Quantronics GroupCEA Saclay, France

B. Huard D. Esteve H. PothierN. O. Birge

Measuring current fluctuations with a Josephson junction

Question : what is P(n) ?

Vb

Atomic contact

Tunnel junction

Diffusive wire

0 n

t >>

I = n e/average current

on time

Counting statistics

I

I(t)

independent tunnel events Poisson distribution

P(n) asymmetric

Noise is more than n²() !

P(n)

5 10 15 20 25 30 35 40 45

108

106

104

102

5 10 15 20 25 30 35 40 45

0.02

0.04

0.06

0.08

n

n

Statistics of the charge passed through a tunnel junction

Gaussian with same n²(t)

Exact (Poisson)lo

g s

calen

n

Experimental implementation of ?

Measure n(t)Gustavsson et al. (2005)

Sample = Quantum DotSee next talk !

Experimental implementation of ?

Measure n(t) Measure properties of I(t) ( I = n() e/

( I(t) - I )3 " squewness "

(0 for Gaussian noise)

Reulet et al. (2003)

Sample impedance 50

Experimental implementation of ?

Measure n(t) Measure properties of I(t) ( I = n() e/

directly measure I(t)Bomze et al. (2005)

Sample impedance »1 M

Experimental implementation of ?

Measure properties of I(t) ( I = n() e/

measure probability that I(t) > Ith+

or that I(t) < Ith-

Current threshold detector

Measurement of current statistics with a threshold detector

P(n)

5 10 15 20 25 30 35 40 45

108

106

104

102

n

I = n() e/

distribution of I

distribution of n()

Measurement of current statistics with a threshold detector

P(I)

5 10 15 20 25 30 35 40 45

108

106

104

102

I = n() e/

distribution of I

distribution of n() I /e

Differences mainly in the tails focus on large fluctuations

Measurement of current statistics with a threshold detector

P(I)

I /et >>

I

Ith+

clic !

5 10 15 20 25 30 35 40 45

108

106

104

102

p+0 = = ( )d

thIP I I

Measurement of current statistics with a threshold detector

P(I)

I /et >>

I

Ith-

clic !5 10 15 20 25 30 35 40 45

108

106

104

102

p-0= = ( )d

thIP I I

Detecting non-gaussian noise with a current threshold detector

P(I)

I /e5 10 15 20 25 30 35 40 45

108

106

104

102 gaussian

p+0, p-

0

2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

gaussianpoisson

p+0 / p-

0

2 4 6 8 10 12 14

5

10

15

20

25

/ e I

/ e I / e I

5 10 15 20 25 30 35 40 45

0.02

0.04

0.06

0.08

P(I)

I /e

Effect of the average current on p+0 / p-

0

p+0 / p-

0

200 400 600 800 1000

1.5

2

2.5

3

3.5

20 000

2000 Increase

I

Current threshold detector reveals non-gaussian distribution

/ e I

The Josephson junction

I

V

I

V2/e

I0

- I0

supercurrent branch

Biasing a Josephson junction V

I

V

I0

- I0

- remains on supercurrent branch as long as |I|<I0

- hysteretic behavior

natural threshold detector

R I vb = R I + V

2/e

[Proposed by Tobiska & Nazarov

Phys. Rev. Lett. 93, 106801(2004)]

vb

is

Using the JJ as a threshold detector

Is+I

I

V

I+ib*

ib

I0

Switching if

I+ib > I0

clic ! I =I0-ib

Rb ib

vb

Josephson junction

Vs

* assuming Is=is

Using the JJ as a threshold detector

I

V

- I0

ib

clic !

Switching if

… or if

I+ib > I0

I+ib < -I0

is

Is+I I+ibRb ib

vbVs

Using the JJ as a threshold detector

I

V

clic !

is

Is+I I+ibRb ib

vbVs

I0

response time=

inverse plasma freq.

0

02 2

1 30 GHz

p I

C

1 ns

Experimental setup

Al

Cu

NS junction

V

is

Is+I

I+ib

Rb ib

vbVs

JJ (SQUID)

ib-is

Is+I

C

Cuse atIs>0.2µA

Rt=1.16 k

Measurement procedure

C=27 pF=180 µeVI0=0.84 µA

t - s I0

s I0 tp

count # pulses on V for N pulses on iband deduce switching rates + and -

…

V

is

Is+I

I+ib

Rb ib

vbVs

ib

C

I =I0-ib=I0(1-s)I0

-I0

Measurement procedure

t - s I0

s I0 tp

…I+ib

ibV

is

Is+I Rb ib

vbVs C

I =I0-ib=I0(1-s)I0

-I0

ib

Vsw

sw

8

4

/

8/

5

P

P

t

Resulting switching probabilities after a pulse lasting tp:

Switching rates

Switching ratesProbability to exceed threshold during "counting time"

p+0 / p-

0

200 400 600 800 1000

1.5

2

2.5

3

3.5

20 000

2000 Increase

I

/ e I

/

/sw 0/ /1 1

1

p

p

t

t

P p

e

0/ p

I =I0(1-s)

p+0, p-

0

2 4 6 8 10 12 14

0.1

0.2

0.3

0.4

0.5

poisson

/ e I

0/ /

1log 1

p

Resulting switching probabilities after a pulse lasting tp:

Switching rates

Switching ratesProbability to exceed threshold during "counting time"

/

/sw 0/ /1 1

1

p

p

t

t

P p

e

0/ p

I =I0(1-s)

0/ /

1log 1

p

0.025 0.05 0.075 0.1 0.125 0.15 0.175

0.1

0.2

0.3

0.4

0.5

1-s

0.23 µA

1.96 µA

p+0, p-

0

p+0 / p-

0

200 400 600 800 1000

1.5

2

2.5

3

3.5

20 000

2000 Increase

I

/ e II0=0.83 µA=0.65 ns

1

4000

I µA

Ie

300I

e

Resulting switching probabilities after a pulse lasting tp:

Switching rates

Switching ratesProbability to exceed threshold during "counting time"

/

/sw 0/ /1 1

1

p

p

t

t

P p

e

0/ p

I =I0(1-s)

0/ /

1log 1

p

1-s

p+0, p-

0

(log scale)

0.025 0.05 0.075 0.1 0.125 0.15 0.175

-10

-8

-6

-4

-2

p+0 / p-

0

200 400 600 800 1000

1.5

2

2.5

3

3.5

20 000

2000 Increase

I

/ e I

p+0

p-0

I0=0.83 µA=0.65 ns

0.23 µA

1.96 µA

0. 49 µA

1.47 µA

0. 98 µA

Resulting switching probabilities after a pulse lasting tp:

Switching rates

Switching ratesProbability to exceed threshold during "counting time"

/

/sw 0/ /1 1

1

p

p

t

t

P p

e

0/ p

I =I0(1-s)

0/ /

1log 1

p

0.025 0.05 0.075 0.1 0.125 0.15

2

4

6

8

10 Increase

I

1-s

0.23 µA

0. 49 µA

0. 98 µA1.47 µA

1.96 µA

1-s

1.96 µA

p+0, p-

0 0.025 0.05 0.075 0.1 0.125 0.15 0.175

-10

-8

-6

-4

-2

0.23 µA

1.96 µA

0. 49 µA

0. 98 µA1.47 µA

p+0 / p-

0

p+0

p-0

I0=0.83 µA=0.65 ns

0.85 0.9 0.95 1s

1 mHz

1 Hz

1 kHz

1 MHz

evian

ledom +

-

Rates ±

0.2

3 µ

A

0. 4

9 µA

0. 9

8 µA

1.47

µA

I s =

1.9

6 µA

0.85 0.9 0.95 1s

1

2

3

4

5

R

evianledom

Ratio of rates

0.23 µA0. 49 µA

0. 98 µA1.47 µA

1.96 µA

I0=0.83 µA=0.65 ns

Switching rates

R

Resulting switching probabilities after a pulse lasting tp:

Switching rates

Probability to exceed threshold during "counting time"

/

/sw 0/ /1 1

1

p

p

t

t

P p

e

0/ p

I =I0(1-s)

0/ /

1log 1

p

I0=0.83 µA=0.65 ns s

Characterisation at equilibrium

0.2

0.4

0.6

0.8

1

sw/ P

s0.87 0.88 0.89 0.9

t - s I0

s I0

…

ib

V

ib

Rb ib

vb

(no current)

C

Characterisation at equilibrium

0

1

s0 1

ideal threshold detectorsw

/ P

0.2

0.4

0.6

0.8

1

sw/ P

s0.87 0.88 0.89 0.9

V

ib

Rb ib

vb

(no current)

C

NOT an ideal threshold detector

JJ dynamicsI

V ibC

irC

q

rC

r

b

C

I

V ir

i i

q

C

Josephson relations :

0 0

0

( )

n

2

si

e

V

II

0

20 0( sin ) cos , with

J J JC E rCE E Is

frictiond

dU

U U

20

Jp

E

C

supercurrent branch : 0

r

JJ dynamicsI

V ibC

r

Josephson relations :

0 0

0

( )

n

2

si

e

V

II

0

2 ( sin ) cos J JC E r noiss CE e term

frictiond

dU

U

20

Jp

E

C

in

U

3 / 2

esc

( )exp

2

4 21

3

( )

( )

p

B

J

T

EU s

k

s U s

s

Escape rate (thermal) :

rC

r

b

C

I

V ir

i i

q

C

(Quantum tunneling disregarded)

0.87 0.88 0.89 0.9s

102

103

104

zH

Fit I0 and T with theory of thermal activation :I0 = 0.83 µAT= 115 mK

Characterisation at equilibrium

s

0.2

0.4

0.6

0.8

1

sw/ P

s0.87 0.88 0.89 0.9

3 / 2

esc

( )exp

2

4 21

3

( )

( )

p

B

J

T

EU s

k

s U s

s

escsw 1 ptP e

Applying a current in the NS junction

sw/ P

s

V

is

Is+I Rb ib

vbVs Rt=1.16 k C

Is=0.98 µA

0.76 0.78 0.8 0.82

0

0.2

0.4

0.6

0.8

1is tuned arbitrarily ! (isIs)

shift on s between the 2 curves

Applying a current in the NS junction

sw/ P

ssw sw10 ( ) P P

V

is

Is+I Rb ib

vbVs Rt=1.16 k C

Is=0.98 µAcount on Npulses=105 pulses

1 sw swpulsesN N P P

(binomial distribution)

1

sw sw

pulses

P PP

N0.76 0.78 0.8 0.82

0

0.2

0.4

0.6

0.8

1

significant difference

- Qualitative agreement with naive model- Small asymetry visible :

0.23

µA

0. 4

9 µA

0. 9

8 µA

1.47

µA

I m =

1.9

6 µA

+ -

with a current in the NS junction

0.85 0.9 0.95 1s

1 mHz

1 Hz

1 kHz

1 MHz

evian

ledom

s I0 (µA)

/

sw 1 ptP e

0.62 0.66 0.7 0.74

100 Hz

1 kHz

10 kHz

100 kHz

I s=

s

with a current in the NS junction

0.85 0.9 0.95 1s

1 mHz

1 Hz

1 kHz

1 MHz

evian

ledom

/ sw 1 ptP e

0.23

µA

0. 4

9 µA

0. 9

8 µA

1.47

µA

I m =

1.9

6 µA

0.62 0.66 0.7 0.74

100 Hz

1 kHz

10 kHz

100 kHz

1log(1 )

1 1

1 log(1 )

sw

p

sw

sw sw

pulses

Pt

P

P PN

0.2 0.4 0.6 0.8 10

2

4

6

8

10

12 pN

swP

I s=

s I0 (µA)s

search at larger deviations ?

+ artifacts

1 1swP

with Q(s)=(r C p(s))-1

1) Modification of T by I2 (shot noise)

Beyond the ideal detector assumption(theory: J. Ankerhold)

( )

( )

B

Seff

p

TsIe Q

CT

sk

I

ibC

r

inis

Is+I

Vs

inoise

4

niBk

ST

r2

noisei sS e I

s

r = 1.6 Best fit of using

with Q(s)=(r C p(s))-1

1) Modification of T by I2 (shot noise)

0.62 0.66 0.7 0.74

Pulse height I I2µA100 Hz

1 kHz

10 kHz

100 kHz

theoryexperiment

0.23

µA

0. 4

9 µA

0. 9

8 µA

1.47

µA

I s =

1.9

6 µA

Qualitative agreement

Beyond the ideal detector assumption(theory: J. Ankerhold)

s I0 (µA)

( )

( )

B

Seff

p

TsIe Q

CT

sk

0.75 0.8 0.85

0.2

0.3

0.4

Tffe(K)

2) Rates asymmetry caused by I3

Beyond the ideal detector assumption

2

0

3exp ( )

s

eff

p

B

I

Tksf

0.62 0.66 0.7 0.74

100 Hz

1 kHz

10 kHz

100 kHz

/

s I0 (µA)

0.23

µA

0. 4

9 µA

0. 9

8 µA

1.47

µA

I s =

1.9

6 µA

is

Is+I Rb ib

vbVs Rt=1.16 k Cis tuned arbitrarily ! (isIs)

shift on s between the 2 curves

2) Rates asymmetry caused by I3

Step 1: shift curves according to theory

Beyond the ideal detector assumption

2

0

3exp ( )

s

eff

p

B

I

Tksf

0.23

µA

0. 4

9 µA

0. 9

8 µA

1.47

µA

I s =

1.9

6 µA

shift from theory

I0s (µA)0.62 0.66 0.7 0.74

100Hz

1 kHz

10kHz

100kHz

/

is

Is+I Rb ib

vbVs Rt=1.16 k Cis tuned arbitrarily ! (isIs)

shift on s between the 2 curves

2) Rates asymmetry caused by I3

Step 1: shift curves according to theory

0.23 µA

0. 49 µA

0. 98 µA

1.47 µA

Is = 1.96 µA

Quantitative agreement

Beyond the ideal detector assumption

s

theoryexperiment

0.75 0.8 0.85

1.2

1.3

1.4

2

0

3exp ( )

s

eff

p

B

I

Tksf

Step 2: compare s-dependence of with theory (using experimental Teff)

0.23

µA

0. 4

9 µA

0. 9

8 µA

1.47

µA

I s =

1.9

6 µA

shift from theory

I0s (µA)0.62 0.66 0.7 0.74

100Hz

1 kHz

10kHz

100kHz

/

Conclusions

JJ = on-chip, fast current threshold detector…

… with imperfections

0.23 µA

0. 49 µA

0. 98 µA

1.47 µA

Is = 1.96 µA

s0.75 0.8 0.85

1.2

1.3

1.4

0.2

0.4

0.6

0.8

1

sw/ P

s0.87 0.88 0.89 0.9

0.2

0.4

0.6

0.8

1

sw/ P

s0.87 0.88 0.89 0.90.87 0.88 0.89 0.9

… able to detect 3d moment in current fluctuations

to be continued …

optimized experiment on tunnel junction

experiments on other mesoscopic conductors (mesoscopic wires)