P. Bertet Quantum Transport Group, Kavli Institute for Nanoscience, TU Delft, Lorentzweg 1, 2628CJ...
Transcript of P. Bertet Quantum Transport Group, Kavli Institute for Nanoscience, TU Delft, Lorentzweg 1, 2628CJ...
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P. Bertet
Quantum Transport Group, Kavli Institute for Nanoscience,TU Delft, Lorentzweg 1, 2628CJ Delft, The Netherlands
A. ter Haar A. Lupascu
J. PlantenbergF. PaauwJ. Eroms
C.J.P.M. HarmansJ.E. MooijI. Chiorescu
Y. Nakamura
Photon-noise induced dephasingin a flux-qubit
G. BurkardD. DiVicenzo
+
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Introduction
zq
QH 2
nI
kH
2
2)( kS
Dephasing ? 2/)0(21 SnT z (weak coupling)
Very slow and strongly coupled fluctuators
Underdamped modes strongly coupled to qubit
E. Paladino et al., Phys. Rev. Lett. 88, 228304 (2002)
M. Thorwart et al., Chem. Phys. 296, 333 (2004)
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Qubit dephased by photon noise
zq
QH 2
aahH pp
ig Temperature T
Dispersive regime : ipq g
Shift of oscillator frequency z
20
Shift of qubit frequency 0n
Coupling
Quality factor Q
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Qubit dephased by photon noise
Photon fluctuations )(tn
Qubit frequency
)()( 00 tnt qq
Dephasing factor
Phase shift t
dtntnt0
0 '))'((2)(
2/)(exp)(exp 2tti
t t
dtdtttC0 0
20 ''')'','()(2exp with 2)''()'()'','( ntntnttC
around n
A. Blais et al., PRA 69, 062320 (2004)Dephasing time T
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2) Thermal fluctuations in the non-driven oscillator
Thermal field :
n
n
n
nnp
11
1)( 11)/exp( kThn p
)exp()1()0,( tnntC )1()2( 2
0
nnT
D. Schuster et al., PRL 94, 123602 (2005)
Cf also M. Brune et al., PRL 76, 1800 (1996)
Qubit dephased by photon noise
1) Oscillator driven by a coherent field
nn
en
!
2
2
)2/exp()0,( tntC
nT
20 )2(2
2n
Measurement induced dephasing
Photon shot noise
Q/0
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Flux-qubit coupled to SQUID plasma mode
Our circuit :Flux-qubit
DC-SQUIDplasma mode
)( bi Ig
Optimal points (with respect to photon noise) whenever
Our measurements : qubit coherence limited by thermal fluctuations in plasma mode
)1()2( 20
nn
T
1) Quantitative agreement with formula
2) Thanks to our circuit geometry, coupling constants
0),(0 xbI
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The flux-qubit
Q
2/0 x
0
2 xQ
Josephsonjunctions
elQJ EEH )(
1 control parameter
Al/AlOx/Al junctions by shadow evaporation + e-beam lithography
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0.48 0.50 0.52
0
40
80F
requ
enc
y(G
Hz)
|0>
|1>
|2>
|3>
Q/2
qubit
Qubit energy levels
EJ=225GHz
EC=7.2GHz
=0.76
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Persistent-current
Property of states |0> and |1> : iH
iIx
i
Useful to measure the qubit state
-0.02 0.00 0.02-300
0
300I (
nA)
I0
I1
Q-
-Ip
+Ip
|0>
|1>|1>|0>
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Fre
quen
cy
(
GH
z)
0.50
5
100
1
Q/2
Two-level approximation
Flux-noise optimal point
0Q
Q
d
df
In the 0 1 basis, xzq
hH
2
2/))(/( Qp eI
(cf Saclay)
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Control of the qubit state
Arbitrary state
Rabi 01tan
h
I xpRabi
sin2
Rotation axis : ()
Angle : tRabi 201
2 )(
x
x+xcos(2t+)
Microwave pulse t
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Our detector : a hysteretic DC-SQUID as on-chip comparator
0 10
4
Ic ( A
)
Sq /0
Persistent-current and detection of the qubit state
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Qubit inductively coupled to SQUID
ix
CxC
iC MI
d
dIII
)()(
IC depends on qubit state (i)
Persistent-current and detection of the qubit state
P(1)Psw
5.4 5.6
0
50
100
Psw
itch
(%
)
Current Ib ( A)
|0>
Theoretical |1>
rela
xatio
n
P(1)=Psw
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Persistent-current and detection of the qubit state
SQUID shunted by a capacitor
PLASMA MODE
shJ
pCLL
2
1
)Re(/2 1 ZCQ shp
)2/1( aahH pp
)(0 aaaa
),( bSqJ IL
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Coupling of the qubit and the plasma mode
Complex : qubit Circ current J Plasma mode currentM dJ/dIb(Ib)
2 different effects :
a) Effective inductive coupling with tunable mutual inductance
b) Flux dependent SQUID Josephson inductance
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SQUID circulating current
-0.5 0.0 0.5
Circ
. cur
rent
JIb/2IC
dJ/dIb=0
dJ/dIb(Ib)
Ideal symmetric SQUID : dJ/dIb(0)=0
Including asymmetries : 0*)( bb
IdI
dJDecoupling current
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Coupling of the qubit and the plasma mode
1) Measurement shift
-0.02 0.00 0.02
Ene
rgy
(GH
z)
21
Cur
rent
(A
)
(e/Ip)
221 )()()()( aaIhgaaIhgH xbxbI
bJb dI
d
LLIg
01 2
1)(
2
22
02 4
1)(
bJb
dI
d
LLIg
2) Coupling hamiltonian
NON RESONANT
inductive Flux-dependent Josephson inductance
)()( bx I
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The sample
Ib
V
Microwave antenna
Csh
G. Burkard et al., cond-mat/0405273
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1k 3k
The setup
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Qubit spectroscopy
timetrig
ger
Ib pulse
read-outt
Microwave pulseat frequency f
Parameters : =5.85GHz, Iq=270nA
5.7 5.8 5.9 6.0
Psw
itch
Frequency (GHz)
B
-0.01 0.00 0.010
5
10
15
20
25
Larm
or fr
eque
ncy
(GH
z)
(x-0/2)/0
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Plasma mode spectroscopy
time
Microwave pulseat frequency f
Ib
Switching probability enhancement if f=p : resonant activation
4.4 4.5 4.6 4.7 4.8 4.9
2.6
2.8
3.0
2.4 2.6 2.80
50
100
0
2
4
Fre
qu
en
cy (
GH
z)
Magnetic field (Gs)
Psw
itch
(%)
Frequency (GHz)
Sw
itch
ing
cu
rre
nt
(A
) Resonant activation peak :
Typical width : 20-50MHz
15050
Q
shxJ
pCLL ))'((2
1
Csh=12pF, L=170pH (design)
P. Bertet et al., Phys. Rev. B 70, 100501 (2004)
bJb dI
d
LLIg
01 2
1)(
2
22
02 4
1)(
bJb
dI
d
LLIg
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Evaluating the coupling constantsbJ
b dI
d
LLIg
01 2
1)(
2
22
02 4
1)(
bJb
dI
d
LLIg
Measure (Ib)
Spectroscopy
0.0 0.3 0.6
-1
0
(G
Hz)
Ib (A)
Ib*
-0.001 0.000 0.001
5.5
6.0
6.5
Fre
quen
cy (
GH
z)
(x-0/2)/0
Ib=0A
Ib=0.6A
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Evaluating the coupling constantsbJ
b dI
d
LLIg
01 2
1)(
2
22
02 4
1)(
bJb
dI
d
LLIg
Measure (Ib)
Spectroscopy
0.0 0.3 0.6
-1
0
(G
Hz)
Ib (A)
Ib*
g1
g2
Ib*0.0 0.3 0.6-0.2
0.0
Cou
plin
g (G
Hz)
Ib (A)
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Frequency shift
xzq
hH
22
21 )()()()( aaIhgaaIhgH xbxbI
)2/1( aahH pp
ac-Zeeman shift. Always >0
Frequency shift 0 due to g1
0.0 0.3-2
0
2
Ib (A)
epsi
lon
(GH
z)
Ib*
-20MHz
+26MHz
0MHz
Frequency shift 0 due to g2
Shift has same sign as epsilon
0.0 0.3-2
0
2
Ib (A)
epsi
lon
(GH
z)
=0
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Frequency shift
0.0 0.3-2
0
2
epsi
lon
(GH
z)
Ib (A) 0=0
Quantitative prediction : optimal point for photon noise
Optimal point for flux/current noise
Optimal pointFor flux-noise
Optimal point for photon noise
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Characterizing decoherence (1) : spectroscopy
5 types of experiments :
Low-power spectroscopy
Rabi oscillations
T1 measurements
Spin-echo measurements
At decoupled optimal point (Ib=Ib*,=0)
5.52 5.56 5.60
68
78
Psw
itch
(%)
Freq F(GHz)
f1,w1
f2,w2
Strongly coupled 2-level fluctuator
)(/2 212 wwt
Ramsey fringes
Thermal photon noise :« high frequency »
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sTRabi 5.1
Psw
itch
(%)
0 1 2
100
200
Pulse duration Dt (s)
Non-exponential because low-frequency noise
Characterizing decoherence (2) : Rabi oscillations
At decoupled optimal point (Ib=Ib*,=0)
Dt
MW=Q
0.00 0.06
60
80
Pulse length (s)
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Characterizing decoherence (3) : T1 measurements
Dt
0 10 20
60
80
Psw
itch
(%)
Delay Dt (s)
- Exponential decay
At decoupled optimal point (Ib=Ib*,=0)
sT 41
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Characterizing decoherence (4) : Ramsey fringes
0.0 0.1 0.2 0.3 0.4
60
80
100
120
140
160
Psw
itch
(%)
Delay between pulses (microseconds) T/2
/2 /2
T/2
MW-Q
Difficult to extractdephasing time …
At decoupled optimal point (Ib=Ib*,=0)
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1.0 1.1 1.2
60
80
Psw
itch
(%)
t (s)
T/2=2.2s
0 1 2 3 4 5
60
70
80
Psw
itch
(%)
t (s)0 4000 8000
0
10
20
Ech
o a
mp
l (%
)
T/2 (ns)
Techo=3.9+-0.1s
Bertet et al., cond-mat/0412485
Characterizing decoherence (5) : spin-echo sequence
t
/2 /2
T/2
T/2/2
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T1 dependence on Ib
-0.2 0.0 0.2 0.4
1
T1 (s
)
Ib (A)
Ib*
Away from Ib*, T1 limited by coupling to measuring circuit
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Spin-echo and t2 dependence on Ib and
-0.001 0.000 0.001
100
1000
5.5
5.6
5.7
Tim
e (n
s)
(e/Ip)
Fre
quen
cy (
GH
z)
Ib=Ib*
g1=0
Techo
t2=2/(w1+w2)
Best coherence :=0 (optimal point)
Ib=0A
g1=80MHz
-0.001 0.000 0.00110
100
1000
5.6
5.8
Tim
e (n
s)
(e/Ip)
Fre
quen
cy (
GH
z)
Best coherence :
=m<0
NOT LIMITED
by flux-noise
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Decoherence due to qubit-plasma mode coupling
-0.001 0.000 0.00110
100
1000
5.6
5.8
Tim
e (n
s)
(e/Ip)
Fre
quen
cy (
GH
z)
m
-0.2 0.0 0.2 0.4
-1
0
m (
GH
z)
Ib (A)
0=0
Dephasing minimum for spin-echo and Ramsey when 0=0
Quantum coherence limited by photon noise
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-0.001 0.000 0.001
100
1000
5.5
5.6
5.7
Tim
e (n
s)
(e/Ip)
Fre
quen
cy (
GH
z)
Ib=Ib*
g1=0
T=70mK, Q=150
Ib=0A
g1=80MHz
-0.001 0.000 0.00110
100
1000
5.6
5.8
Tim
e (n
s)
(e/Ip) F
requ
ency
(G
Hz)
Spin-echo and t2 dependence
Quantitative agreement
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Conclusion
Long spin-echo time (4s) at optimal bias point
Dephasing due to thermal fluctuations of the photon number in an underdamped resonator coupled to the qubit : very general situation
Case of a flux-qubit coupled to the plasma mode of its SQUID detector
By tuning coupling constants, could decouple qubit from photon noise
Quantitative agreement with simple model for spin-echo time
2 questions :
- mechanism for low-freq noise ? (charge or critical current noise ?)
- effect of dispersive shifts in usual spin-boson model ?