1

Paolo Silvestrini

Macroscopic Quantum Coherence and Quantum Computing

Seconda Università di Napoli -Dip. di Ingegneria dell’Informazione CNR - Istituto di Cibernetica “Eduardo Caianiello”

MQC group in Naples

Valentina CoratoCarmine Granata Sara RombettoBerardo Ruggiero Maurizio RussoRoberto Russo

2

Back to basics…

1|0| ba

Fundamental carrier of information: the bit

Possible qubit states: any superposition describedby the wavefunction

“0” “1”or

Fundamental carrier of quantum information: the qubit

Possible bit states:

3

10

122

qbit

1)2/sin(0)2/cos( ie

Z

1

0

X

Y

4

Quantum computationwith chloroform NMR

Deutsch algorithmdemonstrated.

1H

13C

Cl

Cl Cl

5

Five Criteria for physical implementation of a quantum computer

1. Well defined extendible qubit array -stable memory

2. Preparable in the “000…” state3. Long decoherence time (>104 operation time)4. Universal set of gate operations5. Single-quantum measurements

D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” quant-ph/0002077.

6

Physical systems actively consideredfor quantum computer implementation

• Liquid-state NMR

• NMR spin lattices

• Linear ion-trap spectroscopy

• Neutral-atom optical lattices

• Cavity QED + atoms

• Linear optics with single photons

• Nitrogen vacancies in diamond

• Electrons on liquid He

• Josephson devices

– “charge” qubits

– “flux” qubits

– “phase” qubits

• Spin spectroscopies, impurities in semiconductors

• Coupled quantum dots

– Qubits: spin,charge,excitons

7

Dissipation effects (Chakravarty and Leggett PRL 1984, Grabert and Weiss PRL 1984)

Long decoherence time Low dissipation Low T

0 1 2 3 4-1,0

-0,5

0,0

0,5

1,0

P(t* )

t*

R1<R2< R3

P(t)

8

NISTChalmers

NEC

TU Delft

Superconducting Josephson qubits

Schoelkopf et al, Yale

NEW: « atom in cavity » :

(N)

N

12 N

flux

1~2N

phasecharge

charge-phase

quantronium

9

Tilt Washboard Potential

sin

1

22

2

cNdc Idt

d

Redt

dC

eII

0tIN

R

TKtItI B

NN 2

JJR

CIN Idc

-50

0

50

I(A)

-4 -2 0 2 4

V(mV)Stat

o Jo

seph

son

U

Stato resistivo

cos2

tIe

U c

10

-50

0

50

I(A)

-4 -2 0 2 4

V(mV)

11

0

cL

150

LI2

weber1007.2e2

h

rf-SQUID

00

Lx

0

2sin2

n22• Quantizzazione del flussoide

• Effetto Josephson

x rf SQUID

U

2

xL0x 2

1cosU,U

L4U

2

20

0

x

= x

12

Iecc

V o u t

I b i a s

Li

Ip

LP

Is

excitation coil

rfSQUID

dcSQUID

flux transformer

Controls:

rf-SQUID

x• Simmetry of potential

xdc rf

SQUID•Hight of the barrier

Flux transformer

Stacked JJ

• Coupling to the readout dc-SQUID

13

350 m

dc-SQUIDflux

transformer

rf-SQUIDmodulation and feedback coils

flux transformer

rf-SQUIDexcitation

coil

Josephson junction

50 m

14

A. J. Leggett, Prog. Theor. Phys. 69, 80 (1980)

Rf-SQUID Potential

|L> |R>

LAS 2

1,

R+-

I I

15

k

jjkkkjjjj WW

dt

d

Quantizied Energy levels

Tunnel rate

Potential Potential barrier

Wjk

j

Quantum picture

16

U

cx

U

0

U

q eA

2

1

00q

UA

Quantum Tunneling

B

00 K2

TT

RC

LCL

0

2/1

min0

87.012.7

cos1

TK

U

TBeAT

CRAT

1 TKBeRR

0

U

cx

U

Thermal Activation

BKTT

20

0

17

Measurements of Macroscopic Quantum Tunneling out of the Zero-Voltage of a Current-Biased Josephson Junction

M. H. Devoret, J. M. Martinis, and J. Clarke, PRL 55, 1543 (1985)

18

102

103

104

10-2

10-1

1 data

-1/TR-1

Q

P. Silvestrini, R. Cristiano, S. Pagano, O. Liengme, and K. E. Gray,“Effect of dissipation on Thermal Activation in an underdamped Josephson

Junction: First evidence of a Transition between Different Damping Regimes”PRL 60, 844 (1988)

19

25,0 25,2 25,4 25,6 25,8 26,00

3

6

9

01214 10 8 6 4 2

2S/h

I(A)

P(a.u.)

103

106

109

esca

pe

rate

(s-1

)

103

106

109

103

106

109

0 4 8 12

0,8

1,0

1,2

1,4Iexp

Iteo

n

Observation of Energy Level Quantization in Underdamped Josephson Junctions above the Classical-Quantum Regime Crossover Temperature

P.Silvestrini, V.G. Palmieri, B. Ruggiero, and M. Russo, Phys. Rev. Lett. 79, 3046 (1997)

R>20K

20

Observation of Resonant Tunneling between Macroscopically Distinct Quantum Levels

R. Rouse, S. Han, and J. E. Lukens, PRL 75, 1614 (1995)

Resonant macroscopic quantum tunneling in SQUID systems

P.Silvestrini, B.Ruggiero, and Y.Ovchinnikov; PRB 54, 3046 (1996)

21

U

E

0E

RE

1RE Tℓ

cos

2

1UU L

2xo

te

UM

H Lxo cos2

cos2

1

2

1 2

2

2

22

Quantum superposition of distinct macroscopic states

Jonathan R. Friedman, Vijay Patel, W. Chen, S. K. Tolpygo & J. E. Lukens

NATURE | VOL 406 | 6 JULY 2000

23

Coherent control of macroscopic quantum states in

a single-Cooper-pair boxY. Nakamura, Yu. A. Pashkin & J. S. Tsai

NATURE |VOL 398 | 29 APRIL 1999

Manipulating the Quantum State of an Electrical CircuitD. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M. H. Devoret

SCIENCE VOL 296 3 MAY 2002

dec =0.5 s

24

Coherent Temporal Oscillations improvement of Macroscopic Quantum States in a Josephson Junction

Yang Yu, Siyuan Han, Xi Chu, Shih-I Chu, Zhen Wang

SCIENCE 296, 889 (2002)

dec =5 s

Rabi oscillations in a large Josephson-junction qubitJohn M. Martinis, S. Nam, J. Aumentado, andC. Urbina

dec =10 ns

Phys. Rev. Lett 89, 117401 (2002)

25

Coherent Quantum Dynamics of a Superconducting Flux Qubit

I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, J. E. Mooij

dec =20 ns

Science 299, 1869 (2003)

26

Quantum oscillations in two coupled charge qubitsYu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai

NATURE |VOL 421 | 20 FEBRUARY 2003

dec =2.5 ns

dec =0.6 ns

27

0.0 0.5 1.0 1.5 2.0 2.5 3.0

2.4

2.6

2.8

3.0

0.000 0.002 0.004 0.006

4

6

8

10

12

14

16

Ib (A)

g1

e1e0

g0

g2

|g0> |e1>

|g0> |e0>

plasma resonance p

red sideband R

blue sideband b

Res

on

ant

freq

uen

cies

(G

Hz)

/0

qubit Larm

or frequency L

|g1> |e0>

e2

0 1 2 3

0

100

200

300

400

500

time

(n

s)Ibias (A)

T1 T2 Tphi

Coupling a flux qubit and a harmonic oscillator Relaxation-limited dephasingat the optimal point

Dynamics of a flux-qubit coupled to a harmonic oscillator (P. Bertet)

28

Coupling phase qubits

U

UIbI*

2

10|01|

29

Quantronium:-arbitrary robust operations-decoherence fought: echoes, wave driving-new readout developed at Yale

16.30 16.35 16.40 16.45 16.50

30

40

50

60

p (

%)

frequency (GHz)

Corpsecomposite

pulse

0 500 1000 1500

0.3

0.4

0.5

delay (ns)

p

*1 1T T

0 200 400 600 800 1000 12000.2

0.3

0.4

0.5

0.6

pulse length (ns)

p

*2 2T T

0 500 1000 15000.2

0.3

0.4

0.5

0.6

delay (ns)

p

2ECHOT T

combined X&Y rotations0 20 40 60

20

30

40

50

60

p (%

)

delay (ns)

SPIN-LOCKING

Rabi

30

BIFURCATION AMPLIFICATION

• Bifurcation amplifier: sensitive to any input variable coupled to I0

minimal back-action- no on-chip dissipation- efficiently thermalize load- backaction narrow band

31

Demonstration of conditional gate operation using superconducting charge qubit

T. Yamamoto, Yu. A. Pashkin, O. Astafiev, Y. Nakamura, & J. S. Tsai

32

ji

xj

xiij

ji

zj

ziij

xi

ii

i

ziiH

,,

………..

………..

….

….

….

….

Superconducting Adiabatic Quantum Device

NP-Hard Problems: i=ij

ji

zj

zi

i

zinpH

,

33

L. B. Ioffe and M. V. Feigel'man, Phys. Rev. B 66, 224503 (2002)

B. Douçot., M. V. Feigel'man and L. B. Ioffe, Phys. Rev. Lett. 90, 107003 (2003)

Josephson Networks

R. Burioni, D. Cassi, I. Meccoli, M. Rasetti, S. Regina, P. Sodano, A. Vezzani, Europhys. Lett.52, 251, (2000).           R. Burioni, D. Cassi, M. Rasetti, P. Sodano, A. Vezzani, J.Phys.B., 34, 4697, (2001). 

Bose-Einstein Condensation in Inhomogeneous Josephson Junctions Arrays

34

The theoretical model (II)

fingers

Backbone

Eigenvalue equation

)()( iEjH

t is a positive hopping parameter

Axy,x'y' is the adiacency matrix

Axy,x'y' = 1 If xy-x'y' is a link

0 otherwise

'''',

'', ˆˆ yxxyyxxy

yxxy aaAtH

35

There are localized states even for the free Hamiltonian: the ground state decays exponentially along the fingers

G. Giusiano, F. P. Mancini, P. Sodano, A. Trombettoni, Int. J. Mod. Phys. B 18, 691 (2004)

Topology induces new phases at finite temperature for bosons on graphs

The experimental signature of the Bose Einstein condensation is given by the inhomogeneity of Josephson critical currents below the BEC critical temperature

Hamiltonian Solutions for a comb graph

36

Chip design

b)

JJ

50m

The realized arrays have 4mx4m and 5mx5m junctions

8 different chips with different current density were fabricated and tested

The Backbone (BB) and its reference (RBB) have 72JJ

while finger (CF) and its reference (RCF) have 80JJ

a)

CF

BB

RCF

RBB 1mm

37

Measurement

Junctions are connected in series and incresing the bias current the switch to the gap branch of each junction is well visible.

We can count the number of junctions in the array

V-

I+ I-

V+

V-

I+ I-

V+

0 5 10 15 20 25 30

7.5

8.0

8.5

9.0

Cur

rent

(A

)

Voltage (mV)

38

Measurement

Junctions are connected in series and incresing the bias current the switch to the gap branch of each junction is well visible.

We can count the number of junctions in the array

V-

I+

I-

V+

V-

I+

I-

V+

0 10 20

7.5

8.0

8.5

Switches of single junctions (about 2.8mV)

Cur

rent

[A

]

Voltage [mV]

39

Experimental Results on Backbone

Backbone shows a critical current higher than the reference one in particoular at T=1.2K

The gap voltage is the sum of the number of junctions (72JJs)

VI I

V

VI I

V

VI I

V

VI I

V50 100 150 200

0

5

10

15

20

25

30 T=4.2K

Cur

rent

(A

)

Voltage (mV)50 100 150 200

0

5

10

15

20

25

T = 1.2K

Cur

rent

(A

)

Voltage (mV)

40

Experimental Results on Fingers

The CFA shows an increased disuniformity at T=1.2K

50 100 150 2000

5

10

15

CFA

RFA

T = 1.2K

Voltage (mV)C

urre

nt (A

)

V

I

I

V

V

I

I

V

50 100 150 2000

5

10

15

CFA

RFA

T=4.2K

Cur

rent

(A

)

Voltage (mV)

41

Further test

JJ

50m

BBArray before cutting BBArray After cutting

reduced to a Linear array

50m

42

Experimetal Results on Backbone Voltage is normalized to the number of junctions

50 100 150 2000

5

10

15

20

25

30

= RBA

BBA BBA after Cutting RBA

BBA after cut

BBA

T=4.2K

Cur

rent

(A

)

Voltage (mV)50 100 150 200

0

5

10

15

20

25

30

BBA after cut = RBA

BBA

BBA BBA after cutting RBA

T=1.2K

Cur

rent

(A

)Voltage (mV)

VI I

VRBA

VI I

V

BBA

VI I

V

BBA After cut

43

Effect of noise and disuniformity

0 50 100 150 200 250

10

15

20

25

Ic = 23ASigma=4%

T=6kT=8k

T=4kT=2k

Cur

rent

(A

)

Voltage (mV)

Calculated current switchings

Effect of temperature

0 50 100 150 200

10

15

Tn=1K

Ic=20A

T=4.2K

Sigma 12%

Sigma 8%

Sigma 4%

Sigma 2%Sigma 1%

Cur

rent

(A

)

Voltage (mV)

Effect of a Ic disuniformity

44

Fit of data 1.2K - 4.2K

The only free parameter to fit IV curve is the mean Ic:

the sigma is 4% in 4x4 BBACUT and 3% in BBACUT 5x5arrays

45

Fit at all temperature BBA 4x4m2

0 50 100 150 200

5

10

15chip 6 BBA4x4

C

urre

nt (A

)

Voltage (mV)

BBA present a larger disuniformity at T < 5K

46

0.2 0.4 0.6 0.80.4

0.6

0.8

1.0

1.2BBA

BBACUT

Tc = 9.0K

T=5.0K

Chip6 BBA4x4

N

orm

aliz

ed C

urre

nt

Normalized Temperature

Critical Current Temperature Behaviour

Critical Current is measured and normalized to the 64 JJ switching

on a total number of 72JJs

47

Critical Current Temperature Behaviour

0.2 0.4 0.6 0.80.4

0.6

0.8

1.0

T=6.0k

Chip6 BBA5x5

Nor

naliz

ed C

urre

nt

Tc=8.9K

BBA

BBA cut

Normalized Temperature

Critical Current is measured and normalized to the 50 JJ switching

on a total number of 55 JJs

48

Sample Junction area (μm2)

Number of

Junctions

Mean Ic

(μA)

Sigma(%Ic)

Ic BBA / Ic

BBACUT

BBA3 4x4 72 18 4 1.17

BBA4 5x5 72 24 3 1.13

BBA5 5x5 72 25 3 1.11

BBA6 5x5 55 26 3 1.04

BBA6a 4x4 72 17 4 1.07

49

Summary

• We have observed a critical current enhancement along the backbone of a comb-shaped Josephson Junction array

• We have inferred from data its temperature dependence

• At the same time we observed along the finger a critical current reduction away from the backbone

• The whole effect is related to the inhomogenous topology (connectivity)