chalmers nec review of solid state quantum bit circuits two strategies single particle states in...
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ChalmersNEC
REVIEW OF SOLID STATE QUANTUM BIT CIRCUITS
Two strategies
single particle statesin semiconductor structures
global quantum statesof superconducting Josephson circuits
(A) Kane’s proposal : nuclear spins of P impurities in Si
(B) Electrons in quantum dots
(C) Propagating states: flying qubits NIST
SL SR
eeL
w
QHE edge states:LPA (ENS Paris)
B1) Charge: NTTB2) Spin: TU Delft, Harvard,…
U. Of New South Wales
TU DelftSchoelkopf et al, Yale
Quantronics
From charge states to phase states
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First demonstration of coherent oscillations in a double dotd
Coherent charge oscillations in a double dot(NTT, Hayashi et al. 2003)
T2 (charge qubit) ~ 1 ns
But charge too much coupled to the environment ! spin expected better
L
R
LR
LR
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• 1-qubit control:• magnetic (ESR)• electric (modulate effective g-factor)
• 2-qubit coupling: exchange interaction between 2 dots
• Read-out through charge
SL SR
B2) Spin qubits
Initial ideas: Loss & DiVincenzo (1998)
EZ = gBB
EZ = gBB
BZ
J(t) J(t)
Expts: TU Delft , Harvard
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b) 2e spin qubit in a double dot (NEW)
charge readout Two electron spin qubit
GDGD
GSRGSRGSLGSL
Harvard
Charge sensor
0(1,1) (1,1) mS T with QPC
Harvard U. C. Marcus team,
# e indot
Nature, june 2005,& Petta et al., in prep.
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Bloch sphere in (1,1) S - T0 subspace
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(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
2t
(1,1)T0
Measuring Spin Dephasing (T2)
(electrostatic energy difference)
Move from (0,2)S to (1,1) s & let evolve
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(1,1)S
(1,1)S (0,2)S
(0,2)S
ε
2t
(1,1)T0
dephasing causes failure to return to (0,2)
Measuring Spin Dephasing (T2)
spin T2 ~ 10 ns
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BZeeman
BNuclear
BTotal
Short coherence time : 10 ns due to nuclei
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But coherence restored byspin echo
experiment
tflip
tflip
pattern still observed at long times: coherence time TE =1.2 s
model
0 40 ns
40 ns0
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Electron spin resonance in a double dot
Gates ~ 30 nm gold
Dielectric ~ 100nm calixerene
Stripline ~ 400nm gold
team L. Kowenhouven& L. Vandersypen, TU Delft
B0
Expected AC current ~ 1mA Expected AC field ~ 1mT
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ESR detection via Pauli spin blockade
Flip right spin: ↑↑ ↑↓ = S11 + T0 S02
Flip left spin: ↑↑ ↓↑ = -S11 - T0 S02
Flip both spins: ↑↑ (T+ + 2 T0 + T- )/2 S02
Advantages: - low frequencies (B0 > 20mT, f > 100MHz. )
- not sensitive to electric fields (unlike single-dot ESR)
- no confusion with ESR in leads
hfac = g B0
T-
T0
T+
S
Bext
Ene
rgy
31jan06_20
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Pulsed ESR scheme
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Rabi oscillations
pA
F. Koppens et al.
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Way 1
single particle statesin semiconductor structures
global quantum statesof superconducting Josephson circuits
A)Kane’s proposal : nuclear spins of P impurities in Si
B) electrons in quantum dots (charge or spin)
C)Propagating states: flying qubits
Phase qubit
SL SR
eeL
w
Flux qubit
C P Box
Why superconductivity ?Why different flavors?
Solid State Qubits: way 2
Charge-flux qubit
C P Box
Way 2
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NIST NEC, ChalmersTU Delft
Josephson qubits come in different flavors
Yale (Schoelkopf)
(N)
N
12 N
flux
1~2N
phase chargecharge-phase
Single Cooper pair boxes
Saclay, Yale (Devoret)
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Energy spectrum of an isolated electrode
Superconducting stateNon superconducting
state
N
+
2S
singlet ground state
Superconducting helps making qubits
2 2
The Josephson junction
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N
01
01
01 0/1
U1U1
Building blocks for quantum bit circuits
?
?
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Basics of the Josephson junction
ˆN
N N N N
R Lθ = θ -θ 0,2
N=Q/2e θ,N i
RθLθ
single degreeof freedom
θ =N
iˆ
N N+1 ie
1
2N
N
N
ie
2
0
1
2NN
ied
ˆJJ J
N
N N+1 N+H = - = -1 c sE
E2
N o 2Jt
hΔ=
8e RE
Josephson Hamiltonian :
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Phase qubits
J 0J
2
b
qH = - cos - I
C+
2E
: extended phase conjugated of q on CJ
b 0arcsin /II
I 0 >I 1 >I 2 >
12
31 10
i
i
0 0 50.7
0.8
0.9
1
pU
10
2132
pn
nE
E /
1
tilted washboard potential
V
ac
dc
Ib
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1 2
A flux quantum bit : the three junction loop
EJEJ
EJ
k
i
0.5 0
E
Icirc
0
-1
0
1
2t
0.5 /o
+Ip
-Ip
0k
i
x=0/2
2D potential:
(0.5<<1)
Mooij et al. (TU Delft), (1999)
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Single Cooper pair boxes
Vg
CgCJ
N θ
The first ‘working’ qubit
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0 0.5 1 1.5 2 2.5 31
0
1
2
3
4EjEc1.1
Ng
En
erg
y (E
C)
0E
1E
2E
3E
JE 1
0
qubit
BJE >> k T
CJ/EE =1.1
energy spectrum
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Josephson Phase Qubitsat UCSB
UC Santa Barbara
Team of John Martinis
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Phase qubits
23
2/3
000 /1
24II
IU
C
4/1
0
2/1
0
0
122
III
p
I RCLJ
Lifetime of state |1>
0 50.7
0.8
0.9
1
pU
10
2132
pn
nE
E /
1
RC
Up
U()<V> = 0
<V> pulse(state measurement)
I0
= 0/2I0cos nonlinear inductor
I cos I 0j V ) (1/L J
0 sin I I
LJ
2
V 0
1: Tunable well (with I)2: Transitions non-degenerate3: Tunneling from top wells4: Lifetime from R
E0
E1
E2
0
1
2n
n+11000~
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better phase qubit : rf SQUID with dc SQUID readout
J. Martinis team (2003-2005)NIST & SB
Is
I
low noise bias qubit
"sample and hold" readout
U()
~50
00 s
tate
s
“0”
“1”
… fastdeca
y
1
SQUID fluxS
witc
hin
g c
urre
nt
0
Is
I
time
QubitCycle
Qubit Op Meas Amp Reset flux
Measure p1
I 0 >I 1 >
2 2e
0J
xt1 qH = - cos + - 2 +
2L CE
2
T2 : 10-50 ns (?)
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• State PreparationWait t > 1/ for decay to |0>
• Qubit logic with current bias
• State Measurement: U(Idc+Ip) Fast single shot – high fidelity
Josephson-Junction Qubit
|0>
|1>
I = Idc + Idc(t) + Iwc(t)cos10t + Iws(t)sin10t phase
pote
ntia
l
)2( z wsywcx IIH Idc(t)
pulse height of Ip
Pro
b. T
unn
el
|0> : no tunnel
|1> : tunnel 96%
|0>|1>
dt
e 01
Apply ~3ns Gaussian pulse
1.0
0.8
0.6
0.4
0.2
0.00.80.70.60.50.40.30.2
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IC Fabrication
I
IsIwave
100m
Qubit
Al junction process& optical lithography
via junction
SiNxAl
AlAl
Al2O3 substrate
(old design)
Z
X,Y readout
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Qubit Fidelity Tests (2006)Rabi:
Ramsey:
Echo:
T1:
Pro
bab
ility
1 s
tate
Large Visibility! T1 = 110 ns, T ~ 85 ns
~90% visibility
(slightly detuned)
(no detuning)
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SUPERCONDUCTING FLUX QUBITS
Group of Prof. Hans MooijTU DELFT
team Y. Nakamura NEC
team K. Semba NTT
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tunneling: Tin = m exp{ -0.64(EJ/EC)½}
Tout = 1.6 m exp{ -1.5(EJ/EC)½}
Tin
Tout
1 (
2
()
magnetic flux 0.5 o
1 2
persistent-current quantum bit
EJEJ
EJ
barrier scales with EJ,depends on
effective massscales with junction capacitance C
<1 to suppressinfluence of charge noise
for =0.8
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1 2
flux qubits : the three junction loop
EJEJ
EJ
k
i
0.5 0
E
Icirc
0
-1
0
1
2t
0.5 /o
+Ip
-Ip
0k
i
x=0/2
2D potential:
(0.5<<1)
Mooij et al. (TU Delft), (1999)
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SQUID readoutof the flux qubit(readout # 1)
Switching measurement(Ic 200 nA)
w on resonance
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o
switchingcurrent
o-0.5
x10-3
f GHz
Van der Wal et al., Science 290, 773 (2000)
also SUNY (Friedman, Lukens et al.)
First spectroscopy of flux qubits
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Irinel Chiorescu and Yasu Nakamura (NEC, Delft)
pH 5 pH 5
A 20 cI
exI
qqI ,
0~
05.0~
Science 299, 1869 (2003)
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Dephasing: T2Ramsey, T2echo measurement (sample5)
~ 4ns
t/2
readout pulse
~2ns
t/2
~2ns
t
correspond to detuning
readout pulse
Ramsey interference
spin echo
NEC
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Observation of vacuum Rabi oscillationsin a flux qubit coupled to a SQUID resonator
VSQUID
Isw
I
|↓>
|↑>
DC SQUID detector:
VSQUID = 0 qubit state |↑> ≠ 0 qubit state |↓>
qubitcircqubit
n Id
dE,
ISQUID~Mqub,SQIcirc,qubit
ISQUID Ithreshold
6.7μm
6.3μm
MIcirc,qubit
SQUID
qubit
K. Semba et al., NTT
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|e0
|e1
|g1|g0
readout readout qubit state qubit state
excite qubit excite qubit by aby a-pulse-pulse
1 → 21 → 2 3 ⇔ 43 ⇔ 4
shift qubit shift qubit adiabaticallyadiabatically
|e0
|e1
|g1|g0
|e0
|e1
|g1|g0
2 → 32 → 3
|g0
|g 1
|e0
4
shift qubit shift qubit adiabaticallyadiabatically
I I qubitqubit, , LC-oscillator LC-oscillator >> Vacuum Rabi : measurement schemeVacuum Rabi : measurement scheme
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Vacuum Rabi oscillationsVacuum Rabi oscillationsDirect evidence of level quantization in a 0.1 mm large
superconducting macroscopic LC -circuit
J. Johansson et al., cond-mat/0510457 → to appear in J. Johansson et al., cond-mat/0510457 → to appear in Phys. Rev. Lett.Phys. Rev. Lett. (2006). (2006).
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Quantization of Rabi periodQuantization of Rabi period
7.0
1
)2/(1/ |e2
|g2
√2
|e0
|g1|g0
|e1
/
J. Johansson et al., cond-mat/0510457 → to appear in J. Johansson et al., cond-mat/0510457 → to appear in Phys. Rev. Lett.Phys. Rev. Lett. (2006). (2006).
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Readout of single Cooper pair boxes
Vg
CgCJ
N θ
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Hamiltonian and energy spectrum
Vg
CgCJ
2 characteristic energies:
2
C
g J
2e
2E
C C
=
2Jt
h
8eE
R
θ,N i1 degree of freedom: 1 knob: or /g g gN = C V (2e)gV
Nθ
ˆ ˆˆ cos2gC JEH = ( -E N N ) - Hamiltonian:
ˆg
N
J2C
EE N N N N+1 NH = ( -N ) -
2N N+1
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0 0.5 1 1.5 2 2.5 31
0
1
2
3
4EjEc1.1
Ng
En
erg
y (E
C)
0E
1E
2E
3E
JE 1
0
qubit
BJE >> k T
CJ/EE =1.1
energy spectrum
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0 0.5 1
0
1
0 0.5 1
0
1
EjEc1.1
Ene
rgy
(EC)
Ng
Ng
kN
0N
1N
CJ/EE =1.1
expectation value of the box charge:
(measurement of the quantum state)
ˆ ˆˆ cos2gC JEH = ( -E N N ) -
ˆ1ˆC
gg
N N=H
-E
+2 N
Readout through the charge
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Capacitive coupling to a Single Electron Transistor
gV
N
V
I N
( )gq e0 1/2 1
en
V. Bouchiat et al.Quantronics (1996)
-1.5 0 0.5 1 1.5-2
-1
0
1
2
T = 2 0 m K
<N
0>
Ng
Expérience
Théorie
Sans effet Josephson
0 1
20
N NE
EJ/EC=0.1
theory
experiment
Theory with no Josephson effect
too slow ...
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A box with a continuous readout Nakamura, Pashkin &Tsai (NEC,1999)
N
V
2e- or 0
gV
I N
Continuous measurement by
energy relaxation
1n
0n
Ng0.5
DC pulses from Ng=0.25 to 0.5 with duration t
Pulsedurationt (ps)
200 400 600
5
(I pA)
0
First Rabi oscillations
Short coherence time : a few ns2000: WHY??
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Open ports Decoherence
write
0
Readout
1?1
0U1
Decoherence Sources
The main difficulty
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Readout port... lets noise in
1
0
AXFluctuating
environment
A -meter
0 0A 1 1A
01n
( )X tddetuning :
Readout + environment
0101( ) ( )
Xt X t
ndn d
¶¶
=
DEPHASING
[ ] 011 1 0 0A A hXn
- =¶¶
signal (if A measured) :
dephasing and readout closely related !
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... and noise dephases...
2
0 1 ei
012 ( ') ) '( tt dtj p n= ò
( )
tTi te e jjD
-
=
2 11 1 0 0( ))(0A A XSTj p -
-=
if
constant
at low f eq.
( )
r
XS w
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... and depolarises
Fermi Golden Rule
w
w
G
=
= -
G +
01
2
2
2
01
2
10 1
1( )0
(
1
)X
X
R
E
A
SA
S
XS
ww+ 01
relaxationw- 01 0
dephasingexcitation
to summarize:
( ) - -= +G GG 1 11 R RET
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Solving the noise/readout dilemma:connect only at readout time
01 0Xn¶
=¶
Operate qubit at a stationary point:
For readout:
-Move away adiabatically at:
-better: stay there, and apply an ac drive(to be shown later on)
01 0Xn¶
¹¶
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gate 160 x160 nm
A qubit ‘protected’ from decoherence:the quantronium
A general strategy now applied to different Josephson qubits
Vion et al, Science 2002; Esteve&Vion, cond-mat 2005
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UU
0δ = /
δ
ˆ cos cosˆ ˆ 2
g JCE NH (2
N E- ) -
The quantronium: 1) a split Cooper pair box
2 knobs : g gN C U/2e
2 energies:
2
Cisland
EC
2e
2 =
2Jt
h
8eE
R
State dependent persistent currents
k
0 0
ˆ1 H 1 Ei
i
ˆθ,N i1 d° of freedom
N
U
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-1€€€€€2
0
1€€€€€2 0
1€€€€€2
1
-0.25
0
0.25
0.5
-1€€€€€2
0
1€€€€€2
2) protected from dephasing
EJ=0.86 kBKEC=0.68 kBK
1
0
-1€€€€€2
0
1€€€€€2 0
1€€€€€2
1
0
5
10
15
20
-1€€€€€2
0
1€€€€€2
01(
GH
z)Ng
h01
ene
rgy
(kBK
)
Ng
CJE E
01 0Xn¶
=¶
Optima working points exist in many qubits
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-1€€€€€2
0
1€€€€€2 0
1€€€€€2
1
0
5
10
15
20
-1€€€€€2
0
1€€€€€2
I0 Ib
V=0or
V0
U
Ng
3) with a readout junctionfirst readout of
persistent currents with dc switching
switching
no switching
0 1 2 3 4 5 6
200
400
600
800
1000
bia
s c
urr
en
t (n
A)
time (µs)
0
20
40
60
80
100
ou
tpu
t vo
lta
ge
(µ
V)
RF
am
plitu
de
(a
.u.)
1
0discrimination
t
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Rabi precession
1.0 2.0 3.0 4.0
-50
0
50
100
16 GHz
mic
row
ave
outp
ut v
olta
ge (
mV
)
time (ns)
0 50 100 150 200 250
30
40
50
pulse duration (ns)
Rabi oscillationsswitchingprobability (%)
μwA cos( ) t
Note: visibility : <40%
1
µw
X
Y
0
Effectivefield
rotationRabiURF
0
readout
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Optimal point for other qubits?
the flux qubit
Nakamura, Ciorescu,Bertet, Mooij et al. Delft, 2003-2004
pH 5 pH 5
A 20 cI
exI
qqI ,
0~
05.0~
0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54
-50
-48
-46
-44
-42
shift
read-outpoint
operation point
E /
EC
total flux (0)
shift in practice
operating point readout point
coupling to SQUID
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Ib=0
Ib=-0.2
Ib=-0.4
Optimal point for other qubits? the flux qubit
2005: T2=400 ns
pH 5 pH 5
A 20 cI
exI
qqI ,
0~
05.0~
Chiorescu et al, TU Delft 2003Nakamura et al., NEC 2005
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projection fidelity ?
1
0
Fluctuatingenvironment
A -meter
0 0A 1 1A
idealQuantum
measurement:1 1
0 0
Readout: 1
Readout: 0
errors: wrong answer & projection error
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510 515 520 525 530 535 5400
20
40
60
80
100
d ifference
ground state pulse
M agic pointN
g = 1/2
switc
hin
g p
rob
ab
ility
(%
)
b ias current am plitude (nA)
Switching readout fidelity ?
40% contrast (only)
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0 200 400 600
-200.0
0.0
200.0
400.0 µw pulse
plateau
15mK
I b (
nA
)time (ns)
-300 -200 -100 0 100 200 3000.30
0.35
0.40
0.45
0.50
100 ns long plateau
Ip (nA)
Sw
itc
hin
g P
rob
ab
ility
0 200 400 600 800 1000 1200 1400 1600 1800 20000.25
0.30
0.35
0.40
0.45
0.50
T1=730 ns
T1=59 ns
Sw
itc
hin
g p
rob
ab
ility
plateau duration (ns)
Relaxation during readout ramp !
partly explains readout fidelity
T1=730ns
T1=60ns
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switchingdc pulse
simple, but:
rep rate limited by quasiparticlesfidelity <1 due to relaxation qubit reset : NOT QND
tV
t
dc readout
U
Switching readout resets the qubit
resets the qubit
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Towards QND readout ‘at’ optimal point
flux qubit : charge qubit :
SQUID inductance quantum capacitance
Chalmers, Helsinki
charge-phase qubit :
readout junction inductance
Quantum capacitance
C/C
J
0
1
TU Delft Yale, Saclay
0
1
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PULSE IN
PULSE OUT
U
“RF” pulse
dynamics in anharmonic potential
more complex, but:
-better fidelity ?-no reset: possibly QND
switching
dc pulse
simple, but:
-fidelity 40%-qubit reset : NOT QND
t
U
rf readout (M. Devoret, Yale)
dc versus ac readout in the quantronium
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M. Devoret team at Yale I. Siddiqi et al., (2004)
µW Pulse IN
QuBitcontrol
0 1
0
1
OUT
-1€€€€€2
0
1€€€€€2 0
1€€€€€2
1
0
5
10
15
20
-1€€€€€2
0
1€€€€€2
Towards non destructive readout at optimal point with an AC drive
UJop
timal
P
1
001n
Similar dispersive methods developed for other qubits
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M. Devoret team at Yale I. Siddiqi et al., (2004)
µW Pulse IN
QuBitcontrol
0 1
0
1
OUT
UJop
timal
P
1
001n 01
180°
-180°
am
plitu
de
µW drive amplitude
µW
pha
se
State dependent bifurcation
The Josephson Bifurcation Amplifier
Enhanced
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300 K
Quantronium from Yale
Quantronium + JBA SETUP
4 K
0.6 K
30 mK
1.3-2GHz
MS
-20dB
-30dB0 100 200 300 400
0
1
2
3
4
VM (
V)
time (ns)
Q
50
TN=2.5KG=40dB
G=40dB
ILO
demodulator
bifurcation
NO bifurcation
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-6.8 -6.6 -6.4 -6.20.0
0.2
0.4
0.6
0.8
1.0
Bifu
rca
tio
n p
rob
ab
ility
readout µW input power (dB)
0 20 40 60 80 100 120 140 160
gate µW pulse duration (ns)
45-5
0%
Rabi oscillations with the JBA
Contrast : 50%
0
1
100ns 125ns
JBA pulse
(Saclay exprt)
1 0
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100ns 125ns5n
s
20ns
40ns
JBA readout
10ns
gate
100ns
0
1
0
1
0
1
0
1
partially QND
initial finP( , , ral esult)
1
0 0
10
34%
100%
66%
0%
10
25%9%
30%36%
1
10
17%83%
10
0%0%
Notice: relaxation againpartly avoidableby tuning the qubit
0
1
Quantum Non Demolition ? read twice
initial A B
& correlations
Note: results for flux-qubit now available
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Dispersive readout of the flux qubit
detection
Tdetection
Tplateau
switching
time
Iac
A. Lupascu et al.TU DELFT
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Activation rates for different detuning values
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0
0.5
1.0
1.5
2.0
2.5
3.0
Tplateau
=400 ns
Tplateau
=80 ns
I ac
2 (au
)
ln(a/)2/3
F = 775 MHzFres=822 MHz
Tk
U
attsw
Be
2
2
0 1B
acaa I
I
2/32
1
B
acdyn I
IuU
2033
4 pa RC
Iac,bifurcation2
slope=udyn/(kT)
Thy: M. Dykman
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Rabi oscillations with optimal settings
0 20 40 60 800.0
0.2
0.4
0.6
0.8
1.0
Psw
Dt (ns)0 10 20 30 40 50 60 70 80
0
2
4
6
8
10
12
14
Psw
Dt (ns)
0.1 1
0.01
0.1
FR
abi (
GH
z)
Imw
(au)
Dt = length of MW pulse
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Ramsey oscillations with optimal settings
0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
97 %
Psw
Dt (ns)
11 %
Rabi oscillation
0.0 0.1 0.2 0.3 0.4 0.50.0
0.2
0.4
0.6
0.8
1.0
Psw
tRamsey
(ns)
Ramsey: ge-mw= 69 MHz
Ramsey frequency vs detuning
4.50 4.55 4.60 4.650
10
20
30
40
50
60
70
80 F
Ramsey
|Fmw
-Fqubit
|
FR
amse
y (
MH
z)
Fmw
(GHz)
Relatively strong low frequency fluctuations visible in the drift of the Ramsey frequency.
QND data : analysis in progress
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A Circuit Analog for Cavity QED2g = vacuum Rabi freq.
= cavity decay rate
= “transverse” decay rate
L = ~ 2.5 cm
Cooper-pair box “atom”10 m10 GHz in
out
transmissionline “cavity”
Blais, Huang, Wallraff, Girvin & RS, cond-mat/0402216; to appear in PRA
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Cavity QED with a Cooper pair box: first dispersive readout
R. Schoelkopf, A. Wallraff, S. Girvin et al., Yale (2004)
Dispersive readout with out of resonance photons
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Dressed Artificial Atom: Resonant Case
? T01 R
2g
/ R
T
2
1“vacuum Rabi splitting”
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Rabi Oscillations of Qubit
Prf = 0 dB Prf = +6 dB
Prf = 18 dB
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Coherence time measurementswith 2 pulse Ramsey sequence
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CONCLUSION
Solid state qubits at work:
Semiconductor qubits recently demonstrated
Superconducting qubits:qubit control
single-shot readout
Decoherence, QND readout, couplingin next lectures
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The work on
SPEC
ELECTQUANRONICSUM
GROUP
G. ITHIERE. COLLINN. BOULANTD. VION P. ORFILA P. SENATP. JOYEZP. MEESOND. ESTEVE
A. SHNIRMANG. SCHOENY. MAKHLINF. CHIARELLO
1
0
Fluctuatingenvironment
A -meter
0 0A 1 1A
01n
the Quantronium
1µm
boxqp
trap
dc gatedc gateµw
readout junction
2004
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Thanks to
NEC