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

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

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

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)

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

≡ ∆ ∼

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= − −

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

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

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

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

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⎛ ⎞

+ + − − =⎜ ⎟⎝ ⎠

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

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

HIGH FREQUENCY CRYOGENIC FILTERING

-3dB @ 1 MHz

-3dB @ 2 GHz

RF CURRENT BIAS III: Plasma Resonance

kT/EJ

VD

C(nV)

1/Q

Q=ωpRC

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

PHASE DIAGRAM:EXP & THYIN GOOD

AGEEMENT

• Dark region corresponds towell-jumping

• All parameters in predictionmeasured experimentally!

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

−

−

↔

↔

↔

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

⎛ ⎞= − =⎜ ⎟⎜ ⎟

⎝ ⎠

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

DC vs. ACIB = f(I0,α)I0

0

1

0

1

IDC ↔ iRFVDC ↔ φRs ↔ α

ATTRACTORS

0

( ) ( ) ( )sin cost t tδ δ ω δ ω⊥= +

THY

1

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 = −⎜ ⎟

⎝ ⎠

ESCAPE TEMPERATURE

*RFT

*DCT

Ib ~ 0.1 I0*

RFT kω

=

Quantum Saturation at Different Temperature ! * 7.2

pDCT k

ω=

JOSEPHSON BIFURCATION AMPLIFIER

JBA: INPUT COUPLES TO I0- φ (irf,I0) - Pswitch (irf,I0)

- minimal backaction- no on-chip dissipation

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

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

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