Depts. of Applied Physics & PhysicsYale University

expt.K. LehnertL. Spietz

D. SchusterB. Turek

Chalmers UniversityK.Bladh

D. GunnarssonP. Delsing

The Cooper-pair Box as a Quantum Spectrum Analyzer

Rob Schoelkopf

The David and LucilePackard FoundationFunding:

And discussions w/:M. Devoret & J. Martinis

theoryA. ClerkS. GirvinD. Stone

Yale

Cooper-pair Box Coupled to an SET

Box

SET

Vg Vge

Cg Cc Cge

Vds

Box SET Electrometer

Superconducting tunnel junction

Qubit Quantum state readout

Quantum spectrum analyzer

Cooper-pair Box SET Transistor

Nonequilibrium noise source

or

Cooper-pair Box

2(2 )4 4 K

2c

eE

C

20.5 K

4 JJ e R

E

Vg

Vg

n

1 fFg jC C C

14

2 2g

elg

c

CE

e

VE

2ˆ ˆ

2ˆz x

e JlHE E

1

0

z

z

n

n

(e.g. Bouchiat et al., 98)

effB

effB

2ggC V

e0.5

0

1

1

01

0E

2ggC V

e

2JE

Cooper-pair Box as Quasi-spin 1/2

14

2 2g

elg

c

CE

e

VE

1 / 2Zn

Measure charge2

ˆ ˆ ˆ2z x

e JlHE E

2elE

zExcited state

Ground state

2JE

2JE

x

z

x

z

x

n

effB

2elE

a b ca b c

a b c

Continuous Measurement of a Single Spin

Measured continuously by SET

Theory: Cooper-pair box ground state

2e

1e

n

0

0.5

0 1 0.5

Measurement must cause additional dephasinguncertainty principle

Measurement may also mix states, drive transitions from ground state

2ggC V

e

1

1

0 1

0E

2ggC V

e

n

Cooper-Pair Resonance Spectroscopy

2ggC V

e

Vapp

0 1 0.5

=38 GHz

Cg

Vapp=Vg+Vacsint

1

0

38 GHz

2-photonPeak

Peak location

2ggC V

e

0

0-1-2 1 2

0.29

0.25

0 1

E 0 / 2eff

JE

2ggC V

e

0effJE

B

0cos( / )effJ JE E

“SQUID box” to vary EJ

Fit parameters:

1.79 0.004

0.626 0.008C

J

E K

E K

Determination of Box Hamiltonian

Vapp

32 GHz

35 GHz

38 GHz

01

Effects of Voltage Noise on Pseudo-Spin

sinelB E

coselB E

slow fluctuations of B dephasing

resonant fluctuations of B mixing

2

201

1sin

boxmix Vmix

eS

T

01 effB

221

0 cosboxV

eS

T

2el boxeE V effB

2elE

z

x

2JE

01

sin JE

01VS 01VS

0T 0T

0101

absorption emission

Emission and Absorption due to Environment

g

e

01

Box envR /

2 R

1env

envV kT

Se

kT

VS

0

Cg Box

Spontaneous Emission into Environment

2

2

01

1

21sin

envV

eS

T

12%gC

C

01 01( ) 2 (50 )envVS

1 0.1 1 sT

50 envR

01kT 01( ) 0envVS

estimate:

Excited-statelifetime, T1

Vg

Pea

k he

ight

(e)

0 time 10 s

Excited-state Lifetime Measurement of Box

1 1.3 sT 2

ggC V

e

n

0 1 0.5

1

0

0.3e

follow peak height after shift

with continuous measurement

910( ) 5 10 pairs /

boxnS Hz (@ 76 GHz)

Relaxation by Electrometer?

Pea

k H

eigh

t (e

)

Electrometer Operating Point (Vge)

0

0.6 1 1.3 sT

1Peak Height T

0.3

Cg

2e SET

Vg

Cc

e-

Vge

1 2

1 Rabi

TT

Peaks saturate when

Vds

Charging Diagram of SET Electrometer

CgVge /e

eVds

Vge

Electrometer SET:

R = 150 kEC ~ ~ 2.4 K

4

2Ec

0

1e

Electrometer operating pt.on “DJQP” feature

Quantum Shot Noise of DJQP* Process

Excitation Relaxation

Sharp thresholds due to opening & closing of transport channels

Γ

*Double Josephson-quasiparticle cycle: (A. Clerk et al. cond-mat/0203338)

0

log VS

2 2small: ( )

4J

VJ

S

Predicted Effects of DJQP on Box Charge

Qubit acts like a spectrum analyzer of the SET quantum noise!

(A. Clerk et al. cond-mat/0203338)

(see also Aguado & Kouwenhoven, 2000 for double dot)

01

2ggC V

e

n

1

0

01 0

Average box charge

Log

[SV(

)]

SET noise spectrum

on resonanceoff resonance

• Inelastic lifetime is long > 1 s : (and electrometer affects T1 !)

• Cooper-pair box as a “quantum spectrum analyzer”

• RF-SET a good probe of the charge states of box

• Spectroscopic determination of Hamiltonian of box

Conclusions

1 1 01 500,000Q T

( )VS ( )VS

Measures all Noise Classical (symmetric) Quantum (asymmetric)

0

( )VS

p

Gap rise (Vds= 1200 V)

JQP (Vds= 800 V)

Supercurrent (Vds=0)

dseV

2ggC V

e

gg eC V

e

Coulomb Staircase vs. Electrometer Bias

T=20 mK

n

Back-action increases with

electrometer bias

DJQP (Vds= 400 V)

Cooper-pair Staircase vs. Electrometer Bias

Theory: Cooper-pair box ground state

2e

1e

n

0

0.5

0 1 0.5 2ggC V

e

1

sweep gate @ 2e per 100 s

Data:Vds= 350 VVds= 275 VVds= 250 V

Cooper-pair Staircase vs. Josephson CouplingTheory: Cooper-pair box w/ max EJ

2e

1e

n

0

0.5

0 1 0.5 2ggC V

e

1

Data: maximum EJ

minimum EJ

Pea

k he

ight

0/ 0-1-2 1 2

B

0cos( / )effJ JE E

“SQUID box” to vary EJ

Charge States Coupled by EJ

Vapp

2elE

z

2JE

x

Peak location 2

ggC V

e32 GHz

35 GHz

38 GHz

ne=-1 ne=0 ne=1

Ec

E

ggC V

e0-0.5 0.5

/q e

Single-electron Box: Coulomb Staircase

ee

Coulomb Staircase

Thermally broadened

kT/Ec

-1 1

500 mK200 mK 50 mK0

-1

1

First demonstrated by Lafarge et al, ’91

(CEA Saclay)

B 1 T

Ec/4 1.6 KCE

Temperature Dependence in Normal State

2

*2

1

T

0.235 2HWHM

0

2ggC V

e

n

Peak width

Decoherence Time of Box

*2 500 psT

2Power (arb) R

*2 1 nsT

0.265

74 GHz 78 GHz0.2

01 /E h

DJQP Noise, Off-resonance

0 0.5 1 1.5 20

2

-1 -0.5 0 0.5 110

-15

10-10

Her

tz-1

Γ > Γ Population inversion in the qubit.

Ω / ECSA

vg. Q

ubit

Cha

rge

•Move away from the center of the resonance by increasing VDS…

NB

Pea

k he

ight

0/

0-1-2 1 2

0 1

E 0 / 2eff

JE

2ggC V

e

0effJE

B

0cos( / )effJ JE E

“SQUID box” to vary EJ

Charge States Coupled by EJ

Vapp

01

cos( )2

elEt

z

2JE

x

Effects of Voltage Noise on Qubit

xelE

effB

sinelB E

coselB E

2JE

slow fluct. of Bdephasing

resonant fluct. of B mixing

2

201

1sin

boxmix Vmix

eS

T

01 effB

2

210 cos

boxV

eS

T

z 2el boxeE V

2ggC V

e

n

Box State Depends on Electrometer Bias

Vds (V)

250290

1200

0

420470760

Cg

2e SET

SET Box Environment

Spontaneous Emission

Vg

50 envR

CcVds

E

Relaxatione

g

Backaction of SET on Box

boxV

SETV

t

SET

eC

Cm

Cg

Zenv

mSEox

oxb T

b

CV

CV

SETV

effB

2elE

z

2JE

x

Who’s measuring whom?

Measured continuously by SET

Theory: Cooper-pair box ground state

2e1e

m r

g

e

n

0

1

Can Electrical Circuits be ‘Quantum?’

Cooper-pair boxY. Nakamura et al, Nature 1999

New Challenges:

•Understand and minimize decoherence

•Develop efficient quantum readout

New Opportunities:

•Create artificial atoms

•Quantum computation

Macroscopic Quantum Coherence:

( , )E f Q ( , )?QH f

Quantum Circuits for Quantum Computing

Classical bit

values 0 or 1

Information as state of a two-level quantum system

orvalues ,0 10 1

Prediction: a 2,000 bit quantum computer = a conventional computer the size of universe.

Quantum bit (or “qubit”)

superposition:

0

( )VS

The Quantum Spectrum Analyzer

( )VS

( )VS

pCmeas

?

Vbias

Measures all Noise Classical (symmetric) Quantum (asymmetric)

Quantum Computing

Scalable

Coherent

ControllableMeasurable

Cooper-pair boxSQUID’s

Ion TrapsLiquid State NMRNuclear Spins in

Semiconductors

How coherent is a Cooper-pair box?

Outline

•Charge quantization on a normal-metal islandSingle-electron Box

•Superconducting island as quantum two-level systemCooper-pair Box

•Spectroscopy of the Cooper-pair boxSingle-electron Tranistor (SET) measures box

•Box Measures SET Quantum Spectrum Analyzer

Microwaves

Small, Cold and Fast

1 m

Dilution refrigeratorT = 15 mK

MillikelvinsNanometers

Experiment Diagram

Quantum Shot Noise of DJQP* Process

0 0.5 1 1.5 20

2

Avg

. Qub

it C

harg

e

-1 -0.5 0 0.5 110

-15

10-10

Her

tz-1

NBΩ / ECS

Excitation RelaxationΩ= -ECS Ω= ECS

Qubit acts like a spectrum analyzer of the SET quantum noise!

*Double Josephson-quasiparticle cycle: (A. Clerk et al. cond-mat 0203338)

(see also Aguado & Kouwenhoven, 2000 for double dot)

Single Spin ½ Quantum Measurement

NMR of a Single Spin

Box

SET

Vgb Vge

Cgb Cc Cge

Vds

e

ne

The Single-Electron Box

2/c g geE E n C V e

ne to ne+1 electrons

Vg

island

Cj Rj

1 fFg jC C C Cg

2

1 K2c

eE

C

ne=-1 ne=0 ne=1

Ec

E

ggC V

e

Ec/4

Normal tunnel junction

ne=-1 ne=0 ne=1

Ec

E

ggC V

e0-0.5 0.5

/q e

Single-electron Box: Coulomb Staircase

ee

Coulomb Staircase

Thermally broadened

kT/Ec

-1 1

500 mK200 mK 50 mK0

-1

1

First demonstrated by Lafarge et al, ’91

(CEA Saclay)

B 1 T

Ec/4 1.6 KCE

CgeVge

Vds

Ids

Vds

0ge eg VC

e

1

2gegeVC

e

10 nA

1 mV

Single-electron Transistor: Electrometer

Electrometerinput gate

drain

source

SET

Quantum Shot Noise of DJQP* Process

-1 -0.5 0 0.5 110

-15

10-10

Her

tz-1

Ω / ECS

Excitation RelaxationSharp thresholds due to opening & closing of transport channels

Γ

*Double Josephson-quasiparticle cycle: (A. Clerk et al. cond-mat/0203338)

10-5 e/Hz1/2 charge noise

Sub-electron sensitivity for > 100 MHz bandwidth

-10 0 10Time ( s )

0.2 electrons

Electrometer input gate

TransformerSET

RF

Ref

lect

ed p

ower

Measure RF power reflected from LC transformer

Schoelkopf et al., (Science 1998)

Radio-Frequency Single Electron Transistor (RF-SET)

Response to step in Vge

single time trace

e

ne

The Single-Electron Box

2/c g geE E n C V e

ne to ne+1 electrons

Vg

island

Cj Rj

1 fFg jC C C Cg

2

1 K2c

eE

C

ne=-1 ne=0 ne=1

Ec

E

ggC V

e

Ec/4

Normal tunnel junction

Conclusions

•Cooper-pair Box: A quantum two-level systemworst-case coherence

•Box Hamiltonian determined with spectroscopy

•Long excited-state lifetime while continuously measured.

*2 01 100Q T

1 1 01 100,000Q T

1.79 0.004CE 0.626 0.008JE

Gap rise 500 pA

Eye 100 pA

Weak JQP 100 pAStrong JQP 150 pADouble JQP 300 pA

dseV

2ggC V

e

gg eC V

e

Coulomb Staircase vs. Electrometer Bias

T=20 mK

n

Back-action increases with

electrometer current