1 paolo silvestrini macroscopic quantum coherence and quantum computing seconda università di...
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Paolo Silvestrini
Macroscopic Quantum Coherence and Quantum Computing
Seconda Università di Napoli -Dip. di Ingegneria dell’Informazione CNR - Istituto di Cibernetica “Eduardo Caianiello”
MQC group in Naples
Valentina CoratoCarmine Granata Sara RombettoBerardo Ruggiero Maurizio RussoRoberto Russo
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Back to basics…
1|0| ba
Fundamental carrier of information: the bit
Possible qubit states: any superposition describedby the wavefunction
“0” “1”or
Fundamental carrier of quantum information: the qubit
Possible bit states:
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3
10
122
qbit
1)2/sin(0)2/cos( ie
Z
1
0
X
Y
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Quantum computationwith chloroform NMR
Deutsch algorithmdemonstrated.
1H
13C
Cl
Cl Cl
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Five Criteria for physical implementation of a quantum computer
1. Well defined extendible qubit array -stable memory
2. Preparable in the “000…” state3. Long decoherence time (>104 operation time)4. Universal set of gate operations5. Single-quantum measurements
D. P. DiVincenzo, in Mesoscopic Electron Transport, eds. Sohn, Kowenhoven, Schoen (Kluwer 1997), p. 657, cond-mat/9612126; “The Physical Implementation of Quantum Computation,” quant-ph/0002077.
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Physical systems actively consideredfor quantum computer implementation
• Liquid-state NMR
• NMR spin lattices
• Linear ion-trap spectroscopy
• Neutral-atom optical lattices
• Cavity QED + atoms
• Linear optics with single photons
• Nitrogen vacancies in diamond
• Electrons on liquid He
• Josephson devices
– “charge” qubits
– “flux” qubits
– “phase” qubits
• Spin spectroscopies, impurities in semiconductors
• Coupled quantum dots
– Qubits: spin,charge,excitons
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Dissipation effects (Chakravarty and Leggett PRL 1984, Grabert and Weiss PRL 1984)
Long decoherence time Low dissipation Low T
0 1 2 3 4-1,0
-0,5
0,0
0,5
1,0
P(t* )
t*
R1<R2< R3
P(t)
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NISTChalmers
NEC
TU Delft
Superconducting Josephson qubits
Schoelkopf et al, Yale
NEW: « atom in cavity » :
(N)
N
12 N
flux
1~2N
phasecharge
charge-phase
quantronium
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Tilt Washboard Potential
sin
1
22
2
cNdc Idt
d
Redt
dC
eII
0tIN
R
TKtItI B
NN 2
JJR
CIN Idc
-50
0
50
I(A)
-4 -2 0 2 4
V(mV)Stat
o Jo
seph
son
U
Stato resistivo
cos2
tIe
U c
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10
-50
0
50
I(A)
-4 -2 0 2 4
V(mV)
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0
cL
150
LI2
weber1007.2e2
h
rf-SQUID
00
Lx
0
2sin2
n22• Quantizzazione del flussoide
• Effetto Josephson
x rf SQUID
U
2
xL0x 2
1cosU,U
L4U
2
20
0
x
= x
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Iecc
V o u t
I b i a s
Li
Ip
LP
Is
excitation coil
rfSQUID
dcSQUID
flux transformer
Controls:
rf-SQUID
x• Simmetry of potential
xdc rf
SQUID•Hight of the barrier
Flux transformer
Stacked JJ
• Coupling to the readout dc-SQUID
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350 m
dc-SQUIDflux
transformer
rf-SQUIDmodulation and feedback coils
flux transformer
rf-SQUIDexcitation
coil
Josephson junction
50 m
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A. J. Leggett, Prog. Theor. Phys. 69, 80 (1980)
Rf-SQUID Potential
|L> |R>
LAS 2
1,
R+-
I I
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k
jjkkkjjjj WW
dt
d
Quantizied Energy levels
Tunnel rate
Potential Potential barrier
Wjk
j
Quantum picture
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U
cx
U
0
U
q eA
2
1
00q
UA
Quantum Tunneling
B
00 K2
TT
RC
LCL
0
2/1
min0
87.012.7
cos1
TK
U
TBeAT
CRAT
1 TKBeRR
0
U
cx
U
Thermal Activation
BKTT
20
0
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Measurements of Macroscopic Quantum Tunneling out of the Zero-Voltage of a Current-Biased Josephson Junction
M. H. Devoret, J. M. Martinis, and J. Clarke, PRL 55, 1543 (1985)
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102
103
104
10-2
10-1
1 data
-1/TR-1
Q
P. Silvestrini, R. Cristiano, S. Pagano, O. Liengme, and K. E. Gray,“Effect of dissipation on Thermal Activation in an underdamped Josephson
Junction: First evidence of a Transition between Different Damping Regimes”PRL 60, 844 (1988)
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25,0 25,2 25,4 25,6 25,8 26,00
3
6
9
01214 10 8 6 4 2
2S/h
I(A)
P(a.u.)
103
106
109
esca
pe
rate
(s-1
)
103
106
109
103
106
109
0 4 8 12
0,8
1,0
1,2
1,4Iexp
Iteo
n
Observation of Energy Level Quantization in Underdamped Josephson Junctions above the Classical-Quantum Regime Crossover Temperature
P.Silvestrini, V.G. Palmieri, B. Ruggiero, and M. Russo, Phys. Rev. Lett. 79, 3046 (1997)
R>20K
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Observation of Resonant Tunneling between Macroscopically Distinct Quantum Levels
R. Rouse, S. Han, and J. E. Lukens, PRL 75, 1614 (1995)
Resonant macroscopic quantum tunneling in SQUID systems
P.Silvestrini, B.Ruggiero, and Y.Ovchinnikov; PRB 54, 3046 (1996)
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U
E
0E
RE
1RE Tℓ
cos
2
1UU L
2xo
te
UM
H Lxo cos2
cos2
1
2
1 2
2
2
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Quantum superposition of distinct macroscopic states
Jonathan R. Friedman, Vijay Patel, W. Chen, S. K. Tolpygo & J. E. Lukens
NATURE | VOL 406 | 6 JULY 2000
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Coherent control of macroscopic quantum states in
a single-Cooper-pair boxY. Nakamura, Yu. A. Pashkin & J. S. Tsai
NATURE |VOL 398 | 29 APRIL 1999
Manipulating the Quantum State of an Electrical CircuitD. Vion, A. Aassime, A. Cottet, P. Joyez, H. Pothier, C. Urbina, D. Esteve, M. H. Devoret
SCIENCE VOL 296 3 MAY 2002
dec =0.5 s
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Coherent Temporal Oscillations improvement of Macroscopic Quantum States in a Josephson Junction
Yang Yu, Siyuan Han, Xi Chu, Shih-I Chu, Zhen Wang
SCIENCE 296, 889 (2002)
dec =5 s
Rabi oscillations in a large Josephson-junction qubitJohn M. Martinis, S. Nam, J. Aumentado, andC. Urbina
dec =10 ns
Phys. Rev. Lett 89, 117401 (2002)
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Coherent Quantum Dynamics of a Superconducting Flux Qubit
I. Chiorescu, Y. Nakamura, C. J. P. M. Harmans, J. E. Mooij
dec =20 ns
Science 299, 1869 (2003)
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Quantum oscillations in two coupled charge qubitsYu. A. Pashkin, T. Yamamoto, O. Astafiev, Y. Nakamura, D. V. Averin & J. S. Tsai
NATURE |VOL 421 | 20 FEBRUARY 2003
dec =2.5 ns
dec =0.6 ns
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0.0 0.5 1.0 1.5 2.0 2.5 3.0
2.4
2.6
2.8
3.0
0.000 0.002 0.004 0.006
4
6
8
10
12
14
16
Ib (A)
g1
e1e0
g0
g2
|g0> |e1>
|g0> |e0>
plasma resonance p
red sideband R
blue sideband b
Res
on
ant
freq
uen
cies
(G
Hz)
/0
qubit Larm
or frequency L
|g1> |e0>
e2
0 1 2 3
0
100
200
300
400
500
time
(n
s)Ibias (A)
T1 T2 Tphi
Coupling a flux qubit and a harmonic oscillator Relaxation-limited dephasingat the optimal point
Dynamics of a flux-qubit coupled to a harmonic oscillator (P. Bertet)
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Coupling phase qubits
U
UIbI*
2
10|01|
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Quantronium:-arbitrary robust operations-decoherence fought: echoes, wave driving-new readout developed at Yale
16.30 16.35 16.40 16.45 16.50
30
40
50
60
p (
%)
frequency (GHz)
Corpsecomposite
pulse
0 500 1000 1500
0.3
0.4
0.5
delay (ns)
p
*1 1T T
0 200 400 600 800 1000 12000.2
0.3
0.4
0.5
0.6
pulse length (ns)
p
*2 2T T
0 500 1000 15000.2
0.3
0.4
0.5
0.6
delay (ns)
p
2ECHOT T
combined X&Y rotations0 20 40 60
20
30
40
50
60
p (%
)
delay (ns)
SPIN-LOCKING
Rabi
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BIFURCATION AMPLIFICATION
• Bifurcation amplifier: sensitive to any input variable coupled to I0
minimal back-action- no on-chip dissipation- efficiently thermalize load- backaction narrow band
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Demonstration of conditional gate operation using superconducting charge qubit
T. Yamamoto, Yu. A. Pashkin, O. Astafiev, Y. Nakamura, & J. S. Tsai
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ji
xj
xiij
ji
zj
ziij
xi
ii
i
ziiH
,,
………..
………..
….
….
….
….
Superconducting Adiabatic Quantum Device
NP-Hard Problems: i=ij
ji
zj
zi
i
zinpH
,
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L. B. Ioffe and M. V. Feigel'man, Phys. Rev. B 66, 224503 (2002)
B. Douçot., M. V. Feigel'man and L. B. Ioffe, Phys. Rev. Lett. 90, 107003 (2003)
Josephson Networks
R. Burioni, D. Cassi, I. Meccoli, M. Rasetti, S. Regina, P. Sodano, A. Vezzani, Europhys. Lett.52, 251, (2000). R. Burioni, D. Cassi, M. Rasetti, P. Sodano, A. Vezzani, J.Phys.B., 34, 4697, (2001).
Bose-Einstein Condensation in Inhomogeneous Josephson Junctions Arrays
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The theoretical model (II)
fingers
Backbone
Eigenvalue equation
)()( iEjH
t is a positive hopping parameter
Axy,x'y' is the adiacency matrix
Axy,x'y' = 1 If xy-x'y' is a link
0 otherwise
'''',
'', ˆˆ yxxyyxxy
yxxy aaAtH
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There are localized states even for the free Hamiltonian: the ground state decays exponentially along the fingers
G. Giusiano, F. P. Mancini, P. Sodano, A. Trombettoni, Int. J. Mod. Phys. B 18, 691 (2004)
Topology induces new phases at finite temperature for bosons on graphs
The experimental signature of the Bose Einstein condensation is given by the inhomogeneity of Josephson critical currents below the BEC critical temperature
Hamiltonian Solutions for a comb graph
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Chip design
b)
JJ
50m
The realized arrays have 4mx4m and 5mx5m junctions
8 different chips with different current density were fabricated and tested
The Backbone (BB) and its reference (RBB) have 72JJ
while finger (CF) and its reference (RCF) have 80JJ
a)
CF
BB
RCF
RBB 1mm
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Measurement
Junctions are connected in series and incresing the bias current the switch to the gap branch of each junction is well visible.
We can count the number of junctions in the array
V-
I+ I-
V+
V-
I+ I-
V+
0 5 10 15 20 25 30
7.5
8.0
8.5
9.0
Cur
rent
(A
)
Voltage (mV)
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Measurement
Junctions are connected in series and incresing the bias current the switch to the gap branch of each junction is well visible.
We can count the number of junctions in the array
V-
I+
I-
V+
V-
I+
I-
V+
0 10 20
7.5
8.0
8.5
Switches of single junctions (about 2.8mV)
Cur
rent
[A
]
Voltage [mV]
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Experimental Results on Backbone
Backbone shows a critical current higher than the reference one in particoular at T=1.2K
The gap voltage is the sum of the number of junctions (72JJs)
VI I
V
VI I
V
VI I
V
VI I
V50 100 150 200
0
5
10
15
20
25
30 T=4.2K
Cur
rent
(A
)
Voltage (mV)50 100 150 200
0
5
10
15
20
25
T = 1.2K
Cur
rent
(A
)
Voltage (mV)
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Experimental Results on Fingers
The CFA shows an increased disuniformity at T=1.2K
50 100 150 2000
5
10
15
CFA
RFA
T = 1.2K
Voltage (mV)C
urre
nt (A
)
V
I
I
V
V
I
I
V
50 100 150 2000
5
10
15
CFA
RFA
T=4.2K
Cur
rent
(A
)
Voltage (mV)
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Further test
JJ
50m
BBArray before cutting BBArray After cutting
reduced to a Linear array
50m
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Experimetal Results on Backbone Voltage is normalized to the number of junctions
50 100 150 2000
5
10
15
20
25
30
= RBA
BBA BBA after Cutting RBA
BBA after cut
BBA
T=4.2K
Cur
rent
(A
)
Voltage (mV)50 100 150 200
0
5
10
15
20
25
30
BBA after cut = RBA
BBA
BBA BBA after cutting RBA
T=1.2K
Cur
rent
(A
)Voltage (mV)
VI I
VRBA
VI I
V
BBA
VI I
V
BBA After cut
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Effect of noise and disuniformity
0 50 100 150 200 250
10
15
20
25
Ic = 23ASigma=4%
T=6kT=8k
T=4kT=2k
Cur
rent
(A
)
Voltage (mV)
Calculated current switchings
Effect of temperature
0 50 100 150 200
10
15
Tn=1K
Ic=20A
T=4.2K
Sigma 12%
Sigma 8%
Sigma 4%
Sigma 2%Sigma 1%
Cur
rent
(A
)
Voltage (mV)
Effect of a Ic disuniformity
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Fit of data 1.2K - 4.2K
The only free parameter to fit IV curve is the mean Ic:
the sigma is 4% in 4x4 BBACUT and 3% in BBACUT 5x5arrays
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Fit at all temperature BBA 4x4m2
0 50 100 150 200
5
10
15chip 6 BBA4x4
C
urre
nt (A
)
Voltage (mV)
BBA present a larger disuniformity at T < 5K
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0.2 0.4 0.6 0.80.4
0.6
0.8
1.0
1.2BBA
BBACUT
Tc = 9.0K
T=5.0K
Chip6 BBA4x4
N
orm
aliz
ed C
urre
nt
Normalized Temperature
Critical Current Temperature Behaviour
Critical Current is measured and normalized to the 64 JJ switching
on a total number of 72JJs
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Critical Current Temperature Behaviour
0.2 0.4 0.6 0.80.4
0.6
0.8
1.0
T=6.0k
Chip6 BBA5x5
Nor
naliz
ed C
urre
nt
Tc=8.9K
BBA
BBA cut
Normalized Temperature
Critical Current is measured and normalized to the 50 JJ switching
on a total number of 55 JJs
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Sample Junction area (μm2)
Number of
Junctions
Mean Ic
(μA)
Sigma(%Ic)
Ic BBA / Ic
BBACUT
BBA3 4x4 72 18 4 1.17
BBA4 5x5 72 24 3 1.13
BBA5 5x5 72 25 3 1.11
BBA6 5x5 55 26 3 1.04
BBA6a 4x4 72 17 4 1.07
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Summary
• We have observed a critical current enhancement along the backbone of a comb-shaped Josephson Junction array
• We have inferred from data its temperature dependence
• At the same time we observed along the finger a critical current reduction away from the backbone
• The whole effect is related to the inhomogenous topology (connectivity)