superconducting qubits

Solid State Qubits: way 2
Way 1
single particle states
in semiconductor structures
Way 2
global quantum states
of superconducting Josephson circuits
A)Kane’s proposal : nuclear spins of P impurities in Si
Phase qubit
Flux
qubit
B) electrons in quantum dots (charge or spin)
6 / 6 5
C)Propagating states: flying qubits
Charge-flux qubit
C P Box
C P Box
e
Z
/
e
Why superconductivity
Why different flavors

1) Energy spectrum of an isolated electrode
Non superconducting
state
N
Why superconducting qubits
Isolated
electrode
Superconducting state
S
2 ∆
+
singlet ground state
But one is not enough...

Why superconducting qubits
2) The Josephson junction
2 ∆
2 ∆

Building blocks for quantum bit circuits
N
1
0
1
0
1
0
0/1
U 1
U 1

Basics of the Josephson junction
single degree
of freedom
L
R
R
L
[ ]
= ∈ π
1
θ = ∑ H
2π
N
N∈]
2π
0
−LNθ
N
1
LNθ
= ∫ Gθ
H θ
2π
Nˆ
= ∑N N N
ˆ = L
N∈]
∂
∂ N
N=Q/2e∈]
H
Lθˆ
N = N+1
⎡ˆ, ˆN ⎤ =
⎣ ⎦
L
E
J
h
= 8e
2 R
t
Josephson Hamiltonian :
E
∑( N N+1 + N+ 1 N ) E ˆ
Jc s
J
H
J
= - = -
2 N
o θ

Josephson qubits come in different flavors
Ψ (1)
1 2 >>1
1
2
~ 1
1
phase
flux
charge-phase
charge
1,67 TU Delft
Saclay, Yale (Devoret)
NEC, Chalmers
Yale (Schoelkopf)
Single Cooper pair boxes

δ : extended phase
conjugated of q on C J
V
δ
Phase qubits
2
q
H = -EJcosδ
-ϕ0Ibδ
+ 2C
J
1
4 2 , Φ
0 0
∆8 = [ 1−
, / ,
3 2π
ω
S
⎛ 2 2π
⎜
,
=
⎝ Φ0
⎞
1/ 2
0
] 3/ 2
[ ] 1/ 4
0 ⎟ 1 − , ,
0
& ⎠
Ib
dc
ac
I 2 >
I 1 >
I 0 >
( I )
arcsin /I
Γ
L +1 10
3
Γ
L
b 0
tilted washboard potential
Γ 2
Γ 1
( ( − )
Q
( Q
/ = ω
+1 S
0.9
0.8
0.7
Γ0
0 5
δ
ω 10
ω 21
ω 32
∆8 = ω S

Rabi oscillations
t r
ω 10 ω 31
Aumentado, Urbina,
Martinis (NIST), 2002

etter phase qubit : rf SQUID
with dc SQUID readout
δ
2
1 ⎛ φ q
δ π
⎞
⎝ ⎠ 2C
ext
H = -EJcos + ⎜ - 2 ⎟ +
2L φ0
2
low noise bias
I φ
I s
"sample
and hold"
readout
qubit
J. Martinis team (2003-2005)
NIST & SB
T2 : 10-50 ns ()
I 1 >
I 0 >
Switching
current
10 µΑ
0
“0”
U(δ)
Qubit
Cycle
I φ
I s
…
“1”
fast
deca
y
1 Φ 0
SQUID flux
Qubit Op Meas Amp Reset
flux
Measure p 1
~5000 states
time

A flux quantum bit : the three junction loop
J
(0.5

SQUID readout
of the flux qubit
(readout # 1)
Switching
measurement
(Ic 200 nA)
µw on resonance

flux qubit spectroscopy
linewidth
Coherence
time
, SW
(0.4 nA per division)
~
0.498 0.500 0.502
) ext
() 0
)
9.711 GHz
8.650 GHz
6.985 GHz
5.895 GHz
4.344 GHz
3.208 GHz
2.013 GHz
1.437 GHz
1.120 GHz
0.850 GHz
f (GHz)
10
5
0
0
0.8
2
0
0 1 2 3
') res
(10 -3 ) 0
)
Short coherence times !
Van der Wal
TU Delft (2000)

Single Cooper pair boxes
V g
C g
C J
N
7KH ILUVW ZRUNLQJ·
TXELW

Hamiltonian and energy spectrum
V g
N
C g
C J
2 characteristic energies:
⎡ˆ, ˆN ⎤ = L
⎣ ⎦
1 degree of freedom: 1 knob: or /
Hamiltonian:
E
h∆
8e R
J = 2
t
V g
E
C
=
N = C V (2e)
2
g g g
( 2e)
2
( Cg
+ CJ)
ˆ ˆ 2
H = E cosˆ
C( N-N g) -EJ
θ
ˆ
2 EJ
H = E ( N-N ) N N - N+1 N + N N+1
2
C
g
∑( )
N∈]

energy spectrum
4
E =1.1
Ej /E J
Ec
C
1.1
3
E 3
Energy (E C )
2
1
0
E J
E 2
E 1
E 0
≡ 1
≡ 0
1
0 0.5 1 1.5 2 2.5 3
N g
J
qubit
E >> k T
B

Readout of single Cooper pair boxes
V g
C g
C J
N

Readout through the charge
(measurement of the quantum state)
ˆ ˆ 2
H = E ( N-N ) -E
cosθˆ
C
Nˆ
= -
g
J
1 ∂Hˆ
2E
∂N
C
g
+ N
g
Energy (E C )
1
0
E =1.1
Ej /E J Ec C 1.1
1
0 0.5 1
N g
expectation value
of the box charge:
N k
N 1
0
N 0
0 0.5 1
N g

Capacitive coupling to a Single Electron Transistor
E
0
N = 0 + N = 1
=
2
V. Bouchiat et al.
Quantronics (1996)
2
1
0
-1
theory
Théorie
Expérience
experimen
t
Sans effet Josephson
E J /E C =0.1
T=20 mK
Theory with no Josephson effect
N
V g
-2
-1.5 0 0.5 1 1.5
N g
n e
V
I( N )
0 1/2 1
qg( H)
too slow ...

A box with a continuous readout
Nakamura, Pashkin
&Tsai (NEC,1999)
Q =1
Q = 0
N
V g
Continuous
measurement by
energy relaxation
0.5
N g
2e- or 0
DC pulses from N g
=0.25 to 0.5
with duration ∆t
V
I ∝ N
5
I ( pA)
First Rabi oscillations
0
200 400 600
Pulse duration∆t (ps)
Short coherence time : a few ns
2000: WHY

The main difficulty
U 1
write
1
0
D
0
E
1
Readout
Decoherence
Sources
Open ports
Decoherence

Readout port... lets noise in
1
0
O
e!8
0 $ 0 1 $ 1
A -meter
Fluctuating
environment
detuning :
E8 T
EO
T
sO
s8
E8 T
Readout + environment
signal
(if A measured) :
DEPHASING
O
< ! ! > H
s s8
dephasing and readout closely related !

... and noise dephases...
ψ =
0 + e iϕ 1
2
MW S ÔQ
W
GW
LI 6 Z
;
FRQVWDQW
DWORZIUHT
H
L'M
W
W
7
H M
7 M S
$ $ ;
6

... and depolarises
)HUPL*ROGHQ5XOH
* 5
Z
$ 6;
=
*(
$ 6 Z
;
=
7
*
* *
( 5 5
WRVXPPDUL]H
6 ;
Z
H[FLWDWLRQ
GHSKDVLQJ
Z
Z
UHOD[DWLRQ

Solving the noise/readout dilemma:
connect only at readout time
Operate qubit at a stationary point:
sO
s8
For readout:
-Move away adiabatically at:
sO
s8
- better: stay there, and apply an ac drive
v

A qubit ‘protected’ from decoherence:
the quantronium
gate
160 x160 nm
A general strategy now applied to
different Josephson qubits
Vion et al, Science 2002; Esteve&Vion, cond-mat 2005

The quantronium: 1) a split Cooper pair box
U
2 knobs : Ng = CgU/2e
Φϕ 0
Φ
)
1 d° of freedom
2 energies:
E
C
( 2e) 2
h∆
= EJ= 2
2C
8e R
island
⎡ˆN
ˆ ⎤ = L
⎣ ⎦
ˆ ˆ 2 δ
H ≈ E
cos cosθˆ
C(
N- Ng) -EJ
2
t
U
N
i
State dependent persistent currents
î =
1 ∂Hˆ
1 ∂E
=
ϕ ∂δ
ϕ ∂δ
0 0
k

2) protected from dephasing
E J
≈ E
C
sO
s8
20
energy (k B K)
0.5
0.25
0
-0.25
- ccccc
1
2
0
δ/2π
1
ccccc
2 0
1
ccccc
2
N g
1
1
0
hν 01
ν 01 (GHz)
15
10
5
0
- ccccc
1
2
0
δ/2π
1
ccccc
2 0
1
ccccc
2
N g
1
E J =0.86 k B K
E C =0.68 k B K
Optima working points exist in many qubits

3) with a readout junction
first readout of
persistent currents
with dc switching
20
15
10
V=0
or
V≠0
I 0
I b
U
5
0
- ccccc
1
2
U
δ/2π
0
1
ccccc
2 0
1
ccccc
2
N g
RF amplitude (a.u.)
1
200 20
0
0 1 2 3 4 5 6
time (µs)
bias current (nA)
1000 100
800 80
600 60
400 40
output voltage (µV)
1
1: switching
0: no switching
0 discrimination

Rabi precession
microwave output voltage (mV)
100
50
0
-50
A cos( ωW
+ χ)
Z
16 GHz
1.0 2.0 3.0 4.0
time (ns)
switching
probability
(%)
UHDGRXW
URWDWLRQ
ω Rabi = αU RF
Rabi oscillations
50
F
Y
40
X
Effective
field
30
0 50 100 150 200 250
ν µw
Note: visibility :

measuring coherence using Ramsey interferences
0
ω Rabi
ω =ω 01 −ω RF
ω Rabi
Projection Z
π/2 pulse )UHHHYROXWLRQ π/2 pulse
URWDWLRQDOVR
UHDGRXW
Ramsey interferences reveal
decoherence during free evolution

Ramsey interferences in the quantronium
switching probability (%) (%)
45
45
40
40
35
35
30
30
∆t
ν RF = 16409.50 MHz
−∆W/
7
sin(2π
.
Fit
∆ν = 19.84 MHz
T = 500 +/- 50 ns
0.0 0.1 0.2 0.3 0.4 0.5 0.6
0.0 0.1 time 0.2 between 0.3 pulses ∆t 0.4(µs)
0.5 0.6
time between pulses ∆t (µs)
H
ϕ
ν.
W)

microwave output voltage (mV)
100
50
0
-50
Arbitrary transformation of a qubit
using multipulse sequences
1.0 2.0 3.0 4.0
time (ns)
16 GHz
X
0
1
Z
Y
X
F
Y
Effective
field
Remind:
8 = 5 5 5
3 ; 2< 1;
ν µw

Checking combined rotations
0
ω Rabi
Collin et al,
PRL 2004
ω =ω 01 −ω RF
ω Rabi
Projection Z
π/2 pulse )UHHHYROXWLRQ π/2 pulse
URWDWLRQDOVR
UHDGRXW
60
(π/2) X
-(π/2) X
(π/2) X
-(π/2) -Y
switching probability (%)
50
40
30
(π/2) X
-(π/2) -X
(π/2) X
-(π/2) Y
50 MHz detuning
20
0 20 40 60
8 = 53 ;
52 <
51;
feasible
delay between π/2 pulse (ns)
robustness

Towards better rotations: composite pulses
Composite ‘ S ’ CORPSE :
60° X 300 ° -X 420 X ° Collin et al,
6 0
sw itching probability (% )
5 0
4 0
3 0
‘ S ’ CORPSE
corpse: 420°(X )/300°(-X )/60°(X )
sim ple π pulse
ν R abi
= 92 M H z
sw eet sp ot
Single pulse
1 6 .3 0 1 6.3 5 16 .40 1 6 .4 5 1 6 .5 0
frequency (G H z)
Improved robustness to frequency detuning
PRL 2004

Corpse :
test for any initial state
NOT operation: θ x
π Corpse
60
θ x
θ x
+ corpse π X
switching probability (%)
50
40
30
20
180 360 540 720
angle (°)

Optimal point for other qubits
the flux qubit
, E
Chiorescu et al, TU Delft 2003
Nakamura et al., NEC 2005
a )
, H[
PRELI-
, E
, F P$
, J
T
S+
a )
S+
T
MINARY
, E
2005: T2=400 ns

Cavity QED with a Cooper pair box
Dispersive readout with
out of resonance photons
R. Schoelkopf, A. Wallraff, S. Girvin
et al., Yale (2004)

Cavity Quantum Electrodynamics (CQED)

Coherence time measurements
with 2 pulse Ramsey sequence

2 qubit gates
CNOT performed with 2 Cooper pair boxes,
(NEC 2003), without single shot readout
Correlations demonstrated on 2 phase qubits,
(NIST-SB, Science 2005)
2 coupled phase qubits
10 and 01 anticorrelated

Summary :
Different Josephson qubits at work
General strategy to improve coherence: optimal point
Single shot readout, but fidelity