1.0
10
01
switching probability
0.8
0.6
00
0.4
0.2
0.0
11
0
100
Josephson qubits
P. Bertet
SPEC, CEA Saclay (France),
Quantronics group
200
300
swap duration (ns)
400
Outline
Lecture 1: Basics of superconducting qubits
Lecture 2: Qubit readout and circuit quantum electrodynamics
1) Readout by a linear resonator
2) Nonlinear resonators for high-fidelity readout
3) Resonant qubit-resonator coupling:
quantum state engineering and tomography
Lecture 3: 2-qubit gates and quantum processor architectures
Fabrication techniques
small junctions
1) e-beam patterning
2) development
3) first evaporation
4) oxidation
5) second evap.
6) lift-off
7) electrical test
e-beam lithography
e-
Al/Al2O3/Al junctions
O2
PMMA
PMMA-MAA
SiO2
I.3) Decoherence
small junctions
Multi angle shadow evaporation
QUANTRONIUM (Saclay group)
gate
I.3) Decoherence
160 x160 nm
FLUX-QUBIT (Delft group)
I.3) Decoherence
TRANSMON QUBIT (Saclay group)
40mm
2mm
I.3) Decoherence
The ideal qubit readout
relax.
1
a|0>+b|1>
1
0
a
+
b
0
0 ? 1
a
+
b
0 ? 1
a
+
b
0 ? 1
1111100
tmeas << T1
1
0
Hi-Fi
0000000
Quantum Non Demolishing
(QND)
BUT ….HOW ???
SURPRISING DIFFICULT AND INTERESTING QUESTION FOR
SUPERCONDUCTING QUBITS
II.1) Linear resonator
The readout problem
1) Readout should be FAST :
t meas  T1
Ideally, t meas
1m s for high fidelity ( F  1  t meas / T1 )
10ns
2) Readout should be NON-INVASIVE
Unwanted transition caused by readout process
(but full dephasing can’t be avoided !!!)
errors
3) Readout should be COMPLETELY OFF during quantum state preparation
(avoid backaction)
II.1) Linear resonator
Readout by a linear resonator
1D CPW
resonator
Superconducting
artificial atom
R. Schoelkopf, 2004
A. Blais et al., Phys. Rev. A 69, 062320 (2004)
A. Walraff et al., Nature 431, 162 (2004)
I. Chiorescu et al., Nature 431, 159 (2004)
Modern readout methods by coupling to a resonator
(CIRCUIT QUANTUM ELECTRODYNAMICS)
II.1) Linear resonator
Physical realization
L=3.2cm, fn=n 1.8GHz
3mm
50mm
Coupling
capacitor Cc
10mm
20mm
Typical lateral dimensions :
10mm
- 1-dimensional mode
- Very confined : Vcav  10
5
3
- Large voltage quantum fluctuations V0
II.1) Linear resonator
mV
- Quality factor easily tuned by designing Cc
CPB coupled to a CPW resonator
A. Blais et al., PRA 69, 062320 (2004)


Vˆg  V0 aˆ  aˆ   Vgext
Cg
c
Vext
ˆ
nˆ
2
ˆ
ˆ
ˆ
ˆ
Htot  EJ cos  4EC  n  ng   c aˆ aˆ
2
ˆ
ˆ
Htot  EJ cos  4EC  nˆ  ngext   c aˆ aˆ  8(CgV0EC / 2e)nˆ(a  a  )
Hˆ q
Hˆ cav
Hˆ int
2-level approximation + Rotating Wave Approximation
Htot
II.1) Linear resonator

ge
2
 z  c (a a  1/ 2)  g ( a   a  )
Jaynes-Cummings
Hamiltonian
g  2eV0 (Cg / C ) 0 nˆ 1
Strong coupling regime with superconducting qubits
g  2eV0 (Cg / C ) 0 nˆ 1
GEOMETRICAL dependence of g
Easily tuned by circuit design
Can be made very large !
(Typically : 0 – 200MHz)
g  200MHz   ,  100  500kHz
Strong coupling condition naturally fulfilled
with superconducting circuits
(Q=100 enough for strong coupling !!)
II.1) Linear resonator
The Jaynes-Cummings model
d
|g,3>
d
|e,3>
HJ C  
|e,2>
|g,2>
|e,1>
|g,1>
|e,0>
H J C
ge
2
 z  c (a a  1/ 2)  g ( a   a  )
couples only level doublets
|g,n+1> , |e,n>
Exact diagonalization possible
|g,0>
Restriction of HJ-C to
g, n  1
g, n  1
e, n
|g,n+1> , |e,n>
e, n
  
 (n  1)c   / 2

g n 1




g
n

1
(
n

1)



/
2
c


Note : |g,0> state is left unchanged by HJ-C with Eg,0=-/2
II.1) Linear resonator
ge
 c 
The Jaynes-Cummings model
d
|g,3>
d
|e,3>
|e,2>
|g,2>
|e,1>
|g,1>
|e,0>
HJ C  
H J C
ge
2
 z  c (a a  1/ 2)  g ( a   a  )
couples only level doublets
|g,n+1> , |e,n>
Exact diagonalization possible
|g,0>
Coupled states
, n  cosn e, n  sinn g, n  1
E,n  (n  1) c 
, n   sinn e, n  cosn g, n  1
E,n  (n  1) c 
n 
II.1) Linear resonator
 2g n  1 
1
tan1 

2



2
2
4g 2 (n  1)   2
4g 2 (n  1)   2
The Jaynes-Cummings model
1.08
e, n
E/(hc)
1.04
g, n  1
g, n  1
1.00
0.96
e, n
0.92
-5
0
/g
II.1) Linear resonator
5
The Jaynes-Cummings model
1.08
,n
e, n
E/(hc)
1.04
g, n  1
1.00
g, n  1
2g n  1
,n
0.96
e, n
0.92
-5
0
/g
II.1) Linear resonator
5
Two interesting limits
1.08
,n
e, n
E/(hc)
1.04
g, n  1
1.00
g, n  1
2g n  1
,n
0.96
RESONANT
REGIME (=0)
e, n
0.92
-5
0
/g
II.1) Linear resonator
5
Two interesting limits
1.08
,n
e, n
E/(hc)
1.04
g, n  1
1.00
g, n  1
2g n  1
,n
0.96
RESONANT
REGIME (=0)
e, n
0.92
-5
DISPERSIVE
REGIME (||>>g)
II.1) Linear resonator
0
/g
5
DISPERSIVE
REGIME (||>>g)
Two interesting limits
1.08
,n
e, n
E/(hc)
1.04
g, n  1
QUANTUM
STATE
ENGINEERING
1.00
2g n  1
g, n  1
,n
0.96
QUBIT STATE
e,READOUT
n
RESONANT
REGIME (=0)
QUBIT STATE
READOUT
0.92
-5
DISPERSIVE
REGIME (||>>g)
II.1) Linear resonator
0
/g
5
DISPERSIVE
REGIME (||>>g)
The Jaynes-Cummings model : dispersive interaction   g
HJ C /  
ge  
2
 z  (c   z )aa  
with

g2

ge  2 (aa  1/ 2)
2
 z  c a a
the dispersive coupling constant
1) Qubit state-dependent shift of the cavity frequency
c  c   z
Cavity can probe the qubit state non-destructively
2) Light shift of the qubit transition in the presence of n photons
ge  2 n
Field in the resonator causes qubit frequency shift
and decoherence
II.1) Linear resonator
Dispersive readout of a transmon: principle
ωc + χσ z
|0> or |1> ??
II.1) Linear resonator
Dispersive readout of a transmon: principle
Veiωc t
Veiωc t
ωc + χσ z
|0> or |1> ??
=c
II.1) Linear resonator
Dispersive readout of a transmon: principle
Veiωc t
Veiωc t
ωc + χσ z
|a1
|a0
=c
II.1) Linear resonator
|0> or |1> ??
Dispersive readout of a transmon: principle
Veiωc t
Veiωc t
iωc t +
Ve
ωc + χσ z

|a1
|a0
=c
p
2
|0>

p
|1>
0,96
II.1) Linear resonator
1,00
d/c
1,04
|0> or |1> ??
Dispersive readout of a transmon: principle
Veiωc t
Veiωc t
iωc t +
Ve
ωc + χσ z

|a1
|a0
L.O
=c
0
p
or
1 ???
2
|0>

p
|1>
0,96
II.1) Linear resonator
1,00
d/c
1,04
|0> or |1> ??
Typical implementation (Saclay)
5 mm
(f0=6.5GHz)
Q=700
80mm
g  45MHz
40mm
II.1) Linear resonator
2mm
(optical+e-beam
lithography)
Typical setup (Saclay)
MW
meas
I
MW
drive
COIL
Vc
20dB 50MHz
LO
dB
20dB
Fast
Digitizer
Q
G=56dB
A(t)
(t)
300K
G=40dB
TN=2.5K
50
4K
DC-8
GHz
30dB
600mK
1.4-20
GHz
20dB
II.1) Linear resonator
4-8
GHz
50
18mK
Observation of the vacuum Rabi splitting with electrical circuits
(courtesy of S. Girvin)
Signature for strong coupling:
Placing a single resonant atom inside the cavity
leads to splitting of transmission peak
2008
vacuum Rabi splitting
atom
off-resonance
observed in:
cavity QED
R.J. Thompson et al., PRL 68, 1132 (1992)
I. Schuster et al. Nature Physics 4, 382-385 (2008)
on resonance
circuit QED
A. Wallraff et al., Nature 431, 162 (2004)
quantum dot systems
J.P. Reithmaier et al., Nature 432, 197 (2004)
T. Yoshie et al., Nature 432, 200 (2004)
II.1) Linear resonator
28
A. Wallraff et al., Nature 431, 162 (2004)
Qubit spectroscopy with dispersive readout
-120
MW drv
g
Pump TLS
Probe resonator
phase
MW meas
 (°)
-122
-124
-126
Some e
5,25
p
5,30
Drive freq (GHz)
e

g
p
0.95
1.00
/c
II.1) Linear resonator
1.05
5,35
Typical spectroscopy of a transmon + cavity circuit
6.5
01
frequency (GHz)
12
6.0
c
5.5
5.0
0.0
II.1) Linear resonator
0.5
/2p
1.0
1.5
Rabi oscillations measured with dispersive readout
Δt
MW drv
Variable-length
drive
MW meas
Projective
measurement
x 10000
 Ensemble averaging
-108
0
-111
X
 (°)
Y
T2R=316 ns
-114
1
-117
0
II.1) Linear resonator
200
400
600
t (ns)
800
1000
Dispersive readout : the signal-to-noise issue
Veiωc t
Veiωc t
iωc t +
Ve
ωc + χσ z

|a1
|a0
Ideal amplifier
L.O
=c
0
p
or
1 ???
2
|0>

p
|1>
0,96
II.1) Linear resonator
1,00
d/c
1,04
|0> or |1> ??
Dispersive readout : the signal-to-noise issue
Veiωc t
Veiωc t
iωc t +
Ve
Real amplifier
TN=5K
ωc + χσ z

0
p
or
1 ???
2
|1>
0,96
II.1) Linear resonator
No discrimination in 1 shot
|0>

p
|0> or |1> ??
|a0
L.O
=c
|a1
1,00
d/c
1,04
Dispersive readout : the signal-to-noise issue
Veiωc t
Veiωc t
QUANTUMLIMITED
AMPLIFIER ??
iωc t +
Ve
ωc + χσ z

|a1
|a0
Real amplifier
TN=5K
L.O
=c
0
p
or
1 in one single-shot ??
2
|0>

p
|1>
0,96
II.1) Linear resonator
|0> or |1> ??
1,00
d/c
1,04
How to build an amplifier with minimal noise ???
pump
signal in
signal out
Nonlinear resonator
/4
/4
Junction causes Kerr non-linearity
K + 2 2
Hc = ωc a a +
(a ) a
2
+
Resonator can behave as parametric amplifier
K. Lehnert group
M. Devoret group
I. Siddiqi group
II.2) Nonlinear resonator
A nonlinear resonator as quantum-limited amplifier
max
II.2) Nonlinear resonator
M. J. Hatridge, R. Vijay, D. H. Slichter, J. Clarke and I. Siddiqi,
Phys. Rev. B 83, 134501 (2011)
(courtesy I. Siddiqi)
A nonlinear resonator as quantum-limited amplifier
Small
Saturated
signal
II.2) Nonlinear resonator
(courtesy I. Siddiqi)
Signal-to-noise enhancement by a paramp
M. Castellanos-Beltran, K. Lehnert, APL (2007)
(quantum limit on how good an amplifier can be : Caves theorem)
Actually reached in several experiments : quantum limited measurement
II.2) Nonlinear resonator
Qubit and amplifier at 30 mK
OUTPUT
INPUT
DRIVE
II.2) Nonlinear resonator
(courtesy I. Siddiqi)
Individual measurement traces
readout off
readout on
R. Vijay, D.H. Slichter, and I. Siddiqi, PRL 106, 110502 (2011)
II.2) Nonlinear resonator
(courtesy I. Siddiqi)
Bivalued histograms
Single-shot discrimination of qubit state
II.2) Nonlinear resonator
(courtesy I. Siddiqi)
Other strategy : sample-and-hold detector integrated with qubit
pump
/4
/4
Nonlinear resonator
used as threshold detector
II.2) Nonlinear resonator
Other strategy : sample-and-hold detector integrated with qubit
Kerr-nonlinear resonator
/4
/4
pump
H
Pd /Pc = 0.2
0.5
1.0
1.8
I Ic
0.2
L
0.1
0
II.2) Nonlinear resonator
2
0
-2
BISTABILITY FOR   c  3

The Cavity Josephson Bifurcation Amplifier (CJBA)
M. Devoret group, Yale
MW drive : Pd(t) , d
in
JBA: I. Siddiqi et al., PRL (2004)
CJBA: M. Metcalfe et al, PRB (2007)
Non linear resonator
out
H
Pd
Pd
H state
Bistable
region
L
Switching from L to H : BIFURCATION
d
L state
c
II.2) Nonlinear resonator
Stochastic process governed by
thermal or quantum noise.
M.I. Dykman and M.A. Krivoglaz, JETP 77, 60 (1979)
M.I. Dykman and V.N. Smelyanskiy, JETP 67, 1769 (1988)
The Cavity Josephson Bifurcation Amplifier (CJBA)
M. Devoret group, Yale
MW drive : Pd(t) , d
in
JBA: I. Siddiqi et al., PRL (2004)
CJBA: M. Metcalfe et al, PRB (2007)
Non linear resonator
out
H
Pd
Pd
H state
Bistable
region
d
L state
c
Switching probability
L
1,0
0,8
0,6
0,4
0,2
0,0
-36
-35
-34
Power Pd (dB)
II.2) Nonlinear resonator
-33
Readout of transmon with CJBA
MW drive : Pd(t) , d
in
Non linear resonator
out
qubit in |0> or
Pd
H state
L state
2
c|1> c|0>
d
Switching Porbability
Pd
|1>
1,0
0,8
|1>
0,6
0,4
0,2
0,0
|0>
-38 -37 -36 -35 -34 -33
SINGLE-SHOT
QUBIT READOUT
II.2) Nonlinear resonator
Power Pd (dB)
Rabi oscillations visibility
2
h12
1
h01
0
t
tp,12
250ns
400ns
1,0
Pswitch (%)
0,8
0,6
0,4
0,2
TRabi=500ns
0,0
-38
-36
-34 0,0
0,2
0,4
0,6
t (µs)
0,8
Mallet et al.,
Nature Physics (2009)
Single-shot 93% contrast Rabi oscillations
II.2) Nonlinear resonator
See also
A. Lupascu et al., Nature Phys. (2007)