quantum simulation with superconducting...
TRANSCRIPT
Quantum simulation with
superconducting qubits
Hans MooijKavli Institute of Nanoscience
Delft University of Technology
KITP Santa Barbara - Workshop on Quantum Information Science
November 4, 2009
1. introduction superconducting qubits
2. flux qubits
3. quantum simulation with quantum vortices and flux qubits
superconducting order parameter
2o
h
eΦ = loop
2oV
t
ϕπ
Φ ∂=∂
2 2ioi
nϕ π πΦ= +Φ∑
ie ϕΨΨ =
Josephson junction
CIo
100 nm
2242
ooJ
C
E I
eEC
πΦ=
≡
2
(1 cos )2
sin
JC
o
QU E Ee
dQI Idt
ϕ
ϕ
= + −
= +
U
ϕϕϕϕ
circuit with Josephson junctions
ϕ1..ϕn phase differences+constraints
U(ϕi.. ϕn): potential energy
Ci(dϕi/dt)2 terms: kinetic energy
set of conjugate variables: Lagrangian, Hamiltonian
quantization
set of islands with discrete numbers of Cooper pairs
junctions provide off-diagonal coupling of charge states
charge qubit
charge qubit
0.5ΦΦΦΦoe
fluxoid qubit
flux qubit
oscillation qubit
phase qubittransmon
δE
0.5 ng
oscEδ ω=ћδE
0.5 f
ng=CVg/2e f=ΦΦΦΦ/ΦΦΦΦ0
2( ) cosg JCH E n n E ϕ= − −
2ˆ ˆ( ) cosˆ g JCH E n n E ϕ= − −
ˆ,n̂ iϕ
=
(classical Hamiltonian)
0 1 ng→
↑↑↑↑E/EC
0
1
EJ
0
1
EC/EJ>>1
around ng=0.5
other levels extremely high
ˆ { ( 0.5) }/2zx gJ CH E E nσ σ=− + −
quantum Cooper pair box: charge qubit
-0.2 0.2 0.4 0.6 0.8 1 1.2
0.2
0.4
0.6
0.8
1
f
Uind/EL
n=1 n=0
0 1f
E/EL
0
1
∆0
1
around f=0.5
ˆ { ( 0.5) }/2zx LH E fσ σ=− ∆ + −
exp JJ C
C
Ea E E b
E
∆ = −
L fΦΦΦΦo EJEC
∆
E
ϕϕϕϕ
flux qubit EJ>>EC
Phase Qubitplasma oscillation of a current-biased single Josephson junction
2 4 6
-15
-10
-5
5
0.5
0.95
I/Io=0E/EJ
ϕ/2π
2.4 2.5 2.6 2.7
-6.48
-6.46
-6.44
3
2
1
Martinis group Santa Barbara
Nature 459, 546 (2009)
Nature 461, 504 (2009)
Koch et al. PRA 76, 042319 (2007), Schreier et al., PRB 77, 180502 (2008)
Yale group: Schoelkopf, Girvin
transmon: Cooper pair box with strongly reduced EC
coupled to high Q harmonic oscillator
external capacitive shunt
EJmax/h= 18 GHz
ECeff/h= 1.4 Ghz
νpl~ 6 GHz
increasing EJ leads to
- gradually decreasing anharmonicity
- exponentially decreasing dependence of δE on ng
circuit quantum electrodynamics
Nature 460, 240 (2009)
sources of decoherence
driving circuits: gate, bias flux, microwave lines, measurement
engineering design, calculation possible
microscopic defects
1/f noise due to many fluctuators
parasitic two-level systems with same energy splitting
smaller qubits, fewer defects
dephasing times > 1 µs reached in most qubit types
at optimal conditions
level separation 3-15 GHz, operation time 5-50 ns
1 µm
Φ~Φo/2
↓↓↓↓ ↑↑↑↑flux bias Φo/2
currents ± Ip
0
-1
0
1
0.5
↑↑↑↑Icirc
↑↑↑↑E
Φ/Φo→
∆
+Ip
-Ip
0
ground stateexcited state
( )z xH εσ σ= + ∆½( / 0.5)2o o pIε = Φ Φ − Φ
Mooij et al. Science 285, 1036 (1999), Orlando et al. PRB (1999)Van der Wal et al. Science 290 1140 (2000)
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
-1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
ϕϕϕϕ1111/π/π/π/π
ϕϕϕϕ2222/π/π/π/π
ϕϕϕϕ1111/π/π/π/π
α=1.α=1.α=1.α=1. α=0.75α=0.75α=0.75α=0.75αααα
exp JJ C
C
Ea E E b
Eα
∆ = −
αααα
f
Ip= 428 nA∆= 1 GHz
Ip= 360 nA∆= 4.6 GHz
Ip=281 nA∆= 8 GHz
Pswitch
α α α α =0.71 αααα=0.64 α α α α =0.60
20 mΦo
F.G. Paauw et al. PRL 102, 090501 (2009)
( )z xH εσ σ= + ∆½( / 0.5)2o o pIε = Φ Φ − Φ
tunable ∆∆∆∆
controlled-NOT gate
Plantenberg et al. Nature 447,836 (2007)
|11⟩⟩⟩⟩
|01⟩⟩⟩⟩
|10⟩⟩⟩⟩
|00⟩⟩⟩⟩
ππππ
2o
ko
LIπ
Φ=
1,2 1,5 1,8
0
5
10
Vsw (V)
dPH
/dV
sw (
V -
1 )
0 pi/2 pi
A. Lupascu et al. Nature Physics 3, 119 (2007)
oscillator state h ⇔⇔⇔⇔qubit ground stateoscillator state l ⇔⇔⇔⇔ qubit excited state
P(h)P(hh)P(ll)
∆t1 (ns)2 64
∆∆∆∆t2 set to give 2ππππ pulse
P(%)
100
0
50 µµµµm
Pieter de Groot - 2-qubit sample with bifurcative readout
cross-talk between readout systems < 0.1 %
tunable-αααα qubit coupled to low Q oscillator
vacuum Rabi oscillation around symmetry point
Arkady Federov
vortex in 2D Josephson junction array: particle moving in 2D potential EJmass determined by junction capacitance 1/EC
quantum vortices
in 2D or quasi-1D array of Josephson junctions
Bose Einstein condensationPRL 69, 2971 (1992)
quantum interference around charge
PRL 71, 2311 (1993)Anderson localizationPRL 77, 4257 (1996)
Mott insulatorPRL 76, 4947 (1996)
Berezinskii, Kosterlitz,Thouless transition
PRB 35, 7291 (1987)
superconductor-insulator transition EJ/EC
PRL 63, 326 (1989)
arXiv:cond-mat/0108266 (2001)
New J of Phys 7, 181 (2005)
chain of flux qubits: excitation at one end, readout at the other end
1D coupled chains of flux qubitsFloor Paauw, Delft
alfa-qubits
coupling
αααα loop
1i iz xσ σ +
4 1
23
4 degenerate states
1 2
2 degenerate states
trapped vortex in junction array
A B
CD
Iv
Ih
4 degenerate states
A B
Iv
2 degenerate states
ϕϕϕϕi
17 junctions
(12-1) loop equations
6 phase variables
EJ>>EC
if vertical symmetry is assumed: 3 variables
6 charge variables
EC>>EJ
trapped fluxoid: two positions
central junction q times larger
0.25 0.5 0.75 1 1.25 1.5 1.75
0.1
0.2
0.3
0.4
potential energy in phase landscape, symmetric solutionphase difference across central junction imposed, rest optimized
q=1.3
ϕcentral/π
potential energy/EJ
0.25 0.5 0.75 1 1.25 1.5 1.75
q=1.5
1.4
1.3
1.2
1.1
1.0
(U-Umin)/EJ
0.2
0.0
0.4
1.00 0.5 1.5
1.1 1.2 1.3 1.4 1.5 1.61.0 1.2 1.4 1.6
q
Ubar/EJ
0.0
0.2
0.4
0.25 0.5 0.75 1 1.25 1.5 1.75
-0.1
0
0.1
0.2
0.3
0.4
flux tilt
I
quantum calculation (Jos Thijssen et al.)
q=1.3, m=5, symmetric case (3 degrees of freedom)
classical : crosses
↑↑↑↑energy current tilt →→→→
9 degrees of freedom
square array with one trapped fluxoid
4 equivalent fluxoid positions
0.4 0.6 0.8 1.2 1.4 1.6
0.002
0.004
0.006
0.008
0.4 0.6 0.8 1.2 1.4 1.6
0.01
0.02
0.03
0.04
0.05
0.06
0.4 0.6 0.8 1.2 1.4 1.6
0.05
0.1
0.15
0.2
q=0. 6 q=1.0 q=1.4
barriers
center junctions q times larger
than other junctions
1.1 1.2 1.3 1.4 1.5qv=qh=q
0.05
0.1
0.15
0.2
0.25
0.3
Ubar êEJ
barrier as a function of central junction strength
1
3 2
4Ix
Iy
Ix >0: 1 and 4 lower energy
Iy >0: 1 and 2 lower energy
0.4 0.6 0.8 1.2 1.4 1.6
0.05
0.1
0.15
quantum calculation Jos Thijssen, 9 degrees of freedom
energy
tilt induced by current
classical
IV I
IIIII
Vg
W.J. Elion, J.J. Wachters, L.L. Sohn, J.E. Mooij, PRL 71, 2311 (1993)
Quantum interference of vortices around charge, Aharonov-Casher
-3 -2 -1 1 2 3
-4
-2
2
4
Ix
Ix=0, Iy=0 Ix=2, Iy=0
Iy
Ix14
23
1
2
3
4
00
00
e
e
e
e
∆ ∆∆ ∆
∆ ∆∆ ∆
E/∆
E/∆ eigenvector
2 (-1,1,-1,1)
0 (-1,0,1,0)
0 (0,-1,0,1)
-2 (1,1,1,1)
E/∆ eigenvector
3.2 (-1,4.2,-4.2,1)
1.2 (1,-4.2,-4.2,1)
-1.2 (-4.2,-1,1,4.2)
-3.2 (4.2,1,1,4.2)
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1.29
1.30
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.60
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.27 .32 .95 .57
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1.30
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1111
22223333
4444 1111
22223333
4444
a bclassical
nearest neighbor interaction
1.0 4.9 3.3 5.6
4.9 1.0 5.6 3.3
2.1 8.9 1.0 4.9
8.9 2.1 4.9 1.0
− − + + − − + + + + − − + + − −
a=1
b=1
2 3 4
2
3
4
+ +8.9+8.9+8.9+8.9 −−−−1.01.01.01.0
−−−−4.94.94.94.9+2.1+2.1+2.1+2.1
+ −−−−1.01.01.01.0 +5.6+5.6+5.6+5.6
+3.2+3.2+3.2+3.2−−−−4.94.94.94.9
++++
++++
++++
++++
++++
++++
++++
++++
++++
1D chain
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
2D array
++++
++++
++++
++++
++++
++++
++++
++++
++++
1D chain
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
++++
IV I
IIIII
Iy
Ix
4 degenerate statesVg
Berry phase
H(p1,p2)
make closed loop through parameter space
wave function picks up geometric phase and dynamical phase
retrace loop backwards to eliminate dynamical phase, but do something smart to double geometric phase
e-change on island charge?