Implémentation d'un bit quantique dans un circuit supraconducteur / Implementation of a quantum bit in a superconducting circuit

HAL Id: tel-00003511

https://tel.archives-ouvertes.fr/tel-00003511

Submitted on 9 Oct 2003

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Implémentation d’un bit quantique dans un circuit

supraconducteur / Implementation of a quantum bit in

a superconducting circuit

Audrey Cottet

To cite this version:

Audrey Cottet. Implémentation d’un bit quantique dans un circuit supraconducteur / Implementation of a quantum bit in a superconducting circuit. Matière Condensée [cond-mat]. Université Pierre et Marie Curie - Paris VI, 2002. Français. �tel-00003511�

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(nA.K )

-1 0 5 = => 1 I

NE + 7 @ 4 4 A= NFB 5 = => : 3 5 4 ? 3 I 4 ? @ 4 3 4 6 3 0 4 5 ! #! 2 I @ ! # ! #_! & A5 = 21B4 ? A= NNB 3 4 4 A= NLB 3 7 , 3 3 4 3 7 , 3 @ 3 3 6 ; 0

NF

(%)

E

_J

/E

_C

from bottom to top: ng=0, 0.3, 0.4, 0.45, 0.475, 0.5

2

4

6

8

10

1

2

3

4

5

0

0

=1%

A

=3.5%

B

exact model

sinus. approx.

-0.4

-0.2

0

0

2 0

2

-0.4

-0.2

0

0.2

E

J

/E

C

=1, n

g

=0.3

E

J

/E

C

=1, n

g

=0.5

E

_C

E

₀

A

B

E

_C

E

₀

5 = 21 ? , I _{! =} > $J , , = >% 4 ,

"

+"/ &

4 7 , 3 SE>T

+"+

I ; 3 4 9 9 : 3 3 4

)

; 3 4 4 3 H % 4 % 3 : 3 4 % V ? " % ; % % & 4 " % % & 4 " % % , 3 ! 6 SF1T " % % A= NMB

NM

)

" &

3 " & 3

4 " & V ? " " & ;

" & " & & 4 A= N>B

" " & " & & A= L1B

# 4 " " & " & , 3 ! 6 SE>4 F1T " " & " & A= L=B

&

4 , V ? 3 " 4 " , " 5 4 ! / 6 A= NMB A= L=B " , " ? 6 4 &" _{" " " "} & " A= L2B

+"+

8

H 6 4 4 7 , ? 3 7 , 4 9 9 4 3 6 H A= L0B

N>

9

+

8

/ # 4 , ' , ' 3 7 , 4 : , & 4 / # 4 ' 3 4 4 6

(

9

)

/ 4 ? ? A= L2B 5 SF=T " & A= LEB 3 6 2 2 A= LFB 4 A= LNB ? 4 A= LLB ? A= LMB 3 4 3 -4 - - 4 A= L>B - - 4 A= M1B

-L1

+"+ *

)

: 3 ; 3 ? 3 6 3 K :' 4 % " & 4 A= M2B V ? "#4 "# " " & A= M0B V ? 0 V ? 4& 4& " " & " % " % A= MEB

&

8

H 4 A= M2B , V ? " A= M0B A= MEB " A= MFB 4 3 3

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:

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% %

3 V & 7 , 7 4 A5 2 =B I 3 ' , ' A2 =B V 4 3 ! H , : 3 5 ; SE>T " # + (( $, & 4 A2 2B

L0 5 A2 0B : 3 (5

% %

) # 0

:

; 0:

<

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LE 0.0 _0.2 0.4 0.6 0.8 _1.0 1.2 1.4 0.0 0.1 0.2 0.3 0.4 0.5 Q = 0.05, 0.5, 5, and 50 (bottom to top)

<( /2 )2>Rk

(L/C)

1/2

kbT/ h 0

5 2 2 ( @ ! 9 3 ,

Static Regime

Diffusive Regime

Running state

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LF

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0.5 1 1.5 2 2.5 3 0.25 0.5 0.75 1

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/(RI

0

)

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∞

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LL

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M1 V J(µV) I J(nA) 0 2 4 6 8 0 10 20 30 40 Cs Rs Ib Rm RI SQUID array Ib’ V Ej Rc I_J V_J RI Rb L -150 -100 -50 0 50 100 150 -1.0 -0.5 0.0 0.5 1.0 I₀=44.9 nA T=34 mK V J(µV) I_J(nA) 5 2 N 6 % , 4% ( ' / 7 % ( ' (# ' (+ ' < 2 =<2 >% , 7 97 , 5 " ( ' 6 % K%B J 6 < 0A ,% 2 5 K 8 9 = , > 3 A2 =1B ) 9 ( '% K ,% %

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d=0

T=10, 50, 100 mK

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0

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5 2 M I : , 4 "% ? I : %

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MF

-0.05

0.0

0.05

1

2

3

4

5

6

-6

-3

0

3

∆

_ng

∆

_{IJ (nA)}

∆

t (µs)

5 2 =1 & 4 6 % 8 ."H = > ."H = >% = 9 ,>%

MN

% %

"

;*

<

(

V(t)

Ib

Cs

Rs

Vg

C’g

V

J

I

_J

C

_L

200

400

0

10

0

5

V

J

(µV)

I

J

(nA) I

_S L 5 2 == % , % ( ' / < 2 , / 5 ( 97 % 8 < 9+ & & % 6 7-, % 3 5 2 == 4 3 . 4 SE24 2NT 5 2 F4 : 3 V ? 6 5 4 ,

ML 5 4 3 I 3 ( '4 5 : 3 ) 3 4 H , ? ! 7 H S2NT 3 @ ) -4 9 9 3 ! 3 ; ! . ! 4 . A5 2 =2B 4 3 ! ; 3 : ? 3 5 2 =0 A B ! 3 & 3 4 3 , 4 6 ? I 3 ; 6 K4 & K4 6 , ? 3 ; ! 5 2 =0 A B : 3 I 3 ? 3 ; 3 A5 2 =EB4 3 I 3 ? 3 4 ? 4 6 ; 3 , 5 #. ; 3 * # 3 & ' 3 3 4 4 4 ; < % ; ! ; ! A2 =FB

MM 3 ! ) A5 2 =EB # 4 &4 < A2 =NB 3 V & 4 3 9+ & & 4 9+ 6 4 3 & A5 2 =FB , 4 ; ! 3 4 " 3 4 ? 4 6 ? A 0B 4 4 2 / N S2LT % 0 1 2 3 4 5 6 7 0 5 10

I

_p τ = 2 µs

I

b

(

n

A

)

tim e (µ s)

0 5 10

P

1-P

V

(

µ

V

)

5 2 =2 8 = > = >%

M>

1

0

T=20 mK τ =2µs

I

p

(

n

A

)

P

s

0.0

0.5

0 5 10 15 20

0.5

P = 0.5

n

_g 5 2 =0 5 H 6 K4 6 K & = > 4 7 , ! ; ! 4 = > ! = >% ( .4 ; ! 4 ! ,% . %

>1 0,86 0,88 0,90 0,92 0,94 0,96 0,98 1,00 0,0 0,2 0,4 0,6 0,8 1,0 0,0 0,2 0,4 0,6 0,8 1,0 simu 11.73 nA C2=0.75pF simu 13.4 nA C2=0 pF Exp Ip=11.73 nA Exp Ip=13.56 nA Ps (n g ) ng 5 2 =E <, H ; ! 4 ! ) ! ) "%$D 5 , , , % ; , 4 , , % 4 , 4 % 75 76 77 78 79 0 500

1000 Ramping speed: 2 nA/µs_{T=32 mK}

4%*2e N u m b e r o f e v e n ts Time of detection T s (µs) 5 2 =F @ 9+ % 4 , &

>=

% %

2 =L 3 3 3 4 %5 # S2=4 N2T ? 2 ? 0 %5 # / ."H ."H ."H 3 3 A2 =LB 4 6 ; 3 ? ? : 5 4 # 6 4 3 6 4 ; : ? 2 0 3 6 3 ? # 4 4 6 ? 3 : %5 # 6 ?

"

+"/ A

I 3 3 ? 4 7 , 4 ? " 3 6 K 3 H H , A /&B I 3 7 , 5 5 " 6 K 7 ? , ? 3

+"+ B

*

#

A /&B A5 2 =NB 6 4 4 3 4 6 * ! #5 P / ? 4 6 ? 4 3 ; 3 : ? ? /& H : 6 4 ? ? : ? # 3 ? ?4 ? 3 3 ? : 6 ; : 3

>E e-gun DAC Board Sample beam blanker X Y

X

Y

electron microscope

e-10 kV

Al

exposure

development

deposition

lift-off

5 2 =N 9 , 4 , = > 7 9%

>F

-

.

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A

+

)

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>M 5 2 => .4 / , 4 5 2 21 2 4 D , % A 4 , % %

>>

*$4 $

#

3 ? F

!

/ A!.. . B & !.. A KB &

%

5 4 !.. 6 7 , !.. . / # $ 4 3 6 # $ 4 3 6

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3 6*4

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Twisted pair

2 single lines/ coaxial line connection Resistor

Capacitor (R,C) filter

Microfabricated (R,C) filter Copper powder filter

10 Decibels attenuator 10 dB 5 2 2= 5 , % ? 4 ? 4 3 3

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0&5

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=10

0.1

0.2

0.3

0.4

0.5

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0

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0

1

2

3

4

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I

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V

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(

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V

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P

V

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T=20mK

T=1K

T=4K

T=300K

CHIP

Feed Back

Coil

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=1>

P

P

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T = 1K

T = 4K

T = 300K

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voltage divider

(%19)

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Array

P

10

dB

Feed

Back

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Feed Back

Monitor

V

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V

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V

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===

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a

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T=4K

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V

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T=20mK

P

T=300K

twisted pa ir M easure

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V

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R

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1 k

19 k

5 2 01 H 4 D = , 2 2=>%

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VOLUME87, NUMBER13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER2001

Direct Measurement of the Josephson Supercurrent in an Ultrasmall Josephson Junction

A. Steinbach,1_{P. Joyez,}1_{A. Cottet,}1_{D. Esteve,}1 _{M. H. Devoret,}1_{M. E. Huber,}2_{and John M. Martinis}3 1_{Service de Physique de l’Etat Condensé, CEA-Saclay, F-91191 Gif-sur-Yvette, France}

2_{Department of Physics, University of Colorado at Denver, Denver, Colorado 80217} 3_{National Institute of Standards and Technology, Boulder, Colorado 80303}

(Received 10 April 2001; published 10 September 2001)

We have measured the supercurrent flowing through a nonhysteretic, ultrasmall, voltage-biased Joseph-son junction. In contrast with experiments performed so far on hysteretic JosephJoseph-son junctions, we find a supercurrent peak whose maximum Ismax increases as the temperature T decreases. The asymptotic

T 苷 0 value of Ismax agrees with the junction Ambegaokar-Baratoff critical current, as predicted by

theory.

DOI: 10.1103/PhysRevLett.87.137003 PACS numbers: 74.50. +r, 73.23. – b, 73.40.Gk

A Josephson tunnel junction between two superconduct-ing electrodes is a basic quantum nonlinear system [1,2]. For excitation energies much smaller than the supercon-ducting gap D, it can be modeled by the Josephson Hamil-tonian ˆH 苷 2EJcos ˆd where ˆd is the gauge-invariant

phase-difference operator, a purely electrodynamic quan-tity which is 2e兾 ¯h times the space and time integral of the electric field across the junction [2]. The Josephson energy EJ is a macroscopic parameter, which, for BCS

superconductors at temperatures T ø D兾kB and for

sufficiently opaque junctions, depends only on the junc-tion tunnel resistance Rt and D through EJ 苷

h

8e2D兾Rt

[3]. The supercurrent flowing through the junction is given by the Josephson relation IS 苷 共2e兾 ¯h兲 具≠ ˆH兾≠ˆd典 苷

共2eEJ兾 ¯h兲 具sin ˆd典, the average 具· · ·典 being performed on the

degrees of freedom of the electrodynamic environment of the junction. Thus, the highest supercurrent that the junction can sustain is given by the so-called Ambekaokar-Baratoff critical current I0苷

p

2eD兾Rtcorresponding to an

environment for which具sin ˆd典 苷 1. This critical current is easily observed for junctions with a small Coulomb energy

EC 苷 2e2兾C0 ø EJ where C0is the intrinsic capacitance

of the junction [4]. For these junctions, the phase behaves as a good quantum number d which can be driven to the critical value d苷 p兾2. However, for the so-called “ultrasmall” junctions characterized by EC * EJ, which

are considered for applications in quantum information processing [5], the highest supercurrent has always been found experimentally well below the expected value I0

[6 – 9]. Several untested hypotheses have been formulated to explain these results. The average具sinˆd典 may not reach the value 1 because of uncontrolled quantum or thermal fluctuations. Failure of the Josephson Hamiltonian model for ultrasmall junctions could also explain the results, even if it is not directly expected from theory. Note that, experimentally, one cannot simply shunt the two leads of the junction by a small superconducting inductance to impose the phase difference, since it then becomes impossible to check the junction parameters by measuring its quasiparticle current.

The aim of the experiment reported in this Letter was to test the validity of the Josephson Hamiltonian model and the Josephson relation for an ultrasmall junction embedded in a controlled environment which should suppress phase fluctuations.

The principle of our experiment is shown schematically in Fig. 1a: an ultrasmall Josephson junction with criti-cal current I0 and intrinsic capacitance C0 is biased by

a circuit equivalent to a capacitor CB in parallel with an

ideal voltage source VB in series with a resistance RB.

This bias circuit is also equivalent to a current source

IB 苷 VB兾RB in parallel with RB and CB (Thévenin

theo-rem). The average current I through the junction is mea-sured by a current meter in series with the junction. The impedance of the meter is made negligible in comparison with the impedance of the bias circuit. The system is analo-gous to a damped quantum particle with mass C共 ¯h兾2e兲2_,

where C 苷 C0 1 CB, placed in a tilted washboard

poten-tial U共 ˆd兲 苷 2EJcos ˆd 2 IB共 ¯h兾2e兲 ˆd. The damping due

to the resistance RB, assumed to be in thermal

equilib-rium at temperature T, manifests itself also as a fluctu-ating force acting on the particle [2]. We consider only the overdamped regime RB ø

p ¯

h兾共2eCI0兲, for which the

particle mass can be neglected. In this regime, all quan-tum fluctuations of the phase are suppressed, provided that RB ø h兾共2e兲2[10]. Previous experiments have tried

to implement this idealized circuit and determine the su-percurrent maximum ISmax 苷 max关IS共VB兲兴 which should

tend to I0 as T ! 0. However, instead of the pure RB

and CBcombination, all of them had a strongly

frequency-dependent and often ill-characterized impedance Z共v兲. This difficulty arose because the measuring setups involved high input impedance field-effect transistor amplifiers and the dc value of Z had to be made large in order to re-solve the contribution of the junction quasiparticle current to the I共V 兲 characteristics. In practice, the condition for hysteresis Z共v 苷 0兲 ¿ ph¯兾共2eC0I0兲 was inevitable. The

zero-voltage state, in which the supercurrent is measurable, was therefore metastable and was switching to the nonzero voltage state at IB 苷 Isw , ISmax [11]. This switching

VOLUME87, NUMBER13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER2001 a) b) C0 I0 F VB L RL RM RI RI RH RC C_H VB C0 I0 _V CURRENT METER I B R CB 20µm junction R_H CH c) chip SQUID array

FIG. 1. (a) Idealized circuit for measurement of supercurrent of Josephson junction. The junction consists of a Josephson element (cross) in parallel with a capacitor. (b) On-chip high-frequency circuitry contributing to electrodynamic damping of Josephson junction. For clarity, metallic thin films only are represented. (c) Measurement setup schematics including both on-chip circuitry (box in dotted line) and off-chip circuitry. A known fraction of the current through the junction is coupled to a SQUID array (box in dashed line). The backaction noise of the SQUID array is attenuated by filter F.

transition is affected by thermal and quantum fluctuations, and Isw is characterized by a probability distribution. The

average ¯Iswobtained so far for ultrasmall junctions showed

various degrees of reduction compared to I0 [6], ranging

from around 1023I0 to 0.65I0, this largest value having

been obtained using an on-chip impedance [9].

In the present experiment, we have circumvented these problems and measured the current through a junction in the nonhysteretic regime using as the current meter a re-cently developed SQUID series array with 100 dc SQUIDs [12]. The impedance of the current-measuring circuit is so low that we can afford to overdamp conservatively the junction at all frequencies, thereby ensuring that the mea-surement finds the junction in a fully stable state with a controlled absence of quantum fluctuations. The total impedance seen by the junction is equivalent to a pure re-sistor RB 苷 24 6 1 V in parallel with a reactive element

which behaves as a capacitor CB 苷 200 6 20 fF above a

few tens of MHz. In conventional measuring setups, such heavy damping would make the junction current hardly dis-tinguishable from the current through the shunt resistor.

The actual measurement setup implementing the ideal-ized Fig. 1a circuit is shown in Figs. 1b and 1c. It has been designed to minimize deviations of the environmental impedance from the ideal limit Z共v兲21 _{苷 R}21

B 1 jCBv

while maintaining dissipative elements in thermal equilib-rium at a controlled temperature T, as well as imposing an accurate bias current. In order to meet, in a fre-quency range spanning 10 orders of magnitude, these con-flicting requirements, we have engineered an environment which consists of both microscopic on-chip (Fig. 1b) and macroscopic off-chip components (see outside of dotted line box in Fig. 1c). The on-chip components contribute mainly to the high-frequency values of the environmen-tal impedance, which include the junction bare plasma frequency p2eI0兾共 ¯hC0兲 in the tens of GHz range, while

the off-chip components contribute mostly to the low-frequency values, which includes possible Josephson reso-nances in the hundreds of MHz range, as well as the measurement frequencies below a few kHz. The on-chip circuitry consists of a resistance RH 苷 11.8 V in series

with a capacitance CH ⯝ 100 pF, which were fabricated

using a five-layer optical lithography process. The resistor

RH was made from 150 nm thick AuPd with a width and

length of approximately 5 mm by 25 mm. This small re-sistor was in good electrical contact with a large Au pad that served as a thermal reservoir. A single Al-AlOx-Al

Josephson junction was fabricated by e-beam lithography and double angle shadow mask evaporation [13]. We es-timate the capacitance C0 苷 1 fF from the junction area.

The contact resistance resulting from the junction fabri-cation process was RC 苷 12.1 V. The off-chip

compo-nents RM 苷 1.67 V, RI 苷 10 V, and RL 苷 10.1 V were

surface mounted resistors for microwave circuits placed within 5 mm of the junction to minimize the stray induc-tance L⯝ 4 nH of the connection between the off-chip and on-chip circuitry. The role of RM and the two RI’s is

to provide a current divider for minimizing the backaction of the Josephson oscillations inside the SQUID array on the measured junction. A microwave copper-powder filter [14] placed in series with the RI resistors provides further

attenuation at high frequency. Only about 8% of the junc-tion supercurrent was thus coupled to the SQUID array. The biasing circuitry at high temperature was connected to the resistor RL through coaxial lines filtered by a

com-bination of copper-powder filters and miniature cryogenic filters [15], and its action is equivalent to a voltage source

VB in series with RL. The sum of RL, RM, and RC

deter-mines the dc value of the environmental impedance, while the sum of RHand RCdetermines its high-frequency value

(inductor L blocks high-frequency currents). The sample and the low temperature bias circuitry were thermally an-chored inside a copper box bolted to the mixing chamber of a dilution refrigerator.

The junction I共V兲 characteristic is shown in Fig. 2a for

T 苷 34 mK. The superconducting gap is directly

VOLUME87, NUMBER13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER2001 -100 0 100 I(nA) -1.0 -0.5 0.0 0.5 1.0 V(mV) -40 -20 0 20 40 I(nA) -5 0 5 V(µV) a) b)

FIG. 2. (a) Large-scale I共V兲 characteristic of the ultra-small Josephson junction of Figs. 1b and 1c at T 苷 34 mK. (b) Josephson supercurrent peak shown on an expanded voltage scale, at different temperatures. From top to bottom: T 苷 34, 98, 245, 400,622 mK, respectively.

state resistance approaches the asymptotic value RN 苷

6.99 kV at voltages many times 2D兾e. The critical cur-rent is then calculated to be I0 苷 44.9 6 0.5 nA. On first

inspection, the I共V兲 of Fig. 2a appears conventional, with the Josephson current manifesting itself as a vertical line at zero voltage. However, because of the low impedance of our biasing circuit, there is no hysteresis and all points on the I共V兲 characteristic are stable, in contrast with the usual current-bias ramp method giving access only to the posi-tive differential conductance part of the I共V兲 in the best cases. Thus, it is worth stressing that in our experiment, a supercurrent peak, as opposed to a supercurrent branch, is measured for the first time. The detailed structure of the supercurrent peak is shown for several temperatures in Fig. 2b. The higher temperature data show a finite slope around zero bias which is due to phase diffusion in the tilted washboard potential [16]. The full I共V兲 characteris-tic in the pure Ohmic damping case was first calculated by Ivanchenko and Zil’berman [17]:

I共VB兲 苷 I0Im ∑_I 122ibeVB兾 ¯hRB共bEJ兲 I22ibeVB兾 ¯hRB共bEJ兲 ∏ , (1)

where In共z兲 is the modified Bessel function, b 苷 1兾kBT,

VB 苷 V 1 RBI. The more general approach [18],

devel-oped to solve the steady-state Fokker-Planck equation for the phase distribution in a tilted washboardlike potential,

has been proved to yield equivalent results [19]. The ex-pression (1) predicts a supercurrent peak with a maximum which tends to I0 in the zero temperature limit. A

de-tailed comparison between the I共V兲 characteristics mea-sured at three temperatures and the theoretical predictions are shown in Fig. 3 with no adjustable parameters. The close agreement between theory and experiment around the peak maximum shows that the temperature of the electro-magnetic environment which drives the phase dynamics is indeed equal to the experimental refrigerator temperature. The agreement over the whole voltage range, without any spurious resonances, confirms that the impedance of the junction environment is indeed almost constant over the corresponding range of Josephson frequencies. As a check, we have simulated the classical dynamics of a small junc-tion for the exact circuit of Fig. 1b, including the effect of thermal fluctuations. The theoretical I共V兲 curves so ob-tained are negligibly different from those obob-tained with expression (1) with RB 苷 24 V, which indicates that our

experiment implements the ideal bias case satisfactorily. As a further check of the influence of the off-chip bias circuitry on the I共V兲, we have increased L to a value of order 100 nH and observed that the I共V 兲 then developed two metastable branches predicted by our numerical simu-lations and corresponding to chaotic Josephson oscilsimu-lations in the hundreds of MHz range [20].

In Fig. 4, we compare the measured supercurrent peak height ISmaxwith the values predicted from (1) over a large

temperature range [21]. The peak height ISmaxincreases as

the temperature is lowered down to 26 mK, in agreement with the classical theory. However, the agreement between theory and experiment below 200 mK was attained only after the filtering at the input of the SQUID array, as de-scribed in Fig. 1c, had been installed. This indicates the magnitude of the backaction noise produced by this type of amplifier. We do not have a fully convincing explana-tion for the deviaexplana-tions between experiment and theory at

40 30 20 10 0 I(nA) 8 6 4 2 0 V(µV)

FIG. 3. Comparison between the I共V兲 characteristics measured at different temperatures (symbols) and the calculated ones (full lines) using Eq. (1) and I0苷 44.9 nA and R 苷 24 V. From

top to bottom: T 苷 34, 157, and 400 mK, respectively. Dashed line represents the I共V兲 predicted at T 苷 0.

VOLUME87, NUMBER13 P H Y S I C A L R E V I E W L E T T E R S 24 SEPTEMBER2001 40 30 20 10 0

I

P

(nA)

800 600 400 200 0

T(mK)

FIG. 4. Comparison between the measured temperature depen-dence of the maximum supercurrent (open circles) and the pre-dicted one (full line).

the highest temperatures, but the contribution of the quasi-particle current to the damping of the junction, which we neglect in our analysis, may play a role in this regime.

Our experiment thus provides strong experimental evidence that the commonly observed reduction of the maximum supercurrent in an ultrasmall junction is not an intrinsic junction property, but is due to its electrodynamic environment. When the environment is engineered to place the junction in the overdamped regime in a con-trolled manner, the Ambegaokar-Baratoff critical current

I0 can be reached at low temperature, thereby showing

the validity of the Josephson Hamiltonian for ultrasmall junctions. A control of the environment impedance similar to that of our experiment, but with higher resistances, would allow the observation of the strong reduction of the maximum supercurrent by quantum fluctuations, which has been recently predicted by Ingold and Grabert [22] in the case of a resistive environment with R ⬃ RQ.

Our work also provides ground for the application of the single Cooper pair transistor (SCPT) [7] to electrome-try. This device consists of two small Josephson junctions in series. At low temperature, it is equivalent to a single junction whose Josephson energy is modulated with a 2e period by the gate charge coupled to the island formed be-tween the two junctions. This modulation can be exploited for low-noise-temperature electrometry [23], provided that the device supercurrent is measured like in the present ex-periment. The SCPT could operate at high frequencies since intrinsic bandwidths up to 120 and 250 MHz have been demonstrated for arrays with 100 and 30 SQUIDs, respectively [24]. Further work is needed to know how this new type of electrometer competes in fast electrome-try with the recently developed RF-SET [25].

We gratefully acknowledge discussions with J. Imry, G.-L. Ingold, and H. Grabert. This work was partly supported by the European Union through Contract No. IST-10673 SQUBIT and by the Bureau National de la Métrologie.

[1] B. D. Josephson, in Superconductivity, edited by R. D. Parks (Marcel Dekker, New York, 1969).

[2] A. O. Caldeira and A. J. Leggett, Ann. Phys. (N.Y.) 149,

374 (1983).

[3] V. Ambegaokar and A. Baratoff, Phys. Rev. Lett. 10,486 (1963).

[4] A. Barone and G. Paternò, Physics and Applications of the Josephson Effect (Wiley, New York, 1982).

[5] M. F. Bocko, A. M. Herr, and M. F. Feldman, IEEE Trans. Appl. Supercond. 7,3638 (1997); J. E. Mooij et al., Science

285,1036 (1999); D. V. Averin, Solid State Commun. 105,

659 (1998); Yu. Makhlin, G. Schoen, and A. Shnirman, Nature (London) 398, 305 (1999); Y. Nakamura, Yu. A. Pashkin, and J. S. Tsai, Nature (London) 398,786 (1999). [6] M. Tinkham, in Single Charge Tunneling, edited by H. Grabert and M. Devoret (Plenum Press, New York, 1992).

[7] P. Joyez et al., Phys. Rev. Lett. 72,2458 (1994). [8] W. J. Elion et al., Nature (London) 371,594 (1994). [9] D. Vion et al., Phys. Rev. Lett. 77,3435 (1996); P. Joyez

et al., J. Supercond. 12,757 (1999).

[10] H. Grabert, G.-L. Ingold, and B. Paul, [Europhys. Lett. 44,

360 (1998)], have established that for the circuit shown in Fig. 1a, and in the limit RB ø RQ 苷 h兾共2e兲2, the

junc-tion dynamics is the same as if the phase was a classical variable, but with a renormalized EJdepending on both RB

and C苷 CB 1 C0. According to this theory, the quantum

corrections to the bare EJare negligible for our experiment

(note that both RB and CB contribute to driving the

junc-tion into the classical phase regime).

[11] T. A. Fulton and L. N. Dunkleberger, Phys. Rev. B 9,4760 (1974).

[12] R. P. Welty and J. M. Martinis, IEEE Trans. Magn. 27,2924 (1991); M. E. Huber et al., Appl. Supercond. 5,425 (1998). [13] G. J. Dolan and J. H. Dunsmuir, Physica (Amsterdam)

152B,7 (1988).

[14] J. M. Martinis, M. H. Devoret, and J. Clarke, Phys. Rev. B

35, 4682 (1987).

[15] D. Vion et al., J. Appl. Phys. 77,2519 (1995).

[16] J. M. Martinis and R. L. Kautz, Phys. Rev. Lett. 63,1507 (1989); R. L. Kautz and J. M. Martinis, Phys. Rev. B 42,

9903 (1990).

[17] Yu. M. Ivanchenko and L. A. Zil’berman, Sov. Phys. JETP

28, 1272 (1969).

[18] V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22,

1364 (1969).

[19] W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation (World Scientific, Singapore, 1996). [20] K. K. Likharev, Dynamics of Josephson Junctions and

Cir-cuits (Gordon and Beach, New York, 1986), pp. 177–180. [21] Small corrections to the Josephson relation have been ap-plied for temperatures above 600 mK following Ref. [3]. [22] G.-L. Ingold and H. Grabert, Phys. Rev. Lett. 83, 3721

(1999).

[23] A. B. Zorin et al., J. Supercond. 12,747 (1999).

[24] M. Huber et al., IEEE Trans. Appl. Supercond. 11,1251 (2001).

[25] R. J. Schoelkopf et al., Science 280, 1238 (1998); A. Aassime, G. Johansson, G. Wendin, R. J. Schoelkopf, and P. Delsing, Phys. Rev. Lett. 86,3376 (2001).

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