Transcript Measuring And Reducing The 1/F Noise In Josephson Junctions For

Measurements of the 1/f noise in
Josephson Junctions and the
implications for qubits
Jan Kycia, Chas Mugford- University of Waterloo
Michael Mueck- University of Giessen, Germany
John Clarke- University of California, Berkeley
Readout SQUID
The Group
Chas Mugford
1/f noise
Shuchao Meng Lauren Lettress
SQUID-sSET
TES sensors
Jeff Quilliam
Ho: YLiF4
Nat Persaud
Liquifier
Jeff Mason
SQUID NMR
dc-SQUID
Ib
The most sensitive
magnetometer
~1 µFo/ (Hz) 1/2
Io
Io
L
Lin
V
Josephson Junctions
Oxide barrier
Superconductor, 1
2
IC
The superconducting order parameters are:
1 = |1(x)| ei1 , 2 = |2| ei2
The phase difference between the superconductors is  =  2 -  1.
As the two superconductors are brought closer together, allowing electrons to
tunnel, the phases start to interact.
Josephson (1962) predicted a phase dependent energy = -EJ cos,
where EJ =
hIco
h D(0)
2e = (2e)2 RN .
d 2eV
=
dt
h ,
IS = Ico sin (),
Resistively and Capacitively
Shunted Junction Model
EJ
C
I = Icosin  + V + C dV
R
dt
Use V =
R
h d ,
2e dt
IS = Ico sin (),
d 2eV
=
dt
h
EJ =
h D(0)
(2e)2 RN
hC d2
h d
+
+ Ico sin  = I
2
2e dt
2eR dt
Tilted washboard model is the mechanical analog,
Tilt  I
with a particle of mass ~ C, moving along an
position  
axis, , in a potential, U() = -Icocos  - I, with
h d
a viscous drag force,
.
2eR dt
velocity  V
One period = Fo
J
F/2L
The DC SQUID
I
I/2 +J
J
1
I/2 -J
F
Imax
F/Fo
2
Fo/L
1
2
1
2
F/Fo
V
F/Fo
A flux locked loop using a high frequency flux modulation
is used to provide a flux to voltage converter with fixed gain
and large dynamic range.
Lfeedback
Ib
V
Io
Io
L
dV
Lin
1
dF
2
F/Fo
V
Magnified Image of DC SQUID
dc
SQUID
2x2µm
Junctions
Input
coil
Palladium
Shunt
resistor
Shunts provide
required dissipation
but also produce noise.
SQUID as a near-quantum-limited
amplifier at 0.5 GHz
M. Mueck, J. B. Kycia, and John Clarke, APL 78, 967
(2001).
Find “Self Heating” at low
temperatures
Loss of temperature dependence, at low temperatures,
is frequency independent
Wellstood et al found that
self heating can be reduced
by adding a cooling pad
to the shunt resistor.
The Hamiltonian,
H = -EJcos1 -EJcos2 + Ec(Q/e)2
if EJ / EC > 1,
 is a good quantum number,
Q fluctuates.
if EJ / EC < 1,
 fluctuates,
Q is a good quantum number.
Phase fluctuations allow
the particle to diffuse down
the washboard;
d 0  V0
dt
Screened room
Lock-in
Transport
reference
Measurement input.
Circuit
bias
with filters AC
~
 0.1 nA
Copper powder filters
.
. .
300 K
4.2 K
x100
x1000
LC filters
RC filters
Cu filters
Follow design of Martinis,
Devoret, Clarke,
20 mK
Phys. Rev. B, 35, 4682 (1987).
Cu filters
Sample
Temperature and dissipation
dependence of sSET
G ~ g2 ; ohmic
dissipation, g = RK
Rg
(Ingold, Grabert PRL 1999)
T
1/3
g
G ~ 5/3 ; transmission line
T
(Wilhelm, Schön, and Zimanyi)
New configuration
Provides in situ control of EJ , Ec , g and T.
H = -EJ()cos1 -EJ () cos2 + Ec(Q/e)2 + H(R2deg)
1 m
SEM image
courtesy of Dan Grupp.
Rimberg, Ho, Kurdak, Clarke
PRL 1997
Wagenblast, Otterlo, Schon, Zimanyi
PRL 1997
Good review: Leggett, Chakravarty, Dorsey, Fisher, Garg, Zwerger
Rev Mod Phys (1987).
Superconducting Qubits:
Charge based qubits: “Cooper pair box”
Demonstrated Rabi oscillation: Nakamura et al, Nature 398, 786 (1999).
Improved read out scheme, decoherence time ~ 0.5 s (Q = 25,000):
Vion et al, Science 296, 886 (2002).
Flux based qubits:
Demonstrated energy splitting dependence on applied magnetic flux:
Friedman et al, Nature 406 43 (2000), van der Wal et al, Science 290,
773 (2000).Coherent Oscillations observed with a dephasing time of
20 ns and a Relaxation time of 900 ns: Chiorescu et al, Science 299,
1869 (2003).
Phase based qubits:
Exhibited Rabi oscillations between ground state and 1st excited state of a current
biased Josephson junction in its zero-voltage regime: Yu et al,
Science 296, 889 (2002), Martinis et al PRL 89, 117901 (2002).
Sources of Decoherence:
•External flux noise
•Nyquist noise currents in nearby metal objects
•Noise in the measurement scheme
•Motion of trapped charge
•1/f “flicker” noise in the critical current of the Josephson Junction
The goal of our experiment is to measure the level of 1/f noise in the
critical current of a resistively shunted Josephson Junction.
Once the measurement is made, we can:
•Measure the temperature, time, material, and fabrication parameter
dependence of the 1/f noise level.
•Estimate the upper limit of the coherence time of superconducting
qubits due to these noise sources.
•Make optimal qubits by selecting the device configuration to
minimize the noise sources.
Flux Qubit
van der Wal et al, Science
290, 773 (2000).
Chiorescu et al, Science
299, 1869 (2003).
- Small loop with three Josephson junctions produces the
flux qubit.
- Hysteretic DC SQUID is used to read the flux state.
Ramsey Fringes in Flux Qubit
90
Switching probability (%)
80
70
60
A = 9 dB
F0 = 6.607 GHz
Df = 250 MHz
50
0
5
10
15
20
25
30
35
time between two /2 pulses (ns)
I. Chiorescu, Y. Nakamura, C.J.P.M. Harmans, and J.E. Mooij, Science 299, 1869 (2003).
40
45
50
“Quantronium”
D. Vion, A. Aassime, A. Cottet, P. Joyez, H.
Pothier, C. Urbina, D. Esteve, M. H. Devoret,
“Manipulating the Quantum State of an Electrical
Circuit”,
“Phase” Qubit
Decoherence in Josephson Phase Qubits from Junction Resonators
Simmonds, Lang, Hite, Nam, Pappas, and Martinis, Phys Rev Lett, 93, 077003-1, (2004).
Resonances Observed
-- likely due to defects (fluctuators)
abc
d
ef
Decoherence in Josephson Phase Qubits from Junction Resonators
Simmonds, Lang, Hite, Nam, Pappas, and Martinis, Phys Rev Lett, 93, 077003-1, (2004).
1/f Noise: Dutta-Horn Model
Dutta and Horn, Rev Mod Phys, 53, 497 (1981)
Random telegraph signal is produced by random transitions between the
states of a double potential well. Define 1/t1 and 1/t2 as the probability of a
transition from state 1 to 2 and 2 to 1 respectively. If 1/t = 1/t1 = 1/t2 then
the power spectrum is a Lorentzian of the form
S(f)  t / [1+(2ft)2]
If the transitions are thermally activated then the characteristic time is
given by
ti = toexp(Ui/kBT), where 1/to is an attempt frequency.
S(f,T) is linear in T because the kernel moves through the distribution of
RTS’s as the temperature varies, selecting only those processes that have
characteristic frequencies in the window of interest.
Mechanism Behind 1/ƒ Critical Current
Fluctuations in Josephson Junctions
The currently accepted picture of the mechanism behind critical current
fluctuations involves traps within a Josephson junction.
An electron is trapped in the tunnel barrier and is subsequently released.
While trapped, the barrier height and hence critical current is modified
temporarily.
For a junction of area A the change in critical current is modified by the
change in area due to an electron DA. DIc=(DA/A)Ic
s.c.1
barrier
height
barrier
s.c.2
electron
trapped
I
no electron
trapped
trap
z
V
Dephasing due to current fluctuations and critical
current fluctuations
Critical current fluctuations with a l/f spectral density are potentially a
limiting source of intrinsic decoherence in superconducting qubits..
W = the frequency of oscillation between
the +0.5Fo and –0.5Fo state.
Dale Van Harlingen et al, PRB (2004).
Methods of Measuring 1/ƒ Noise of the
Critical Current of a Josephson Junction
•
Critical current fluctuations have been measured in the non-zero voltage state.
F.C.Wellstood, PhD thesis, University of California, Berkeley 1988.
B.Savo, F.C.Wellstood,, and J.Clarke, APL 49, 593 (1986).
V.Foglietti et al., APL, 49, 1393 (1986)
R.H.Koch, D.J. van Harlingen, and J.Clarke, APL, 41, 197 (1982).
F.C. Wellstood, C. Urbina, John Clarke, APL, 5296, 85 (2004).
Fred Wellstood’s Thesis
Berkeley
•
Is the 1/f noise the same when the junction is in the zero voltage state? We
measured the critical current fluctuations using the same SQUID operated
as an RF SQUID in the dispersive regime.
Comparing different junctions:
Invariant noise parameter
Normalize current noise spectrum to the critical current
Choose T = 4.2K and 1Hz.
2
S I2 (1Hz ,4.2 K ) I c
But this does not take into account junction area.
For a junction of area A and if the area blocked by a single trap is DA,
then change in critical current for a single fluctuator is DIc = (DA /A)Ic
If n is the number of traps per area, then the critical current spectrum
should scale as:
SI2 ~ n A (DA /A)2Ic2 = n DA2 (Ic2 /A)
Van Harlingen et al found that all values of n DA2 remarkably similar for all
measured junctions.
SI2 scales as (Ic2 /A)
Scaled quantity invariant quantity
(van Harlingen et al. PRB 2004)
Wellstood et al.
average value
of 6 junctions
26
Lukens et al.
IEEE 2005
Also see “slower
than linear” T
dependence
Measuring 1/ƒ Noise Due to Critical Current Fluctuations
in the Non-Zero Voltage State
Readout SQUID
DC SQUID and read-out SQUID circuit
•The sample SQUID is voltage biases.
•The readout SQUID measured the current running though the 2W resistor.
•Fluctuation in the critical current leads to a redistribution of the currents
flowing through the junction and the resistor.
rf tight - low field superconducting
sample container
Readout
SQUID
rf tight SMA
connectors
Sample SQUID
Superconducting
lead shield
rf tight copper sample
container
Coaxial µ-metal shields
1/f noise in DC biased junction
Noise Power (F0 /Hz)
Applying Current Bias Reversal
DC current bias method
-10
10
w/o bias reversal
bias reversal
8
6
2
10
4
2
-11
8
6
2
1
3
4
5
6 7 8 9
Frequency (Hz)
10
Current bias reversal eliminates 1/f noise,
therefore this 1/f noise is not due to flux noise.
Critical current fluctuations
due to a single fluctuator
Ic = 2.5uA
DIc = 0.65nA
This corresponds to a trap radius of ~ 5.6nm
Reading out an rf SQUID in the Dispersive Regime
Vrf
ƒmod
rf SQUID and FET amplifier circuit
- A tank circuit is driven off-resonance with a 360-MHz current of fixed amplitude.
- The tank circuit voltage is read out with a low noise amplifier cooled to 4.2K.
- Fluctuations in the critical currents of the two junctions modulate the SQUID
inductance and thus the resonant frequency of the tank circuit.
Comparing the zero-voltage noise
measurement method to the non-zero voltage
noise measurement method
10
-9
rf 1.6 K
rf 4.2 K
dc 4.2 K
SF (F0 /Hz)
4
-10
2
10
2
10
4
2
-11
4
2
10
-12
2
1
3
4 5 6 78
2
3
10
Frequency (Hz)
•No difference between the measurements.
•The 1/f noise is temperature dependent.
Annealing
Study
Annealing lowers
critical current
and lowers noise
Comparison of Noise Parameter
Best Sample
Van Harlingen et al. 12
Wellstood et al.
26
Lukens et al.
1.5
Conclusion
We have demonstrated that the l/f noise in a dc SQUID due to critical current
fluctuations has the same magnitude in the zero voltage and non-zero voltage
regime.
Thus, the levels of critical current l/f noise measured by others in the nonzero
voltage state should pertain to qubits operated at zero voltage.
Measured noise of different junctions, reduce 1/f noise.
Future Experiments
Temperature dependence of 1/f noise down to dilution refrigerator temperatures.
The dispersive method has no dissipation - best for low temperatures.
We can cut away the shunt resistors to see if they are somehow responsible for
noise.
Continue varying processing parameters.
Study dissipation is submicron Josephson junction.
New Device will allow the in situ control
of EJ, EC, and dissipation.
Temperature and dissipation
dependence of sSET
Outline
- Describe how Josephson Junctions and SQUIDS work.
- Describe how superconducting qubits work.
- Explain why 1/f noise is relevant to superconducting
qubits.
- Present results on 1/f noise measurements.
B.L.T. Plourde, J. Zhang, K.B. Whaley, F.K.W., T.L. Robertson, T. Hime, S. Linzen, P.A.
Reichardt, C.E. Wu, and J. Clarke PRB 70, 140501(R) (2004).
Tunable coupling via curent
Bias
current:
 F 
I B = 2 I c cos  
 sin 
 F0 
Screening
current:
 F 
I s = I c sin  
 cos 
 F0 
Extra flux at constant bias
• directly increases screening
• increases γ → indirectly reduces
screening
rf SQUID
Two kinds of behaviour are observed in rf SQUID loops depending
of the “SQUID hysteresis parameter” rf. The difference is seen in
the applied flux e vs the flux threading the loop .
F
F0 1
The SQUID hysteresis parameter is defined as:  rf =
2LI C
F0
rf <1
0
-1
• If rf <1 the SQUID is dispersive. Ic is never exceeded
• If rf >1 the SQUID is hysteretic or dissipative.
R
Fe
IS
F
L
0
F
F0
1
Fe
F0
rf >1
1
0
-1
0
1
Fe
F0
Abstract
Critical current fluctuations may be a major source of intrinsic
decoherence of qubits made from Josephson junctions.
We have measured the 1/f noise due to critical current fluctuations
in macroscopic ( area  2  2 m2 ) Josephson junctions.
We have exploited two ways for measuring critical current
fluctuations, one way where we directly measure changes in the
critical current of a voltage biased junction, and a second way in
which we measure 1/f flux noise in an rf SQUID running in the
dispersive mode. With both methods, we find the magnitude of the
critical current fluctuations, at a temperature of 4.2K, to be Ic/Ic 
10-5 at a frequency of 1 Hz.
The Bloch sphere
Convenient representation of the two-state Hamiltonian
and state
 x

 = y

 z





H =  Beff  
Beff
Tank Circuit Coupled to Josephson Inductance
Using the Josephson relations:
J = J1 sin  and
 2eV
=
t

A Josephson element can be described as a nonlinear inductor by
deriving the relationship V = d (LJ ( I ) I )
Where:
dt
arcsin( I / I1 )

L J (I) = L1
where L1 =
I / I1
2eI1
When a junction is inserted into a superconducting loop it’s behaviour
affects the total inductance of the loop.
dF
e
The effective inductance of a SQUID can be approximated by Leff =  di


The flux threading the loop is F = F e + Li and the circulating supercurrent i =  I c sin  2 F 


1
F0 

Combining these three it follows that: Leff = L 1 +

  rf cos( 2F / F 0 ) 
Coupling a SQUID loop to a inductor of a tank circuit yields an effective
~
tank circuit resonance modified by the SQUID loop for rf<<1: LT  LT 1   2  rf cos 2F e
(
)
The flux qubit
 = 0.8
(GHz)
E =  2 + D2
5
f
F
10
660 ± 60 MHz
0
0.500
φ

D eff
1  I p ( F  F0 / 2)
Hˆ eff = 

D eff
-I p ( F  F0 / 2) 
2
0.501
F
0.502
(F0)
0.503
Evidence for superposition of
macroscopic states
C.H. van der Wal, A.C.J. ter Haar, F.K.W., R.N. Schouten, C.J.P.M. Harmans, T.P. Orlando, J.E. Mooij, Science 290, 773 (2000).
Measuring 1/ƒ Noise Due to Critical Current Fluctuations in the
Zero Voltage State
Using an rf SQUID in Dispersive Mode
I =
F
 I 
F0

L+
arcsin 

2I
Ic 




L =
2LI C
1
F0
LJ
F 0 / 2I c
LJ = 
F 0 / 4 I c
; I = 0, F = 0
; I = Ic , F = F0
Applying an external flux gives rise to a circulating current which in turn
modifies the inductance of the junctions.
Fluctuations in the critical current Ic appear as equivalent to flux noise.
Operating the SQUID in the dispersive regime means that the screening
current imposed by an applied flux is always smaller than the critical current
Ic of a junction.
Critical Current Noise Specific Measurement Techniques
The spectral density components of low frequency SQUID noise are represented by.
2
2




V
V
 V 
 S I 0 ( f ) + 2
 S I 0 ( f )
SV ( f ) = 
 S F ( f ) + 
 F 
  (I 01 + I 02 ) 
  (I 01  I 02 ) 
2
SF(f): flux noise due to motion of flux vortices.
SI(f): critical current noise – in-phase fluctuations are represented by the
second term and the out-of-phase component is represented by the third.
AC flux modulation with lock-in detection rejects only the in-phase component
of the critical current noise, furthermore it does not affect noise due to flux
motion.
Reverse bias scheme will eliminate out-of-phase fluctuations in the critical
current but does not affect out-of-phase fluctuations due to flux motion.
Thus ac modulation with reverse bias will eliminate in-phase and out-of-phase
fluctuation due to critical current fluctuations. Therefore if excess noise due to
motion of flux vortices exists, the out-of-phase component will still be observed
Lab at Waterloo
• Dilution refrigerator (Base temperature 12 mK)
• e-beam lithography (for fabrications of sub-micron devices)
• Optical lithography (for fabrications of large number
of micron scale multi-layered devices)
• Measurement electronics (low noise environment, low 1/f noise)
Conclusion of sSET work