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

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

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

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

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

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

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

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

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

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

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

e-beam lithographye-beam lithography

suspended-bridge & shadow evapolation suspended-bridge & shadow evapolation

Thermometer

Cavity

Ibias lineVm line

Microwave line

To mixing chamber

A loop

Samples

Sample and Cavity

NTT Atsugi

DC measurement

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

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

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

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

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 )

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

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

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

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

↓

↑

Spectroscopy

EJ = 312 GHz, EC = 3.8, = 0.7

83CJ EE

= 2.6 GHz

after averaging

w/o averaging0.001M 2.4 GHz

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

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=

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

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

SpectroscopySpectroscopy

ground state

excited state

DC measurement

Pulse measurement

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

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

Readout after averaging

Expected Current

( canonical ensemble average )

qubit / 0

DC measurement

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

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

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

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)

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

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

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

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

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

                                                     

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

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

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