lecture i: metastable states of the josephson junction · 2019. 12. 18. · lecture i: metastable...
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LECTURE I: METASTABLE STATES OF THE JOSEPHSON JUNCTION
IRFAN SIDDIQI
Department of Applied PhysicsYale University
7-27-2005
Boulder Summer School
R. VijayM. MetcalfeE. BoakninL. FrunzioC. RigettiM.H. Devoret
Daniel ProberRobert SchoelkopfSteve Girvin
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OUTLINELecture I: Metastable States of the Josephson Junction• Josephson junction dynamics• Non-linear Josephson inductance• DC / RF current biased junction
- metastable states- escape dynamics
• Bifurcation amplification
Lecture II: Bifurcation Readout for Superconducting Qubits• Quantum information and superconducting qubits• Quantronium qubit
- DC switching readout- RF bifurcation readout
• Coherence measurements• Information flow • Stark shift spectroscopy and relaxation
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GENERALIZED FLUX
B
( ) ( )mag surfacet B t dAΦ = ⋅∫
( ) ( )magperimeter
d tE t dl
dtΦ
⋅ = −∫
circuit element1 2
+ -
Egs.
( ) ( )2
1
tt E t dl dt
−∞′ ′Φ = − ⋅∫ ∫
12 ( )t
V t dt−∞
′ ′= ∫
R L C JosephsonTunnel Element
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JOSEPHSON TUNNEL JUNCTION
left
ll
ie ϕψ=Ψ right rrie ϕψ=Ψ
superconductor superconductor
barrier
V
IDC EFFECT AC EFFECT
0( ) ( )
2d dV
e dt dtϕ ϕϕ∆ ∆= =0 sin( )l rI I ϕ ϕ ϕ= ∆ = −
• Supercurrent with V=0• I0 is material dependent (nA-mA)
• Fixed V, I oscillates• Magnitude I0 , ν=2eV/h (483 MHz/µV)
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THE NON-LINEAR JOSEPHSON INDUCTORI0
= =
CJL
( )1( )I t tL
= Φ
( )0 0( ) sin[ / ]I t I t ϕ= Φ
( ) ( )3
0 0
0 0
...6
tI Itϕ ϕ
Φ⎛ ⎞= Φ − +⎜ ⎟
⎝ ⎠
I0
Gauge Invariant Phase DifferenceJosephson Inductance
0
0
( )TJBCS
RL BCSIϕ
π≡ =
∆ 0mod 2δ ϕ δ π
ϕΦ
≡ ∆ ∼
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THE EQUATION OF MOTION
0( ) sinJV dVI t C IR dt
δ= + +
222 00 0 0 02
( )
sin ( ) 0Jmass U
drag
d dC I I tdt R dt
δδ
ϕδ δϕ ϕ δ ϕ∂
∂
+ + − =
0
0p
J
IC
ωϕ
=
p JQ RCω=
0 0 0( ) cos ( )U I I tδ ϕ δ ϕ δ= − −
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DC CURRENT BIAS I: I-V Curve
0DCV ϕ δ=
T=216mKQ = 12
Al / AlOx / Al
0
0
: 0
: 0DC
DC
I I
I I
δ
δ
< =
> ≠
superconducting
dissipative
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DC CURRENT BIAS II: Metastability & Switching
I0=1.43 µA
( )0 1 0( )
, , exp2
pDC
kT
UI I TkT
ω
ωπ→
>>
∆⎛ ⎞Γ = −⎜ ⎟⎝ ⎠
( )3/ 2
0 00
0 50 K
2 2, 13
DCDC
IU I I Ie I
u ≈
⎡ ⎤ ⎛ ⎞∆ = ⋅ −⎢ ⎥ ⎜ ⎟
⎝ ⎠⎣ ⎦
1µA
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DC CURRENT BIAS III: Macroscopic Quantum Tunneling (MQT)
Martinis et al, PRB 35 (1987)
* :7.2
Up kTT T ek
ω −∆> = Γ ∝
* : constant7.2
pT Tk
ω< = Γ =
thermalactivation
MQT
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RF CURRENT BIAS I: SOFTENING POTENTIAL
( ) sin( )RFI t I tω=
max( ) sin( )t tδ δ ω γ= +
0( )V t ϕ δ=
• Frequency decreases w. drive amplitude
• For ω
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RF CURRENT BIAS II: TWO DYNAMICAL STATES
• If , bistability
• Dynamical states differ in oscillation amplitude & phase
32
p
Qω
ω∆ >
22 32 00 0 0 02 sin( ) 06J RF
d dC I I tdt R dt
ϕδ δ δϕ ϕ δ ϕ ω⎛ ⎞
+ + − − =⎜ ⎟⎝ ⎠
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EXPERIMENTAL SETUP
10mK
Challenges
• Fully coherent microwavemeasurements
• High speed data acquisition
• Precision circuit design(remember LJ ~ 300pH)
special capacitors
• Ultra-low noise wide-band electronics
new cryo filters
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Al
Si3N4
27.3 pF
Cu
JUNCTION + MICROWAVE CAPACITOR
1000 nm Cu
20 nm Cr
35 nm Cr15 nm Ti
Silicon Substrate
Al
1 mm
200 nm Si3N4
=Cu
Alεr=7.5
Al/Al203/AlJunction (1.17 µA)
METALLICUNDERLAYER
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HIGH FREQUENCY CRYOGENIC FILTERING
-3dB @ 1 MHz
-3dB @ 2 GHz
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RF CURRENT BIAS III: Plasma Resonance
kT/EJ
VD
C(nV)
1/Q
Q=ωpRC
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STRAY REACTANCES
Lstray Lbond = 2.8 nH
LJ = ϕ0 / I0 = 0.28 nH
C
2 20
1 14 2 strayp
C C Lf e Iπ
= +
Lstray = 0.003 nH
C = 27.3 pF
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PHASE DIAGRAM:EXP & THYIN GOOD
AGEEMENT
• Dark region corresponds towell-jumping
• All parameters in predictionmeasured experimentally!
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THE DRIVEN PENDULUM
ωω < ω0
0 2g
ω = πθ
Regime Drive Dynamical States
harmonic weak (θmax>> 2
V
δ θ
θ
↔
↔
10
1
0
J
p
I l
C gω ω
−
−
↔
↔
↔
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DYNAMICS OF DYNAMICAL SWITCHING
2 µs20 ns
• N = 600,000
• ∆φ = 74 deg
• HysteresisIB and IB’ correct ! ( )3/ 23/ 2 0
16 13 3B
I Iα α⎡ ⎤= −⎢ ⎥⎣ ⎦1 0.122
p
ωαω
⎛ ⎞= − =⎜ ⎟⎜ ⎟
⎝ ⎠
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SWITCHING HISTOGRAMS
• No latching• 40 ns rise + 20 ns settle• τm = 20 ns
Set by SNR
• Latching• 40 ns rise + 20 ns settle• τm = 300 ns
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DC vs. ACIB = f(I0,α)I0
0
1
0
1
IDC ↔ iRFVDC ↔ φRs ↔ α
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ATTRACTORS
0
( ) ( ) ( )sin cost t tδ δ ω δ ω⊥= +
THY
1
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1-D METAPOTENTIAL
( ) ( )3/ 22
30 0
0
64, , 1 118 3
10K
dyn RFRF
B
dyn
iU I i IIe
u
α α α⎛ ⎞⎛ ⎞⎡ ⎤ ⎜ ⎟∆ = − − ⎜ ⎟⎢ ⎥ ⎜ ⎟⎣ ⎦ ⎝ ⎠⎝ ⎠
≈
( )0
1/ 222
(2 ) 35
4
0MHz
13 3
a
RFa p
B
iRCI
ω αω
ω π
⎛ ⎞⎛ ⎞⎡ ⎤ ⎜ ⎟= − ⎜ ⎟⎢ ⎥ ⎜ ⎟⎦ ⎝ ⎠ ⎠≈
⎣ ⎝i
( )3/ 2 03/ 2 016 1 0 1
3.
3BI I Iα α⎡ ⎥⎦
≈⎤= −⎢⎣0 1
exp2
dyndyn a U
kTωπ→
⎛ ⎞∆Γ = −⎜ ⎟
⎝ ⎠
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ESCAPE TEMPERATURE
*RFT
*DCT
Ib ~ 0.1 I0*
RFT kω
=
Quantum Saturation at Different Temperature ! * 7.2
pDCT k
ω=
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JOSEPHSON BIFURCATION AMPLIFIER
JBA: INPUT COUPLES TO I0- φ (irf,I0) - Pswitch (irf,I0)
- minimal backaction- no on-chip dissipation
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AMPLIFICATION
DEVICE INPUT CIRCUIT
OPERATION
SQUIDFluxLoop
DC Switching
Quantronium
Single Cooper Pair Transistor
DC Switching
SQUID JBAFluxLoop
RFBifurcation
Quantronium+ JBA
Single Cooper Pair Transistor
RFBifurcation
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SQUID JBA
I0 = 1.16 µA
B=B0
0 1
EXPI0 = 1.17 µA
B=0
10
EXP
• fidelity = 80% @ 250mKfor ∆I0 = 10nA
• predict > 95% @ T=60mK (single shot)
10 MHzRep. Rate
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SUMMARY
• RF DRIVEN JOSEPHSON JUNCTION- non-linear plasma resonance- metastable states- escape dynamics
• NOVEL QUANTUM SATURATION
• BIFURCATION AMPLIFICATION- observe predicted sensitivity- SQUID JBA- QUBIT READOUT Next Lecture
LECTURE I: METASTABLE STATES OF THE JOSEPHSON JUNCTIONOUTLINEGENERALIZED FLUXJOSEPHSON TUNNEL JUNCTIONTHE NON-LINEAR JOSEPHSON INDUCTORTHE EQUATION OF MOTIONDC CURRENT BIAS I: I-V CurveDC CURRENT BIAS II: Metastability & SwitchingDC CURRENT BIAS III: Macroscopic Quantum Tunneling (MQT)RF CURRENT BIAS II: TWO DYNAMICAL STATESEXPERIMENTAL SETUPJUNCTION + MICROWAVE CAPACITORHIGH FREQUENCY CRYOGENIC FILTERINGRF CURRENT BIAS III: Plasma ResonanceSTRAY REACTANCESTHE DRIVEN PENDULUMDYNAMICS OF DYNAMICAL SWITCHINGSWITCHING HISTOGRAMSDC vs. ACATTRACTORS1-D METAPOTENTIALESCAPE TEMPERATUREJOSEPHSON BIFURCATION AMPLIFIERAMPLIFICATIONSQUID JBASUMMARY
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