MEASURING QUANTRONIUM QUBIT WITH CAVITY BIFURCATION

AMPLIFIER

Michael Metcalfe

E. Boaknin, V. Manucharyan, R. Vijay, C. Rigetti, I. Siddiqi, L. Frunzio and M. H. Devoret

(Qlab, Department of Applied Physics, Yale University)

Acknowledgements: D. Prober, R.J. Schoelkopf, S. Girvin

Monday evening seminarSeptember 11th 2006

CONTENTS1. Introduction: - Quantum computing

- Implementations- Quantum circuits- Quantronium

2. Cavity bifurcation amplifier (CBA) :- Non-linear readout - Implementations- Bistability

3. Quantronium with CBA:- Spectroscopy/Gate modulations- Rabi/Ramsey oscillations

4. Outlook: - Coupled qubits (See Chad’s talk)- Multiplexed C.B.A’s

QUANTUM COMPUTATIONQuantum computation:

Unitary evolution of initially prepared n-qubit state and its subsequent measurement

Qubit:Just a 2-state system (effective spin – ½)

0 1ϕ α β= +

0

1

ener

gy 2

01

3

Isolate two energy levels

QUANTUM PARALLELISM

N-qubit state:2 1

0

N

kk

kCψ−

=

= ∑

Unitary operation,U:

2 1

0

N

kk

U C kUψ−

=

= ∑

2N operations in parallel!!

0 0 00 0 00 0 0 0 0 0

0 0 00 0 00 0 0 0 0 1

0 0 00 0 00 0 0 0 1 0

k

1 1 11 1 11 1 1 1 1 0

1 1 11 1 11 1 1 1 1 1

QUBIT REALIZATIONSTrapped ions NMR

chloroform molecule

Long qubit coherenceLow environmental couplingScalability difficult

semiconductor dots superconducting circuits

Shorter qubit coherenceLarge environmental

couplingScalability easierElectrical access

QUANTUM CIRCUITS

L

φ 2 2

2 2LH q

Cφ= ++q

-qC

[ ], iqφ = h

BE k Tω∆ = h

Ene

rgy

1GHzω

10mKT

Microfabricationand cryogenics

Equal level spacing Need non-linear element

NON-LINEAR ELEMENT: JOSEPHSON JUNCTION

Model:

SCInsulator

SC

Josephson current: Circuit symbol:

( ) ( )02sin Jei t I t⎡ ⎤= − Φ⎢ ⎥⎣ ⎦h

ϕ∆SEM:Inductance:

( )0

12 cos

L i eI ϕ= =∂ ∆∂Φ

h

3.5µmNon-Linear Inductor!

COOPER PAIR BOX

0

1

2

E/E

CP

1 2 3

LJ,EJ

Vg

Cg

~ ( )( )

222CP

g JC Ce

E =+

Island( ) ( ) ( )2

cosg CP g JH N E N N E θ= − −)) )

2g g

g

C VN

e= , N iθ⎡ ⎤ =⎣ ⎦

) )

0Ng

SPLIT COOPER PAIR BOX0

1

CgVgδ

SEM image of Quantronium

Gate

Readout

SET4µm

( ) ( ) ( )21 cosg CP g JH N E N E

iδ θ δ

θ∂⎛ ⎞ ⎡ ⎤= − −⎜ ⎟ ⎣ ⎦∂⎝ ⎠

))

( ) ( )CPB z z x g xH h h Nδ σ σ= +CPJ EE <<

Artificial 2-level atom

ENERGY LEVELSEigenstates, Ek: Mathieu Functions

( )δ

δϕ

δ∂

∂=

,1),(i0

kgk

g

NEN

Average loop current:

( )

Ecp = 17GHz, EJ0 = 15GHz

With inductance:

( ) 1

2

2

20

1

0

,11,−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂=

δδ

ϕδϕδ gk

g

NEINL

Sweet spot

Sweet spot:

0=∂∂=

∂∂

g

kk

NEE

δ

Ng=1/2

NON-LINEAR READOUT

Vd

Vdrive

R

C

L

Non-linear LC oscillator

w

linear LC oscillator

|Vou

t| V

LJ

Vd

NON-LINEAR READOUTNon-linear LC oscillator

Vd

R

C

L

|Vou

t|

Vd

stable

unstable

stable

LJ

Mapping

0

1

Quantronium Qubit0

1

E/E

c

Energy levels

Cg

Vg

g gC V e

0

1

2

IMPLEMENTATIONSQuantronium qubit with Josephson bifurcation amplifier (JBA)

∼

2µm

(I. Siddiqi et. al, PRB 2006)

Quantronium qubit with cavity bifurcation amplifier (CBA)

Qubitin

out

Resonator

0 1

Qubit

NEW

SAMPLEresonator

1) Precisely control non-linearity2) Engineer freq, Q3) No dissipation on-chipcapacitors

3mm

10mm

100µm

Nb

10µm

Al

MICROWAVE RESPONSEBistable states

1.8 GHz 9.8 GHz

OBSERVATION OF BISTABILITY

Re[Vout]

Im[V

out]

τm-2µs

2µs 3µs 10ns

countscounts

WHY: MULTIPLEXED QUBITS

Coupling line Qubit in resonator

Multiplexed readout and gate lines

C.B.A. WITH QUANTRONIUMSEM image of Quantronium in CBA

10µm

CPW ground

Island + 2 junctions

Readout junction

CPW ground

GATE MODULATIONS

Readout in linear regime

Move to bistable regime:

More gainFlux (rads)

Fit is with Ecp = 17.0GHz and EJ0 = 15.1GHz

Gat

e ch

arge

(Coo

per p

airs

)

Flux (rads)

SPECTROSCOPY

Fit: Ecp = 17.0GHzEJ0 = 15.1GHz

ReadoutSpectroscopy pulse

Varying frequency

RABI OSCILLATIONSReadout Y-rotation

0

1qubit pulse

τA

,τ

A, (Volts)

A,

A2 (dBm)

BEST RABI OSCILLATIONS

RELAXATION TIME, T1

π pulse Readout

0

1τω

T1=1.4µs↔1.8µs

τω (µs)

RAMSEY FRINGESπ/2 pulse τ Readoutπ/2 pulse

0

1

200ns ≤ T2 ≤ 800ns4 4

01 22 2.10 7.10Q Tϕ πν= = ↔

MEASURED CONTRASTπ pulse Readout

V

0

1

Readout relaxation Reduced contrast

TOMOGRAPHYPreparation

pulseReadout

Tom. pulse

h

h φ

φ

0 0 12

+ 0 12− 01

Initial states

NOW: SLOTLINE NON-LINEAR RESONATOR

Optical image of device

200µm

Gate

Slotline

SEM image of qubit

4µm

Advantages: 1. Completely fabricated using e-beam lithography2. Gate and readout lines separated3. Excite ± mode in slotline for readout

NEAR FUTURE: COUPLED QUBITS

QUBIT 2

Optical image of coupled qubit with slotline readoutSee Chad’s MES upcoming…..

Ug1

Ug2

QUBIT 1

Cc

QUBIT 2

QUBIT 1

Ug2

Ug1

Cc

LONGER TERM: MANY COUPLED MULTIPLEXED QUIBTS

3mm10mm

Nb

Multiplexed resonators

preliminary data:

amplitude response

CONCLUSION

• Developed fast, non-dissipative RF readout• Single shot qubit readout feasible• Scalable architecture• Coupled qubit experiments in progress…

PREDICTED CONTRAST

0 0.4nAI∆ =0 0.3µAI =

2GHz12mK

Freq ∆ω

2GHz 1.4MHz 0.4nA 0.5nA

10GHz 20MHz 6.4nA 5nA

0measI∆ 0

theoryI∆

V

Sw

itchi

ng p

roba

bilit

y

68%

V (volts)

Single shot possible

RAMSEY TOMOGRAPHYMap of state during Ramsey experiment (c.f. Martinis, PRL)

Slides after this are additional

RANDOM EQUATIONSJosephson Junction:

( ) ( )

( )

( )

spotSweet 0field transverseE

fieldZeeman

22E

where

1

cos1

CP

21

z

2

⇒=↔

↔

−==

+−=<<

∂∂=

−⎟⎠⎞

⎜⎝⎛ −

∂∂=

cont

J

gJCPcont

J

xcontzzCPB

CPJ

Jgcg

X

E

NEEXE

XEHEE

iN

ENi

ENH

σσ

θ

θθ

))

Cooper Pair box: Loop current/Inductances:

( )

( ) [ ]

( )

( ) ( )2

2

20

1

0

0k

00

21

,11,L

:inductancewith

,1),(i

:iscurrent loop average theHence

ˆ1ˆ,ˆ1,

2

ˆˆ K

island onto tunnelling pairscooper ofnumber where

2,

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

=⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

∂∂=

∂∂=

∂∂=−=

⎟⎠⎞⎜

⎝⎛ +

=

−=

δδ

ϕδϕδ

δδ

ϕδ

δϕϕδ

δ

gk

kg

NEINg

NgEN

HHKi

NgI

NN

dtKdeNgI

)

))

))

[ ]

( )J

Jr

Ce

EeQN

iN

eNQ

22E

where

cos2E H

:JJ ofHamilton ,

:JJ across diff phase SC

with associatedFlux 2

:JJ through tunnelingCharge

2

CJ

2

CJ

0

=

−⎟⎠⎞⎜

⎝⎛ −=

==Φ

=

θ

θθϕ

NON-LINEAR OSCILLATOR

LJ.I0V

R C L

R cost d NqL q q V t VC

ω+ + = +&& &

Linear resonance

( )20

1R 1 cos1 /

t J d NqL q q L q V t VC q I

ω⎛ ⎞⎜ ⎟+ + + − = +⎜ ⎟−⎝ ⎠

&& & &&&

V

w

Non-Linear resonance

Readout scheme

0

B-Field

Biaspoint

I0’’

I0’

I0’’ < I0’

Vdrive

|Vou

t|

V

Previous workCQEDQuantronium qubit with JBA

I. Siddiqi et. al, PRB 2006 A. Wallraff et. al, Nature 2004

Flux qubit with linear resonator and JBA

A. Lupascu et. al cond-mat/0601634

Experimental observation of two oscillator states

-60

-40

-20

0

20

40

2.22.01.81.61.41.21.0

Voltage (V)

Pha

se (d

egre

es)

V

time

Microwave responseBistable states

1.8 GHz 9.8 GHz

Bistability

-1

0

1

-1 0 1

counts

-1

0

1

-1 0 1

counts

-1

0

1

-1 0 1

counts

-1

0

1

-1 0 1

counts

|Vou

t|

Vdrive

AB

C D

A

DRe[Vout]

Im[V

out]

B

C

Readout scheme

0 1

Biaspoint

STATE 0

STATE 1

0

OS

CIL

LATI

ON

A

MP

LITU

DE

Power in

JBA

Ti

Al capacitor pads

Bonding pads

SiCrCuCr

• Complicated fabrication• Same physics

4.2K

260mK

300KIN OUT

2w2w

Arb

Source Source

Cold RussianAmplifier

MitiqAmplifier

Mini-CircuitAmplifier

Stepattenuator

Ecosorbfilter

Chocolatefilter

TIME RESOLVEDMEASURMENT

ESCAPE RATES

For : aω Γ

exp2

a

B esc

Uk T

ωγπ

⎛ ⎞∆= −⎜ ⎟⎝ ⎠

Thermally activated escape:

Where:

1/ 222 1 /3 3a bω β β= ΓΩ − 3

2/ 264 1 /

9 3 Jt

Jb

LL

UQ

E β β⎛ ⎞⎜

Ω −⎟⎠

∆⎝

=

( )2/3ln( / 2 ) /a bvsω πγ β βPlot: Tesc

ESCAPE RATESMETHOD

400

300

200

100

0

coun

ts

1.431.421.411.401.39

V ( Volts )

V

time

time

phase not switched

switched

( )1( )

( )v V

v V V

P vdVV Ln

V dt P vγ ≥

≥ +∆

⎡ ⎤⎢ ⎥= ⎢ ⎥∆⎢ ⎥⎣ ⎦

∑∑

ESCAPE RATES

6

5

4

3

2

1

0

[ Ln

( ωa /

2 π

γ )

]2/3

2.22.12.01.91.8

V 2 ( Volts2

)

Intercept: Vb

Slope: Tesc

U0/KBT = 420

U0 = 220K

Tesc = 520mK

Tphysical= 260mK

Useful tool

2D-HISTOGRAMS

τm-2µsReal (arb)

Imag

inar

y (a

rb)

Real (arb)

Imag

inar

y (a

rb)

-50

0

50

-50 0 50

latching

-50

0

50

-50 0 50

No latching

2µs 3µs 10ns 2µs 2µs 11µs 10ns

τm-10µs

CAVITY BIFURCATION AMPLIFIER

( )20

1R 1 cos1 /

t eff J d Neff

qL q q L q V t VC q I

ω⎛ ⎞⎜ ⎟+ + + − = +⎜ ⎟−⎝ ⎠

&& & &&&

Veff

Reff

Ceff

Leff

LJ.I0

Parameters: Z0 = 50Ω;

ω0 ~ (1.8Ghz)2π;

COUT ~ 50fF;

Leff ~ 7nH;

Ceff ~ 1.5pF;

Reff ~ 0.03Ω;

Q ~ 2000

THE JOSEPHSON TUNNEL JUNCTION:AN ATOM-LIKE CIRCUIT ELEMENT

TO WHICH YOU CAN ATTACHWIRES ...

S I STUNNEL

JUNCTIONEJ-N(2e) +N(2e)

Cj

[ , ] iNθ =))θ

NON-LINEAR INDUCTOR

)sin( ϕ∆= cs IIZero voltage supercurrent:

Model:SC

Insulator

SC

JOSEPHSON JUNCTION

C R

tdtvt

J ′′=Φ ∫∞−

)(e2

0h=Φ

0

)(Φ

Φ=∆ tJϕ

h

eVdt

d 2)( =∆ϕApplied voltage: → alternating current

Free energy stored in junction: dtVIW s∫=

eIE C

J2h=constEF J +∆−= )cos( ϕ

02eILJ

h=

Non-Linear Inductor!

ENERGY LEVELS OF AN ISOLATED JUNCTION

( )2/ 2 cos( )JCPH E N q e E θ= − −

))HAMILTONIAN: Z >> RQ = h/4e2

CHARGINGENERGY

COUPLING ENERGY, UJE/ECP

0

1

2

3

qe 2e

UJ

0

BLOCH WAVEFUNCTIONS

-π πθ

E

E-BEAM

WRITE PATTERN WITH SEM

3.5µmOPTICAL IMAGE

OF RESIST

SHADOW-MASK EVAPORATION

PLEXIGLAS (PMMA)

COPOLYMER(MMA 8.5)

SILICON

2” SI WAFER

SPIN ON RESIST

2” SI WAFER

SPIN ON RESIST

DICE WAFER INTO SMALL CHIPS

SHADOW-MASK EVAPORATION

PMMA

MMA

Bridge

Si

Undercut

Develop in MIBK:IPA, 1:3, for ~40s and rinse in IPA

2nd Al evaporation

CROSS SECTION:

Al

Lift off resist with unwanted

Al with Acetone (60-90ºC)

3.5µm

SEM IMAGE OF AL/AL2O3/AL JUNCTION

Oxidation: 15%O2, 85% Ar. 1-10T, 5-20mins

Al

First angle Al

NON-LINEAR OSCILLATOR

LJ.I0V

R C L

R cost d NqL q q V t VC

ω+ + = +&& &

Linear resonance

( )20

1R 1 cos1 /

t J d NqL q q L q V t VC q I

ω⎛ ⎞⎜ ⎟+ + + − = +⎜ ⎟−⎝ ⎠

&& & &&&

V

w

Non-Linear resonance