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)