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
Introduction : Josephson circuits for quantum physics
From a fundamental question (25 years ago) ….
CAN MACROSCOPIC « MAN-MADE » ELECTRICAL
CIRCUITS BEHAVE QUANTUM-MECHANICALLY ????
M.H. Devoret, J.M. Martinis and J. Clarke, PRL 85, 1908 (1985)
M.H. Devoret, J.M. Martinis and J. Clarke, PRL 85, 1543 (1985)
YES THEY CAN
Discrete energy levels
… to genuine quantum information and quantum optics on a chip
… AND
MUCH MORE
TO COME !!
M. Hofheinz et al., Nature (2009)
Q. state engineering and
M. Neeley et al., Nature (2011)
Tomography
3-qubit entanglement
Outline
Lecture 1: Basics of superconducting qubits
Lecture 2: Qubit readout and circuit quantum electrodynamics
Lecture 3: 2-qubit gates and quantum processor architectures
Outline
Lecture 1: Basics of superconducting qubits
1) Introduction: Hamiltonian of an electrical circuit
2) The Cooper-pair box
3) Decoherence of superconducting qubits
Lecture 2: Qubit readout and circuit quantum electrodynamics
Lecture 3: 2-qubit gates and quantum processor architectures
Real atoms
 rˆ , pˆ   i
V (r )  
Hydrogen atom
e
2
quantization
4 0 r
Energy (a.u)
|3>
|2>
|1>
E01=E1-E0=hn01
« two-level atom »
|0>
x (a.u.)
I.1) Introduction
Hˆ 
2
pˆ
2m
 V ( rˆ )
Real atoms
 rˆ , pˆ   i
V (r )  
Hydrogen atom
e
2
quantization
2
pˆ
Hˆ 
4 0 r
2m
E ( t )  E 0 cos 2n t
H I (t )   d E (t )
laser
+ spontaneous emission G
Spectroscopy
(weak field)
 V ( rˆ )
Optical
Bloch
equations
Rabi oscillations (short
pulses, strong field at n=n01)
n01
|0>
|0>
FWHM
G/2
fR 
P(1)
P(1) (a.u)
1
|1>
Frequency n (a.u)
I.1) Introduction
1
2
0
0
 1

Time (a.u)
d E0
h
 (t ) 
Electrical harmonic oscillator
x
k
k
0 
m
0 
Q (t ) 
f
1
+q
-q
LC
m
H ( x, p ) 
kx
2
2

p
2
H ( , Q ) 
|4>
|3>
|2>
|1>
|0>

Q
2
2C
Quantum regime ??
0
X,f
I.1) Introduction
2
ˆ , Qˆ   i



 xˆ , pˆ   i
E

2L
2m

t


V ( t ') dt '
t

i ( t ') dt '
LC oscillator in the quantum regime ?
2 conditions :
E
kT   0
|4>
|3>
|2>
|1>
|0>
X,f
E
|4>
|3>
|2>
|1>
|0>
At T=30mK :
hn 0
8
kT
Q  1
OK if dissipation negligible
X,f
I.1) Introduction
C  pF n 0  5G H z
Typic : L  n H
Superconductors at T<<Tc
Microwave superconducting resonators
Q=4 1010
at 51GHz
and at 1K
Tc(Nb)=9.2K
Quantum
regime
S. Kuhr et al., APL 90, 164101 (2006)
T<<Tc : dissipation negligble at GHz frequencies
I.1) Introduction
Necessity of anharmonicity
|4>
|3>
|2>
|1>
|0>
E
f
+q
-q
0
X,f
How to prepare |1> ?
Need non-linear and non dissipative element : Josephson junction
I.1) Introduction
Basics of the Josephson junction
 (t ) 

Q (t ) 
The building block of
superconducting qubits

θ=
/2e
θR
N = Q /2 e 
Josephson DC relation : I  I C sin 
d

2 e dt
Q
C
B. Josephson, Phys. Lett. 1, 251 (1962)
P.W. Anderson & J.M. Rowell, Phys. Rev. 10, 230 (1963)
S. Shapiro, Phys. Rev. 11, 80 (1963)
I.1) Introduction


V ( t ') dt '
t

i ( t ') dt '
m od(2  )= θ R -θ L   0, 2 
θL
Josephson AC relation : V 
t

 (t ) 
Basics of the Josephson junction

Q (t ) 
The building block of
superconducting qubits

θ=
/2e
t


V ( t ') dt '
t

i ( t ') dt '
m od(2  )= θ R -θ L   0, 2 
θL
θR
N = Q /2 e 
Josephson DC relation : I  I C sin 
d
Josephson AC relation : V 
2 e dt
NON-LINEAR INDUCTANCE
POTENTIAL ENERGY
I.1) Introduction
LJ ( I ) 
E J ( )  
IC
2e
Classical variables ??

Q
C
2 eI C 1   I / I C

2
cos    E J cos 

Hamiltonian of an arbitrary circuit
=
=
HAMILTONIAN ???
Correct procedure described in :
M. H. Devoret, p. 351 in Quantum fluctuations (Les Houches 1995)
G. Burkard et al., Phys. Rev. B 69, 064503 (2004)
G. Wendin and V. Shumeiko, cond-mat/0508729
M.H. Devoret, lectures at Collège de France (2008) accessible online
Hamiltonian of an arbitrary circuit
=
=
HAMILTONIAN ???
1) Identify the relevant independent circuit variables
2) Write the circuit Lagrangian
3) Determine the canonical conjugate variables and the Hamiltonian
Hamiltonian of an arbitrary circuit
node
branch
=
=
Identifying the relevant independent circuit variables
1) Choose reference node (ground)
Hamiltonian of an arbitrary circuit
=
=
Identifying the relevant independent circuit variables
1) Choose reference node (ground)
2) Choose « spanning tree » (no loop)
Hamiltonian of an arbitrary circuit
d
c
=
a
b
e
=
Identifying the relevant independent circuit variables
1) Choose reference node (ground)
2) Choose « spanning tree » (no loop)
3) Define « tree branch fluxes »  i ( t ) 
t
 V ( t ') dt '

Hamiltonian of an arbitrary circuit
=
3
c
a
=
2
b
d
e
1
5
Identifying the relevant independent circuit variables
1) Choose reference node (ground)
2) Choose « spanning tree » (no loop)
3) Define « tree branch fluxes »  i ( t ) 
 4=  b +  d
t
 V ( t ') dt '

4) Define node fluxes = sum of branch fluxes from ground
Hamiltonian of an arbitrary circuit
3
=
c
a
2
b
d
e
1
=
Write Lagrangian
 4=  b +  d
5
L (  i ,  i )  L el (  i )  L p o t (  i )
taking into account constraints imposed by external biases (fluxes or charges)
1
2
L pot  E J cos   2   1  
ext
  1   2   ext 
2L
2
Hamiltonian of an arbitrary circuit
3
=
c
a
2
b
Conjugate variables :
Qi 
5
L
 i
Classical Hamiltonian H (  i , Q i ) 
Quantum Hamiltonian
 4=  b +  d
e
1
=
or
d
Q
H ( ˆ i , Qˆ i )
nˆ i  Qˆ i / 2 e
ˆ
with
ˆ
H ( i , n i )
ˆi  ˆ i ( 2 e / )
i
i
With
L
ˆ , Qˆ   i

 i i
ˆi , nˆ i   i


Different types of qubits
Cooper-pair boxes
NIST
Santa Barbara
0 .1  1  5 0
 50
Charge qubit/Quantronium/Transmon
-1
I.2) Cooper-Pair Box
0
/(rad)
1
4
100mm2
.04mm2
Ep (a.u)
Ep (a.u)
Shape
Of the
Potential
Energy
.01 to 0.04 mm2
10
Ep (a.u)
Junctions
sizes
Phase qubits
TU Delft
MIT
Berkeley
NEC
Saclay
Chalmers
NEC
Yale
ETH Zurich
E J / EC
Flux qubits
-2
0
m/
2
-2
0
 (rad)
2
4
Different types of qubits
Cooper-pair boxes
NIST
Santa Barbara
0 .1  1  5 0
 50
Charge qubit/Quantronium/Transmon
-1
I.2) Cooper-Pair Box
0
/(rad)
1
4
100mm2
.04mm2
Ep (a.u)
Ep (a.u)
Shape
Of the
Potential
Energy
.01 to 0.04 mm2
10
Ep (a.u)
Junctions
sizes
Phase qubits
TU Delft
MIT
Berkeley
NEC
Saclay
Chalmers
NEC
Yale
ETH Zurich
E J / EC
Flux qubits
-2
0
m/
2
-2
0
 (rad)
2
4
The Cooper-Pair Box
E C ,E J
Vg
1 degree of freedom  θˆ , Nˆ   i
1 knob
Ec 
(2 e )
2
« charging energy »
2C 
ˆ = E (N
ˆ - N ) 2 - E cos ˆ
H
C
g
J
The split CPB
E C ,E J
θ1
N1


θ
Vg
θ2
inductance
N2
L
small
2 d° of freedom
 θˆ , Nˆ   i
 1 1
 θˆ , Nˆ   i
 2 2
or
ˆ + Nˆ

N
1
2
ˆ ˆ ˆ ˆ
  =θ 1 +θ 2 , K =
2

2 knobs
2
Hˆ = E C ( Nˆ - N g ) - E J cos
ˆ
2
cos ˆ 
ˆ -θˆ


θ
ˆθ= 2 1 , Nˆ = Nˆ - Nˆ
i

1
2 
2



   0 ˆ
2L

2
L

2
0
EJ

i

The split CPB
E C ,E J
N
Vg
1 d° of freedom
 θˆ , Nˆ   i


2 knobs

2
ˆ
ˆ
H = E C ( N - N g ) - E J cos cos ˆ
2
tunable E J
δ=

0
Energy levels of the CPB
E
ˆ (N ,  ) = E ( N
ˆ - N ) 2 - E (  )cos ˆ
H
g
C
g
J
Solve either in charge basis |N> ( N 
k
k 
)
( N g ,  ),  k ( N g ,  )
c
k ,N
N
N
EJ
2
ˆ
H = E C (N - N g ) N N 2
Diagonalize
I.2) Cooper-Pair Box
 ...

...

 ...

 ...
 ...

...
E C (-1 - N g )
2
  N +1
N  N N +1 
N
...
...
... 
-E J /2
0
...
2
-E J /2
E C (0 - N g )
- E J /2
0
-E J /2
E C (1 - N g )
...
.. .
...
2


... 

...


... 
... N = -1
N = 0 N =1 ...
Energy levels of the CPB
E
ˆ (N ,  ) = E ( N
ˆ - N ) 2 - E (  )cos ˆ
H
g
C
g
J
… or in phase basis |>
(    0, 2 

k
( N g ,  ),  k ( N g ,  )
2
)
k 
 d 
k
( ) 
0
ˆ (N ,  ) = E ( 1  - N ) 2 - E (  ) cos 
H
g
C
g
J
i 
Solve Mathieu equation
EC (
I.2) Cooper-Pair Box
1 
i 
- N g )  k ( ) - E J (  ) cos   k ( ) = E k k ( )
2
Two simple limits : (1) E J (  )
EC
(charge regime)
Ej Ec 0
2
2
Energy
EJ/Ec=0
1
1
0
0
c0 ( N )
c 1( N )
00
1
0.5
0.5
11
1.5
1.5
22
2
1 0
1
2
3
2
0
N
0

0.5
 1 ( )
3
3 3 Ng
0.5
 0 ( )
3
1
2.5
2.5
2
1 0
1
2
3
N
2
0
0

Two simple limits : (1) E J (  )
EC
(charge regime)
2
EJ/Ec=0.1
Energy
E2(Ng)
1
E1(Ng)
E0(Ng)
0
Ng=0.01
0
0.5
1
1.5
2
3 Ng
2.5
1
0.5
c0 ( N )
 0 ( )
3
2
1 0
1
2
2
0
3
1
0.5
c 1( N )
 1 ( )
3
0
2
1 0
1
2
3
2
0
0
QUBIT
Two simple limits : (1) E J (  )
EC
(charge regime)
2
EJ/Ec=0.1
Energy
E2(Ng)
1
E1(Ng)
E0(Ng)
0
Ng=0.5
0
0.5
1
1.5
2
3 Ng
2.5
1
0.5
c0 ( N )
 0 ( )
2
0
3
2
1 0
1
2
3
0.5
1
c 1( N )
 1 ( )
2
0
3
0
2
1 0
1
2
3
0
QUBIT
From E J (  )
E C to E J (  )
Energy
EJ/Ec=0.5
EJ/Ec=2
2.0
3
1.5
2
1.0
1
0.5
0.0
0
0.5
1
1.0
0.0
Ng
0.2
0.4
0.6
0.8
1.0
2
0.0
Ng
0.2
0.4
0.6
0.8
1.0
0.8
1.0
EJ/Ec=10
EJ/Ec=5
2
2
Energy
EC
0
0
2
4
2
6
4
0.0
0.2
0.4
0.6
0.8
1.0
Ng
8
0.0
0.2
0.4
0.6
STILL A QUBIT !
Ng
Two simple limits : (2) E J (  )
(phase regime)
EC
J. Koch et al., PRA (2008)
EJ/Ec=10
2
0
2
4
6
8
0.0
0.2
1.0
0.5
0.0
c0 ( N )
0.5
1.0
1.0
0.5
c 1 ( N ) 0.0
0.5
1.0
I.2) Cooper-Pair Box
0.4
0.6
0.8
1.0
Ng
2
 0 ( )
0.5
4
2 0
2
0
4
 1 ( )
0
2
0.5
4
2 0
2
4
0
0
Experimental spectrum of a transmon
J. Schreier et al., PRB (2008)
One-qubit gates
Ng(t)=DNgcost

TRANSMON QUBIT

ˆ =E N
ˆ - N (t )
H
C
g

2
- E J cos ˆ
ˆ =E N
ˆ 2 - E cos ˆ - 2E D N co s  tN
ˆ
H
C
J
C
g
transmon
Two-level
approximation

 01 (  )
2
drive
z
  R cos  t x
 R  2 EC 0 N 1 DN g
I.2) Cooper-Pair Box
One-qubit gates
=01
f0

Rotation :
X
TRANSMON QUBIT
|0>
Z
Y
I.2) Cooper-Pair Box |1>
X
One-qubit gates
=01
f0
f0f

Xf
Rotation :
X
TRANSMON QUBIT
|0>
Z
Y
f
Xf
I.2) Cooper-Pair Box |1>
X
One-qubit gates
=01
f0
f0f

Xf
Rotation :
X
TRANSMON QUBIT
|0>
Z
Z rotation
Y
f
Xf
X
All rotations on Bloch sphere
Fidelity ?
99%
J.M. Chow et al., PRL 102, 090502 (2009)
I.2) Cooper-Pair Box |1>
Decoherence
Ng+dNg(t)
d(t)
Noise in Hamiltonian parameters
DECOHERENCE
MAJOR OBSTACLE TO
QUANTUM COMPUTING
I.3) Decoherence
Decoherence in superconducting qubits
(Ithier et al., PRB 72, 134519, 2005)
Pure dephasing
Relaxation
(Spontaneous emission)
i(t)
 01 (  i )
1
 01
0
t
1
 0  e
0
D  , 
1
0
H

f (t )  e
1
T2
T1
1
 G1 

2
D  ,  S  ( 01 )
2
environmental density of
modes at qubit frequency
I.3) Decoherence
1
  01

1
i ( t )
 G2 
G   D
D ,z 
i ( t )
e
G1
2
2
 ,z
 G 2t
 G
S  (0 )
Low-frequency
noise
Decoherence
Ng+dNg(t)
d(t)
Noise in Hamiltonian parameters
DECOHERENCE
Origin of the noise ???
I.3) Decoherence
Decoherence
Ng+dNg(t)
d(t)
Noise in Hamiltonian parameters
R
DECOHERENCE
R
Origin of the noise ???
1) ELECTROMAGNETIC
Low-frequency : Johnson-Nyquist due to thermal noise
High-frequency : spontaneous emission (quantum noise)

I.3) Decoherence
Under control
Decoherence
e-
Ng+dNg(t)
Spin flips
d(t)
Noise in Hamiltonian parameters
R
DECOHERENCE
R
Origin of the noise ???
2) MICROSCOPIC
Low-frequency noise
well studied :
High-frequency (GHz)
microscopic noise
I.3) Decoherence
Flux noise
Charge noise
3
S N g ( )
(10 )

6
2
S  ( )
(10  0 )
TOTALLY UNKNOWN !!

2
Decoherence
e-
Ng+dNg(t)
Spin flips
d(t)
Noise in Hamiltonian parameters
R
DECOHERENCE
R
Origin of the noise ???
2) MICROSCOPIC
Low-frequency noise
well studied :
CPB in charge regime
Transmon
High-frequency (GHz)
microscopic noise
I.3) Decoherence
Flux noise
Charge noise
3
S N g ( )
T2
T2
(10 )

6
2
S  ( )
(10  0 )

10  100 ns
T2
1  100 m s
1  10 ms
T2
1  100 m s
TOTALLY UNKNOWN !!
2
State-of-the-art coherence times
T1=1-2ms
T2=1-3ms
Schreier et al., PRB 77, 180502 (2008)
I.3) Decoherence
Very recent breakthrough: transmon in 3D cavity
H. Paik et al., quant-ph (2011)
T1=60ms
T2=15ms
ULTIMATE LIMITS ON COHERENCE TIMES UNKNOWN YET
I.3) Decoherence
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
END OF FIRST LECTURE