson-junction Loop:Flux Dependence of the States
Classical states = persistent currents of
opposite sign. Degenerated at f = 0.5
Quantum tunnelling “anti-crossing”
Symmetric and antisymmetric superposition of the macroscopic
persistent currents Quantum ground state |0> Classical states
Quantum first excited state |1>
<Iq>/Ip
E0 (1) E Level splitting
/0
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Sample FabricationSample Fabrication
Qubit and a detector dc-SQUID
NTT Atsugi
Josephson junctionsAl / Al2O3 / Al
Junction areaSQUID : 0.1 x 0.08 m2
Qubit : 0.1 x L m2, ( = 0.8 ) L = 2 ~ 0.2
Loop size SQUID ~ 7 x 7 m2
Qubit ~ 5 x 5 m2
Mutual inductance M ~ 7 pH
• e-beam lithography
• Shadow evaporation
• Lift-off
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e-beam lithographye-beam lithography
suspended-bridge & shadow evapolation suspended-bridge & shadow evapolation
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Thermometer
Cavity
Ibias lineVm line
Microwave line
To mixing chamber
A loop
Samples
Sample and Cavity
NTT Atsugi
DC measurement
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R.T.
4.2K
1.2K
0.8K
0.4K
10mK
1 2 3 4 5
Twisted Constantan wire
100
HP 10dB
200
connectors
Heat anchor for outer shield
Sample box
•No on-chip capacitor and resistor •No on-chip control line•Change twisted wires to thin coaxial
cables to introduce dc-pulse
V
2.4mm connectors
Flexible coaxial cable
200 200
Loop antenna~ 1mm above the sample
HP 20dB
RF lineVII
10 nF
Through capacitor
attenuator
resistance
DC measurement
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I b
t
Vm
t
0
70 ~ 100 μsec
0
~ 100 nA
Readout through a dc-SQUIDReadout through a dc-SQUID
Vth(~30μV)
4~6 nA
Sweep Ib ( 140 Hz )
Tilt SQUID potential
Record each switching
when Vm = Vth~ 30 V
as a function of Isw
VmI b
qubit
IswIsw
Isw
Isw
~ 7 ms
DC measurement
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Readout with a dc-SQUIDReadout with a dc-SQUID
)cos(20
extcsw II
-1000
-500
0
500
1000
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8Voltage (mV)
Cur
rent
(nA)
Current is swept
I(V) curve
Isw(/ 0) curve
Magnetic field is swept
DC measurement
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Qubit step in the SQUID IQubit step in the SQUID Iswsw
Qubit switches its current sign
Flux in SQUID changes through M
SQUID Isw changes
Step on the Isw(/ 0)
M
dc-SQUID
Qubit
Φ
qubit / 0
DC measurement
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Josephson junctions : Al / Al2O3 / Al
Junction area : SQUID 0.2 x 0.2 m2
qubit 0.2 x L m2, L=0.3, 0.5, 1.0
SQUID
QubitI
Loop size : Lqubit = 5.1, 9.7, 19.0
(m) LSQUID = 6.3, 10.9, 20.2
LSQUIDLqubit
Parameter dependence of the qubit step
( , Ej, Ec )
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Two energy scale Ec, ETwo energy scale Ec, EJJ
H = H = EEcc - - EEJJ cos cos - - IIexex [n, [n,]=i]=i
Josephson energy : Josephson energy : EEJJ
charging energy : charging energy : EEcc =(2ne) =(2ne)22/(2C/(2CJ J ))
kkBBT << T << EEJJ <<<< E Ecc < < → charge qubit
kkBBT << T << EEcc <<<< E EJJ < < → phase 、 flux qubit
energy energy
Phase difference
-Number of
tunneled pair n
Pair tunneling
superconductor superconduct
orTunnel barrier
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QB# 5Junction area = 0.1 m2
Loop size : Lqubit = 9.7 m LSQUID = 10.9 m
QB# 8Junction area = 0.1 m2
Loop size : Lqubit = 19.0 m LSQUID = 20.2 m
~ 0.4GHz ~ kBT )
kBT~25mK
QB# 4Junction area = 0.06 m2
Loop size : Lqubit = 9.7 m LSQUID = 10.9 m
QB# 7Junction area = 0.06 m2
Loop size : Lqubit = 19.0 m LSQUID = 20.2 m
~ 2GHz > kBT )
QB# 6Junction area = 0.2 m2
Loop size : Lqubit = 9.7 m LSQUID = 10.9 m
QB# 3Junction area = 0.2 m2
Loop size : Lqubit = 5.1 m LSQUID = 6.3 m
~ 2MHz << kBT )
Qubit energy splitting
qubit / 0
qubit / 0qubit / 0qubit / 0
qubit / 0qubit / 0
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Calculated qubit energy level
Ej=544 GHzEc=1.6 GHzEj/Ec=338
Ej=280 GHzEc=3.2 GHzEj/Ec=87
Ej=130 GHzEc=5.4 GHzEj/Ec=24
=2 GHz
=0.4 GHz
=2 MHz
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Optimal operation point for SQUIDQubit signals appear at half-integer
Sensitivity of dc-SQUIDdepends on magnetic fields
We can achieve excellentresolution at f = 1.5
↓
↑
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Spectroscopy
EJ = 312 GHz, EC = 3.8, = 0.7
83CJ EE
= 2.6 GHz
after averaging
w/o averaging0.001M 2.4 GHz
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Qubit signals at different SQUID modulationQubit signals at different SQUID modulation
S/N depends on SQUID Isw
qubit and SQUID to be crossed
at small Isw
|>|>
|>
|>
design
T = 25 mK
DC measurement
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H 1
2
, ( f 0.5) , f ext /0
E0(1) () 2 2 , E 2 2
0 a L b R
1 b * L a* R
ˆ I p L L R R I p
I p 0 0 | ˆ I p | 0
( a2 b
2)I p
2 2I p
I p 1 1 | ˆ I p | 1
(a2 b
2)I p
2 2I p
L
R
Quantum ground state |0> Classical states
Quantum first excited state |1>
<Iq>/Ip
E0 (1) E Level splitting
/0
f=
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TkI
ee
Ie
IeII
Bp
Ep
E
pE
p
thermalp
E
2tanh
1
1
22
22
2222
10
Quantum ground state |0>
Classical states
Quantum first excited state |1>
<Iq>/Ip
E0 (1) E Level splitting
/0
Boltzman Distribution
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Schematic qubit energy spectrum Schematic qubit energy spectrum
0.49 0.50 0.51
-10
-5
0
5
10
15
Ene
rgy
(GH
z)qubit /
0.4 0.5 0.6-100
0
100
Ene
rgy
(GH
z)
qubit/
)(
)(
2
1
f
f
5.00qubit f
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SpectroscopySpectroscopy
ground state
excited state
DC measurement
Pulse measurement
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Readout without averaging
Single shot measurement into { l0>, l1> } bases
The <Iq> step shape does not change.
Only the population changes.
qubit / 0
DC measurement
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Close-up of Isw, T=25 mK
f f = 1.50102
Histogram is well separated !
0.001M 2.4 GHz
counts
counts
qubit / 0
DC measurement
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Readout after averaging
Expected Current
( canonical ensemble average )
qubit / 0
DC measurement
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Experimental setupExperimental setupR.T.
4.2K
1.2K
0.8K
0.4K
29mK
1 2 3 4 5
Thin coaxial cable 0.33 mm
HP 10dB
Samplecavity
Flexible coaxial cable
Terminator50
RF lineSLP-1.9
Weinschell10dB
On-chip strip line
Meanderfilters
VVII
V + V -I +I -
RF in
Sample cavity
RFin : 2 attenuatorsRFout : terminator
+ attenuatorDC : LP filter + Meander filter
RF in
Pulse measurement
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Multi-photon transition betweenMulti-photon transition between superposition of macroscopic quantum states superposition of macroscopic quantum states
E 0
(1)
1.5101.5051.5001.4951.490
qubit
/
h
< I
P >
T
1.5101.5051.5001.4951.490
qubit
/
1 1
1
12
3
32
233
2
2h
+
ー
( ) /√2 ground state
( ) /√2 1st excited state
Multi-photon transition
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Multi-photon spectroscopyMulti-photon spectroscopy
SQUID readout
-2
-1
0
1
2
d I
SW (
nA
)
1.5041.5021.5001.4981.496
qubit /
0
RF : 3.8 GHz
-10 dBm
1
12
23
2
1
0
-1
-2
d I
SW (
nA
)
1.5041.5021.5001.4981.496
qubit /
0
RF : 3.8 GHz
0 dBm
1
1 2
2
3
=0.86GHz
1-photon
2 -photon
Multi-photon transition
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110
100
90
80
70
I SW [n
A]
1.49121.4905 1.49421.4935
qubit / 0
1.49721.4965
data fitting
Multiphoton absorption at 9.1 GHz
single
off
10 dBm12 dBm
0 dBmPRF =- 21 dBm
12 dBm
off
off
9.6 dBm
13.2 dBm
doubletriple
RF Power dependence
20[dBm]]dBm[RF
RF10PI
500
400
300
200
100
0
HW
HM
[M
Hz]
43210IRF (arb. units)
singledoubletriple
3.0
2.0
1.0
0.0
amp
litu
de
[nA
]
43210IRF (arb. units)
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9100MHz
212
212
2
1222
1-
2
1222
1Amp
2
1][s HWHM
][rad/s HWHM
TT
TT
T
TT
T
TT
n
n
nn
nn
20rf
rf
rf10
)(P
nn
I
IsJ
power microwave : [dBm]
constant coupling :
point degeneracy at the splittingenergy : [MHz]5602[rad/s]
timedephasing : [s]
timerelaxation : [s]
dipth -n of ampletude : Amp
dipth -n of maxima halfat width half : ][s HWHM
rf
2
1
-1
P
s
T
Tn
n
----- (3)
------------------ (4)
)( rf00 IsJ
Multi-photon transition
Peak width vs MW intensityPeak width vs MW intensity
Bloch Kinetic Equation
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180 ns ~1μs
resonant microwave
Ib DC pulse
time
Pulse measurement schemePulse measurement schemerepetition: 3kHz ( 333 s)
SQUID switch
Non-switching
Pulse measurement
h
E ext )(
ext
I bias
Vout +
Ibias + SQUID Ibias -
Vout -
MW discrimination of the switching event
Non-switching event Switching event
V th
Switchingevents
Non-switchingevents
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55
50
45
40
35
30
25
Pro
bab
ilit
y [%
]
210
Delay Time [ s]
T1 = 1.6 s
data exp-fit
Relaxation time TRelaxation time T11
9.1 GHz 1 s pulse
030304_1 (1,2)FQB2
Ib pulse height 1.474 V, Trailing height ratio 0.6
1 s
500 ns3 s
delay time
0.49 0.50 0.51
-10
-5
0
5
10
15
Ene
rgy
(GH
z)
qubit /
Ground state
1st excited sate
MW
Pulse measurement
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Trailing height ratio 0.7
600 s
150 ns
Resonant MW pulse width
11.4 GHz
Quantum Oscillation : Rabi oscillationsQuantum Oscillation : Rabi oscillations
pulse width ( ns )
sw
itc
hin
g p
rob
ab
ilit
y (
% )
MW amplitude (a.u.)
R
ab
i fr
eq
ue
nc
y
( M
Hz
)
Dephasing time ~ 30 ns
Pulse measurement
NTT Atsugi
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SummarySummary
Future planFuture plan
• Spectroscopy of MQ artificial 2-level systemSpectroscopy of MQ artificial 2-level system• Qubit readout without averaging (DC)Qubit readout without averaging (DC)• Multi-photon transition between superposed MQ statesMulti-photon transition between superposed MQ states• Coherent quantum oscillation ( Rabi oscillation )Coherent quantum oscillation ( Rabi oscillation )
• TT11 ~1.6 ~1.6 s, Ts, T22 RabiRabi ~ 30 ns ~ 30 ns
• Ramsey, Spin echo Ramsey, Spin echo • Two qubit fabrication and operation Two qubit fabrication and operation • MQC with single shot resolutionMQC with single shot resolution
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NTT Basic Research Laboratories Hirotaka TanakaShiro SaitoHayato NakanoFrank DeppeTakayoshi MenoKouich Semba
Tokyo Institute of TechnologyMasahito Ueda
Yokohama National UniversityYoshihiro ShimazuTomoo Yokoyama
Tokyo Science UniversityTakuya MouriTatsuya Kutsuzawa
collaborators collaborators
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エネルギー固有状態を one-shot measurement で見た。
RL qubit
の時、
を測っている。12222
z)(1
2p
)(12 p
を測っているのではない。これを測ると、
0.50/
と の superposition は、生きている。
L R
LL 0 と の間のsuperposition は死んでいる。
LL 1
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0.5
time domain で真ん中に出る理由Qubit の磁場の量子力学的平均値を取っているからQubit の磁場は z のはず( projection) 。
intSQUIDqubit HHHH を使って、 time-dependent な Schrödinger方程式を解き、 SQUID の switching currentを求めると、 EJ/EC が小さくなると、ピークは1つ、反対に EJ/EC が大きくなると、ピークは2つになる。
0.5
ピーク1つ
ピーク2つ