Decoherence of a qubit:
-during free evolution
-during driven evolution
-at readout
The Quantronium
dc gate
1
dc gate
0 A0
µw
n01
0
qp
trap
readout junction
1A1
box
Fluctuating
environment
A -meter
1µm
AC
drive
UM
Daniel ESTEVE QUAN
ELEC RONICS GROUP
T
SPEC
of Michel DEVORET
YALE
DECOHERENCE DURING FREE EVOLUTION
qubit
relaxation
dephasing
1

AZ X
2  1 / 2  
dX (t )
noise
dw01 (t )
DEPHASING
dj (t) =
т
w01 (t ')dt'
The quantronium: 1) a split Cooper pair box
2 knobs : Ng  CgU/2e
U
δ = /0
1 d° of freedom


ˆ i
θ̂,N
 
2 energies:
2e 

EC 
2Cisland
2
EJ =
h
8e 2R t
ˆ  E (N
ˆ - N )2 - E cos  cosˆ
H
C
g
J
2
δ
N
U
i
State dependent persistent currents
ˆ
1 H
1  Ek
î 

0 
0 
2) protected from dephasing
¶ n 01
= 0
¶X
EJ  EC
0.5
15
0.25
1
0
0
-0.25
hn01
n01(GHz)
energy (kBK)
20
10
5
01
- €€€€€
2
1
- €€€€€
2
0
0
/2p
1
1
€€€€€
2
readout
+ environment
0
1
€€€€€
2
/2p
1
1
€€€€€
2
0
Ng
dX (t )
¶ n01
dn01(t ) =
dX (t )
¶X
weaker dephasing at optimal point
1
2
€€€€€
Ng
EJ=0.86 kBK
EC=0.68 kBK
Readout of
persistent currents
with dc switching
3) with a readout junction
V=0
or
V0
20
t
15
10
Ib
I0
5
01
- €€€€€
2
U
0
1
2
€€€€€
Ng
1000
100
800
80
600
60
400
40
200
20
output voltage (µV)
0
bias current (nA)
1
€€€€€
2
RF amplitude (a.u.)
/2p
1
1: switching
0: no switching
1
0
0
1
2
3
time (µs)
4
5
6
0
discrimination
microwave output voltage (mV)
Qubit control: Rabi precession
100
Aμw cos(t   )
50
0
16 GHz
-50
1.0
2.0
3.0
switching
probability
(%)
4.0
time (ns)
0
Rabi oscillations
50
40

X
Y
1
nµw
Effective
field
rotation
Rabi  aURF
30
0
50
100 150 200 250
pulse duration (ns)
Readout fidelity ?
switching probability (% )
100
Magic point
N g = 1/2
80
60
p pulse
ground state
40
difference
20
0
510
515
520
525
530
535
bias current amplitude (nA)
40% contrast (only)
540
Qubit
arbitrary transformations
manipulation
dn 01
U  R3 X R2Y R1X
t
adiabatic
frequency pulses
for Z rotations
Z
0
Y
X
1
robust transformations
Composite ‘ p ’ CORPSE :
60°X 300°-X 420 X°
switching probability (% )
60
50
‘ p ’ CORPSE
40
30
16.30
corpse: 420°(X)/300°(-X)/60°(X)
Single
simple
p pulse pulse
nRabi=92 MHz
sweet spot
16.35
16.40
16.45
frequency (GHz)
16.50
Decoherence sources in the quantronium circuit

e_
ˆ = E (N
ˆ - N )2 - E cos  /2  cosˆ
H
C
g
J
B
Vg
0 ˆI 1
1
ˆ 1 0
0N
0
2 
4
0.2
0.8 0.6 0.4
optimal point
Ng=1/2 , =0
20
6
n 01/ = 0
15
10
7.5
5 nA
2.5
0
0
n 01/Ng = 0
Ng
minimum relaxation due to 
no dephasing
no current
n01(GHz)
Ng drive
10
5
01
- €€€€€
2
0
/2p
1
1
2
€€€€€
0
1
€€€€€
2
Ng
Decoherence in the Quantronium

e_
01 (Ng , )
a
1
B
U
0
environment
+
b
ˆ = E (N
ˆ - N )2 - E cos  /2  cosˆ
H
C
g
J
Pure dephasing
i  dt
a 0   e  01 1
Relaxation
2
P
2
2
1
ˆ
= 0 N 1 SNg (01 )
p T1
2
 0 ˆi 1 S (01 )
dj (t) =
+т
e
0 ˆi 1  0 if balanced junction !
i dj
т
0
¶ w01
dl (t)dt
¶l
1 ¶ 2 w01 2
dl (t)dt
2 ¶l 2
t2 2
wt
= exp[s т Sl n ( w) sinc2 ( )dw]
2
2
not necessarily exponential
e i dj (Tj ) = e - 1
1
 S (  0)
T
Model for dephasing: charge and phase noise
e_
ˆ = E (N
ˆ - N )2 - E cos  cosˆ
H
C
g
J
2
Vg

Ng ou
(linear
coupling)


Ramsey
Echo

B
(t )  t 2  S ( ) sin c 2 (
t
2
)d
1
t
t
(t )  t 2  S ( ) [1  cos( )]sin c 2 ( ) d 
2
2
4
Spectral
density
Relaxation of the Quantronium
n 01 (GHz)
13.81
15.76
16.41 GHz
16.66
switching probability (%)
2.0
=0
Ng = 1/2
T1 (µs)
1.5
1.0
p
60
t
50
P0
40
30
T1=0.5µs
0
1
t (µs)
0.5
0.0
1
 Nmodes (n 01 )
T1
-0.2
|/2p|
-0.1
0.0
|Ng-1/2|
0.1
T1: 0.3-2 ms
2
Ramsey interferences
0
Rabi
 01RF
p/2 pulse
Rabi
Free evolution
(rotation also)
Projection Z
p/2 pulse
AZ X
Ramsey interferences reveal decoherence
of free evolution during the delay
readout
Characterizing dephasing: 1) decay of Ramsey fringes
best ones:
switching
switching
probability
probability
(%)
(%)
45
45
nRF = 16409.50 MHz
t
  t / T
sin(2p .
Fit e
n = 19.84 MHz
T2 = 500 +/- 50 ns
40
40
n . t)
35
35
30
30
0.0
0.0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
time between pulses t (µs)
time between pulses t (µs)
0.5
0.5
0.6
0.6
typical sample
Fit with the linked cluster expansion:
( Makhlin Shnirman, Paladino, Falci)
static approximation
switching probability p
0.6
=0
Ng = 1/2
0.4
T2 ~ 300 ns
0.2
0
200
400
600
delay t between p/2 pulses (ns)
Comparing fits
“static” approximation
( Makhlin Shnirman, Paladino, Falci)
Simple exponential
gaussian noise model
500 ns
Coherence time
n 01
0
Ng
  0
Ng N 1/ 2
g
1,2
n 01
0

20
15
10
0,8
5
01
- €€€€€
2
0
/2p
1
1
€€€€€
2
0
0,4
1
€€€€€
2
Pswitch
n01(GHz)
away from
optimal point
Nc
N  0.53
0,0
g
0.028
0.028
0.028
-0,4
0
200
time delay (ns)
400
0
200
400
time delay (ns)
Characterizing dephasing: 2a) phase detuning pulses
50
p/2X
t1
p/2X
switching probability (%)
40
T2 ~ 200 ns
30
20
0
100
200
300
time t1 (ns)
400
500
/2p=6.3%
45
p/2X
p/2X
t1=270 ns
T2, ~ 60 ns
40
100
200
time t2 (ns)
300
t2
Characterizing decoherence: 3) resonance linewidth
switching prob (%)
switching prob (%)
20
25
20 25 30 35
16.8
16
14
16.6
12
16.5
16.4
10
f / f0 = 0
Ng = 1/2
-0.3
-0.2
 / 2p
-0.1
0.0
0.5
0.4
Ng
16.3
Frequency (GHz)
Frequency (GHz)
16.7
Microwave output voltage (mV)
switching probability (%)
5) Probing the dynamics: spin echo experiments
Ramsey Echo
50.5MHz
60
50
50
0
-50
p
p/2
0.0
2.0
4.0
6.0
p/2
8.0 10.0 12.0 14.0 16.0 18.0 20.0
time (ns)
40
30
0
200
400
600
delay between p/2 pulses (ns)
800
1000
Direct mapping of echo amplitude
50
switching probability (%)
60
T2 ~ 220 +/- 50 ns
p/2
40
p
p/2
30
50
0
500
1000
p/2
p/2
40
30
p/2
TEcho ~ 500 +/- 50 ns
0
500
1000
time between p/2 pulses (ns)
TE  T2
1500
low frequency noise
p
p/2
Echo decay away from optimal point
=0
0.4
Ng = 0.500
Ng = 1/2
switching probability p
0.5
0.4
0.3
0.2
0
500
0.3
1000 1500
0.4
 / 2p = 0.023
0
500
0.484
1000 0
0.4
0.3
0.2
500
0.3
1000 1500
0.4
0.079
0
100
200
0.452
300 0
500
1000 1500
0.3
0.435
0.164
0.2
0
50
100
0
500
1000 1500
0.419
0.248
0
50
100
0.4
0.3
0.3
0.2
0.3
0.4
0.3
0
500
1000 1500
delay t between p/2 pulses (ns)
Comparison exp vs model
Coherence times (ns)
500
noise spectral densities
Spin echo
Free decay
500
S
100
Gaussian
model
SNg
100
1/
1/


0.5MHz
4MHz
10
10
-0.3
-0.2
-0.1
0.0
|/2p|
Conclusion: decay times ok, not time dependence
0.05 0.10
|Ng-1/2|
non gaussian character of noise ??
See G. Ithier et al.: Decoherence in a quantum bit Superconducting circuit circuit,
preprint
Closer look at charge and phase spectral densities:

Ng
Vg
Phase noise
Charge noise
4
1x10
3
[S()]

1x10
2
1x10
Partly
external
1x10
0
1x10
-1
1x10
-2
1x10
1/f
Ng
1
-3
1x10
1/f
-4
1x10
-5
1x10
-6
1x10
-7
1x10
-8
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
10
4
10
5
10
6
10
[] (Hz)
1x10
7
-3
-2
-1
0
1
2
3
4
5
6
7
10
10 10 10 10 10 10 10 10 10 10 10
[] (Hz)
Cut-off
at .5 MHz
Decoherence: driven evolution versus free evolution
Bloch-Redfield description
1
Free
1
 2   
2
1  S (01 )
  S z (  0)
Driven at Rabi
1  cos 2 
sin 2 
 
1 
n
2
2
2
2
3

cos

sin

*2 
1  cos 2   
n
4
2
*
1
n  S z (  Rabi )
See preprint on decoherence G. Ithier et al.
Determination of T*1 : Spin locking
p/2X
switching probability (%)
55
p/2X
p/2X - LockY 24 MHz - p/2X T*1 ~ 600 ns
50
p/2X
Ramsey 7MHz T2 ~ 250 ns
aY
p/2X
45
40
35
p/2X - LockY 24 MHz - 3p/2X T*1 ~ 600 ns
30
0
500
1000
1500
delay between p/2 pulses (ns)
T  T1
*
1
1  n
 
2
*
1
Determination of T*2
:
Decay of Rabi oscillations with Rabi frequency
0.6
nR0 = 2.2 MHz
4.0 MHz
7.8 MHz
15 MHz
30 MHz
61 MHz
0.5
0.4
switching probability p
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0.6
0.5
0.4
0.3
0.2
0
500
1000
0
500
pulse length  (ns)
1000
Decay of Rabi oscillations with frequency
3  cos 2 
sin 2 
2
2
1  cos   
n
4
2
*
Tn (1-100MHz)  1µs
1000
1000
nRabi 0 = 15.4 MHz
800
600
600
400
400
T*2 ~ 480 ns
200
0
200
1
10
100
Rabi frequency (MHz)
0
T2  T2
*
20
n (MHz)
40
0
T*2 (ns)
T*2 (ns)
800
T2
decoherence in the rotating frame ?
a 0  1
0
Y
X
55
switching probability (% )
lab frame:
Z
Ramsey decay:
50
45
40
35
T  = 300 +/- 30 ns
detuning=50MHz
30
T2=300ns
25
1
0
200
rotating frame:
a 0*   1*
Z
I1*>
*>
I0
drive
switching probability p
0.6
400
600
800
Delay between p/2 pulses (ns)
0.5
0.4
0.3
T2*=480 ns
0.2
0
200
400
600
800
pulse length (ns)
Conclusion: more robust qubit encoding in the rotating frame
1000
1200
Decoherence at readout: projection fidelity ?
ideal
QND readout:
1
1
0
0
Readout: 1
Readout: 0
errors: wrong answer + projection error
1
0
0 A0
1A1
A -meter
Fluctuating
environment
Decoherence :dc versus rf readout
t
dc
readout
V
t
U

dc pulse
 switching
simple, but:
rep rate limited by quasiparticles
-qubit reset : NOT QND
resets
the qubit
Decoherence :dc versus rf readout
PULSE IN
rf readout
(M. Devoret,
Yale)
PULSE OUT
t
U
U

dc pulse
 switching
“RF” pulse

  dynamics in anharmonic potential
simple, but:
more complex, but:
-fidelity 40%
-qubit reset : NOT QND
-better fidelity ?
-no reset: possibly QND
Phase oscillations in a state dependent anharmonic potential
(I. Siddiqi et al., PRL 93, 207002 (2004))
Drive
Qubit control
port

I0
g
Ui
C
V
Ur
Vg
Output
LC oscillator
GUr
The Josephson Bifurcation Amplifier :
0
OSCILLATION
AMPLITUDE
1
latching
180°
0
1
1
-180°
0
MICROWAVE DRIVE AMPLITUDE
1.5
Frequency (GHz)
Microwave readout setup
MicroWave
Generator
V
RFin
LO
demodulator
S
300 K
M
G=40dB
Pulsing
I
Q
TN=2.5K
G=40dB
-20dB
4K
-20dB
-30dB
LP
3.3GHz 600 mK
-30dB
4 kW
1 kW
20 mK
HP
LP
Directionnal
1.3GHz 2GHz coupler
Sample from
Yale
50 W
50 W
frequency:
1.4GHz
Rabi oscillations
5ns
Readout contrast?
100ns
tin>150ns
Bifurcation
probability
1.0
1
0.8
P
0.7
0.6
0.6
0.5
0.4
0.4
0.3
0.2
0.2
0
0.1
0
20
40
60
80
100
120
140
160
Pulse duration (ns)
0.0
-7.0
-6.8
-6.6
-6.4
-6.2
-6.0
Microwave power (dBm, top)
Readout :
50% contrast
(Yale: 60%)
(best dc switching: 60%)
QND readout ?
no pulse OR p pulse
on qubit
on qubit
Readout 1
Readout
1
Readout 2
1
0
analysis yields for a single readout:
1
1
p 0.34
p 0
0
p 1
1
0.25
0.09
Answer 1
Answer 0
p 0.66
0
0.17
0.83
Answer 1
Answer 0
partially QND
0
0.30
Answer 1
0.36
Answer 0
(Yale & Saclay)
QND readout with an ac drive at optimal point ?
flux qubit
charge qubit
quantronium
Ek
Ik 

Ek
Qk 
N g
Ek
Ik 

SQUID inductance
  2 Ek 
Lk  
2 




box capacitance
1
TU Delft,
Helsinki (for SSET)
  2 Ek
Ck  
 N 2
g

(in progress)
Chalmers



1
JBA
  2 Ek 
Lk  
2 




1
Yale
partially
Saclay QND
Readout fidelity & QND readout are (still) issues
This work on :
the Quantronium
dc gate
1
dc gate
0 A0
µw
n01
0
qp
trap
1A1
box
Fluctuating
environment
A -meter
readout junction
1µm
QUAN
ELEC
UM
TRONICS GROUP
SPEC
Appl. Physics
YALE
G. ITHIER
E. COLLIN
P. ORFILA
P. SENAT
P. JOYEZ
D. VION
P. MEESON
D. ESTEVE
A. SHNIRMAN
G. SCHOEN
Y. MAKHLIN
F. CHIARELLO
Karlsruhe
Landau
Roma
I. SIDDIQI
F. PIERRE
E. BOAKNIN
L. FRUNZIO
R. VIJAY
C. RIGETTI
M. METCALFE
M. DEVORET
remind: 10-4 error rate on qubit gates…, QND useful but not mandatory
Yale quantronium sample
2mm
Continuous measurement
100
-30dB
50
-23dB
Dephasing (°)
0
-50
-100
-150
-200
1.2
1.3
1.4
Frequency (GHz)
1.5
1.6
Qubit in
ground state
Quantum Non-Demolition Fraction

pulse
Readout 1 (R1)
5ns 100ns 20ns
125ns
Readout 2 (R2)
20ns 30ns
Ps(R1)
Ps(R2)
Ps(R2/R1)
13.3%
Ps(R1R2
)
3.3%
no  pulse
17.6%
 pulse
61.3%
28.0%
19.1%
31.1%
T1=1.3 s
QND Frac. 
18.8%