Noise and decoherence in the Josephson Charge Qubits Oleg Astafiev, Yuri Pashkin, Tsuyoshi Yamamoto,...
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Transcript of Noise and decoherence in the Josephson Charge Qubits Oleg Astafiev, Yuri Pashkin, Tsuyoshi Yamamoto,...
Noise and decoherence in the Josephson Charge Qubits
Oleg Astafiev, Yuri Pashkin, Tsuyoshi Yamamoto,Yasunobu Nakamura, Jaw-Shen Tsai
RIKEN Frontier Research SystemNEC Fundamental Research Laboratories
Outline
• The Josephson charge qubit
• Single-shot readout with charge trap
• Measurements of energy relaxation
• Charge fluctuators and energy relaxation
The Josephson Charge Qubit
b
g
C
ennU
2
2 22
Charging energy (for Cooper pair):
Josephson energy: EJ
EJ
Cb >> Cg
E
ng =Vg Cg /e
Reservoir
BoxCb
Cg
2e2
Cb
Control gate
ng =VgCg
e
Degeneracy
>> kT
2e2
Cb
>> EJ
0 2 31 4
The Hamiltonian
sincos2 xz
EH
C
neU g
2222
JEUE E
EJ
tan
2/sin
2/cos
2
E
2/cos
2/sin
2
E
Eigenstates
Eigenenergies
|0
|1 2
10t = 0:
t > 0: 2/2/
2
1 titi JJ ee
2/sin12/cos0 tit JJ
2
cos11
tP J
Coherent Oscillations
J
J
E
P
t
1
0
E
t
|12
|12
-pulse: J t =
q
EJ
0
1
2
3
4
01234
0
1
2
3
4
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
0 10 20 30
I 1
(pA
) I 2
(pA
)
I 1 (p
A)
I 2 (p
A)
t (ns)
13.4 GHz
9.1 GHz
f (GHz)
Cooper-Pair BoxCooper-Pair BoxCooper-Pair BoxCooper-Pair Box
GateGateGateGate
SQUIDSQUIDSQUIDSQUID
Probe Probe JunctionJunctionProbe Probe
JunctionJunction
1m
Probe Probe JunctionJunctionProbe Probe
JunctionJunction
Cooper-Cooper-pair Boxpair BoxCooper-Cooper-pair Boxpair Box
GateGateGateGate
SQUIDSQUIDSQUIDSQUID
Al/AlOAl/AlOxx/Al tunnel junctions/Al tunnel junctionsAl/AlOAl/AlOxx/Al tunnel junctions/Al tunnel junctions
Pulse induced current in SQUID – box – probe junction
circuit is measured
0+1 I = 2e ||2/Tr
Tr
Control pulse sequence
t (ns)0 1
2e
Cooper-pair box
JE
qp1 JE
Final state read-out
A pair of qusiparticles tunnels through the probe junction biased to Vb 2/e
CC EeVE 322
ee
qp1
qp2
+ probe
Single-shot Readout
Coherent oscillations
Quasiparticle tanneling (when the trap is biased to 2/e)
Reservoir
Box
Control gate
qubit
CbCts
SET
Readout circuit
Cs
I
Cbt
Trap
Readout gate
Ct
t
readout:
control:
Pulses
C
C
C
C C
s t
b t
s
b t
S E T
Tr a pR e s e r v o ir
R ea dou t g a te
S E T g a te
C on tro l ga te
B o x
1 m
Tra
p ga
te
Bo
x ga
te
• Measurement circuit is electrostatically decoupled from the qubit
• Final states are read out after termination coherent state manipulation
Reservoir
Box
SET
Trap
gate
Readout with control -pulses
0 5 10 15 200
100
2000
4
I SE
T (
pA
)
t (ms)
11 Q
t(e)
1 11 1 0 1 0 1
tot
switch
N
NP 1
Readout pulse
Control -pulse
ng
I
0.0 0.2 0.40.00.20.40.60.81.0
P
tc (ns)
0 .8
0 .1
0.2 0.4 0.6 0.8 1
0.75
0.80
0.85
0.90
0.95
t (ns)
b
P
Quantum Oscillations
q
(e)
= /
-pulse
0 2 4 6 80.2
0.4
0.6
0.8
P
tc (ns)
Degeneracy
Crossectiont
q
0 500 1000 1500 20000
50
100
150
200
250
N (
coun
ts)
t delay (ns)
Relaxation to the reservoir
Readout
td
Control -pulse
220 exp(-t/288)+32
T1res = 288 ns
Ntot = 327
Reservoir
Box
SET
Trap
Relaxation to the Trap
0 100 200 300 400 5000
50
100
150
200
250
300
N (
coun
ts)
t width (ns)
Control -pulse
twidth
Teff = (1/T1res + 1/T1
trap)-1 = 31 ns
ReadoutReservoir
Box
SET
Trap
90.031288
2881
11
11
nsns
ns
TT
TP
restrap
res
Readout efficiency
0 500 1000 1500 20000.0
0.2
0.4
0.6
0.8
P
t delay (ns)
0 2000 4000 6000 80000.2
0.3
0.4
0.5
0.6
0.7
P
t (ps)
N0+ N Exp[-t/] N
0 = 152
= 5380 ps
90.000 P
Reservoir
Box
SET
Trap
Two-level Systemas a Quantum Noise Spectrometer
Two-level systemTLS
Environment
zxz tUE
H sincos2
Electrostatic energy noise
Charge basis:
Eigenbasis:
tan = EJ
E
U
EJ
U
z
x
transitions
dephasing
Dephasing Transitions
U
U22 UEE J
sincos2
xzz tUE
H
E
Charge qubit q charge noise spectral density: Sq()
deqqS iq 0
2
1SU() = (2e/C)2Sq()
1 = 22
SU()Relaxation rate: sin2
Dephasing: 1
0
2
2
2
cos
dSU
SU
Dephasing
RelaxationExcitation
0 100 200 300 400 5000.0
0.2
0.4
0.6
P
ta (ns)
T1 time measurements
ng
E
0
1
0 1
ta
P(1) exp(-ta/T1)
time
timereadout pulse
Control -pulse Adiabatic pulse
0 1 2 3 4 5
10
100
T1 (
ns)
Vp (V)
Degeneracy
T1 time vs Gate Voltage
10
100
100 50 0 -50
0.8 0.6 0.4 0.2 0.0 -0.2 -0.4
T1 (
ns)
B = 8 Gs (Ej = 3.7 GHz) B = 5 Gs (Ej = 6.0 GHz) B = 0 Gs (Ej = 8.1 GHz)
E (GHz)
qg (e)
15 20 25 30 351E-3
0.01
0.1
1
E = 400 eV (off degeneracy)
(
ns-1
)
EJ (eV)
E = EJ (degenercy)
~E2
J
EJ-dependences
Degeneracy
Off degeneracy
C
C
C
C C
s t
b t
s
b t
S E T
Tr a pR e s e r v o ir
R ea dou t g a te
S E T g a te
C on tro l ga te
B o x
1 m
Tra
p ga
te
Bo
x ga
te
Coupling to Environment through Electrical Leads
Coupling to gates:
3107.1600
1 aF
aF
C
C
b
g
Coupling to SET:
31051000600
10030
aFaF
aFaF
CC
CC
tb
tsbt
Measured relaxation time can not be explained by coupling to the external environment through electrical leads
-1.0 -0.5 0.0 0.5
107
108
Readout SET normally in ON state OFF state
(
s-1)
q (e)
Effect of the measurement SET
1 10 100107
108
109
SE/22 (
s-1)
(GHz)
c
1/f
f
The noise derived from 1 time
mKk
T cc 100
1 = 22
SU(0)
sin2
dt
St U
2
2
2 2/sin2cos
0
ExpI
2
2ln 2
2
21 tCe
ExpI
CeT
2ln 12
2
2
2
2T
tExpI
T2-2
0 100 200 300 400 500 600
I
t (ps)
I = I1(1-cos(t)exp(-(t/T
2)2/2))-I
0
T2 = 300 ps
Classical Quantum Noise
Quantum f-noise ( > 0): Classical 1/f-noise:
(kTeff)2
Do low frequency 1/f and high frequency f noises have common origin?
1/f
f
SU()
kT/ emissionabsorption
Relaxation through Fluctuators
• Dephasing is caused by 1/f noise of charge fluctuators with activation energy less than kT
• Fluctuators with activation energy of ( >> kT) accept qubit excess energy
kTE
fSq 2
1 10 10010-5
10-4
10-3
10-2
Si (
pA
2 /Hz)
f (Hz)
3x10-2pA2/Hz
Low frequency 1/f noise
Temperature dependences of the 1/f noise
1 10 10010-5
10-4
10-3
10-2
Si (
pA
2 /Hz)
f (Hz)
3x10-2pA2/Hz
T=0.055K
1 10 100
10-3
10-2
10-1
Si (
pA
2 /Hz)
f (Hz)
8x10-2pA2/Hz
T=0.5K
1 10 10010-4
10-3
10-2
10-1
Si (
pA
2 /Hz)
f (Hz)
3x10-2pA2/Hz
T=0.9K
-8 -6 -4 -2 0 2
0
5
10
15
20
25
30
35
40
45
I (p
A)
Vg (V)
0.90 K 0.85 K 0.80 K 0.75 K 0.70 K 0.65 K 0.60 K 0.55 K 0.50 K 0.45 K 0.40 K 0.35 K 0.30 K 0.25 K 0.20 K 0.15 K 0.10 K 0.055 K
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
5
6
71/
2 (1
0-3e)
T (K)
EC = 110 eV
1/2 = 6x10-3eT
Standard qubit on 400 nm thick Si3N4
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
10
12 current measurements
T2
*
= 13x10-3 eT
(
10-3
e)
T (K)
EC = 50 eV
CeT
2ln2
0 100 200 300 400 500 600
I
t (ps)
I = I1(1-cos(t)exp(-(t/T
2)2/2))-I
0
T2 = 300 ps
1/f noise in superconducting – normal SETs
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
10
12
14
B = 0T B = 0.2T (normal)
= 17x10-3 eT
EC = 125 eV
(
10-3
e)
T (K)
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
41/
2 (1
0-3 e
)
T (K)
EC = 110 eV
1/2=4x10-3 eT
GaAs
SET on GaAs substrate
SET on Al2O3
0.0 0.2 0.4 0.6 0.8 1.00
2
4
6
8
10
(
10-3
e)
T (K)
B = 0 (superconductivity) B = 0.4T (normal)
EC = 320 eV
1/2 = 7x10-3 eT
Si
Al Al2O3
SET island
Large area SETs
0.0 0.1 0.2 0.3 0.4 0.50
5
10
15
20
1/2 (
10
-3 e
)
T (K)
EC = 80 eV
Box: 3x2.3 m2
1/2= 43x10-3 eT
1/f noise properties from experiments
• does not depend on substrate type• noise appears in oxide of Al(?)• scales with SET size (area?)• saturation level at low temperatures depends on current
Basic properties of the 1/f noise caused by bistable fluctuators
22
,
s
0.1 1 100.1
1
10
S
2S
dsPS ,
kT
exp0
dPkT
dP
1P
S()
Qubit
TLS(fluctuators)
Environment at T > 0
Qubit island TLSfluctuators
C
eUq
2
The qubit is coupled to environment through charge degree of freedom
1/f noise
2313
12
3
Environment at T > 0
kT13
013 exp 031
kTDV ij
ijenijij 2coth1
2
12 2
022 2PkTqSq
kT13
012 exp
kT23
021 exp
high frequency cutoffof the 1/f noise
01312 , PP If , then
31
2
130
2 enDV
2221122112
21122112
14,,
s
0
23132112
0
2313 ,,, dsPS
k
kkV 1113
2
13
2
12
3
13231
31
11
ee
e
130
2
13
2 PV
130
2
13
2 ePV
Qubit relaxation (excitation)
22
22
qC
eV
2
22
C
eSq
02 40 PqSq
022 2PkTqSq
02 40 PqSq
kT
2
1/f low frequency noise: f high frequency noise:
Crossover frequency:
Same fluctuators contribute in the 1/f noise and the quantum f noise
Constant distribution of two energy parameters for the fluctuators is required
kT
exp0 d
kTd
021, PP 21
210
2210
ddPkTddP
02
0
23132112
0
0
2,, PkTdsPS
20
2, kT
AdsAS
AP
Two energy parameters:
Single energy:
Single energy (TLS)
12
1
2
Environment at T > 0
enDV 22
21 sin2
22sin
2coth1
2
kTDV ij
ijenij
0
High frequency cutoff
12
tan
C
eqV
2
A
P
1/f noise: << kT f noise:
< 105 Hz 1010 1011 Hz
Different TLS contribute in 1/f and f noises
Conclusion
We have demonstrated single-shot readout using charge trap
Energy relaxation of the qubit has been measured
The energy relaxation is caused by quantum f noise which has crossover frequency with 1/f noise at kT/
Nearly T2 dependence of the 1/f noise has been observed