Measuring Quantum Coherence in the Cooper-Pair Box

Depts. of Applied Physics & PhysicsYale University

YaleLafe SpietzRyan HeldBen Turek

Rob Schoelkopf

Chalmers UniversityKevin Bladh

David GunnarssonPer Delsing

Measuring Quantum Coherence in the Cooper-Pair Box

Konrad Lehnert

The David and LucilePackard FoundationFunding:

And discussions w/: M. Devoret, S. Girvin, A. Clerk, K. Nguyen

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:

Quantum Computing

Scalable

Coherent

ControllableMeasurable

Cooper-pair boxSQUID’s

Ion TrapsLiquid State NMRNuclear Spins in

Semiconductors

How coherent is a Cooper-pair box?

Box

SET

Single Electron Transistor Measuring Box

Box

Al/AlOx/Al junctions; 50 x 50 nm

e-beam lithography;

double-angle evaporation

Tc ~ 1.5 K

Vg Vge

Cg Cc Cge

Vds SET Electrometer

Superconducting tunnel junction

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

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

Single Spin ½ Quantum Measurement

NMR of a Single Spin

Box

SET

Vgb Vge

Cgb Cc Cge

Vds

CgeVge

Vds

Ids

Vds

0ge eg VC

e

1

2gegeVC

e

10 nA

1 mV

Single-electron Transistor: Electrometer

Electrometerinput gate

drain

source

SET

10-5 e/Hz1/2 charge noise

Sub-electron sensitivity for > 100 MHz bandwidth

-10 0 10Time ( ms )

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

Microwaves

Small, Cold and Fast

1 mm

Dilution refrigeratorT = 15 mK

MillikelvinsNanometers

Experiment Diagram

Continuous Measurement of a Single Spin

Measured continuously by SET

Theory: Cooper-pair box ground state

2e

1e

n

0

0.5

0 1 0.5Measurement must cause additional dephasing

uncertainty 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

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:

0.895 0.002

0.313 0.004C

J

E K

E K

Determination of Box Hamiltonian

Vapp

32 GHz

35 GHz

38 GHz

01

2

*2

1

T

0.235

2HWHM

0

2ggC V

e

2Power (arb) R

0.5 Photon Peak Heightn

Peak width

Peak height 50% saturated

*1 2

1 =R

TT

Saturation of the Cooper Pair Resonance

*2 1 nsT

2Power (arb) R

0

*2 1 nsT

0.265

37 GHz 39 GHz0.2

01 /E h

n

t<0

t<0 t>0

t=0.4 ms

t=1.6 ms

t=20 ms

n

0.50

Pea

k he

ight

(e)

gV

time

0

0

time 10 ms

0.15 e

Excited-state Lifetime

1 1.3 sT m

10 ms

0.3e

0

1e

2

cg

dgC V

e

dcg ggVV V

1

Cg

2e SET

SET Box Environment

Spontaneous Emission

Vg

50 envR

CcVds

E

Relaxatione

g

Cg

2e

Vg

Box

Spontaneous Emission into Environment

2

2

1 01

1

21sin

V

eS

T

gC

C

( ) 2 (50 )V

S

1 0.5 5 sT m

50 envR Spontaneous Emission:Fermi’s golden rule

effB

2elE

z

2JE

x

Electrometer Input Impedance

Pea

k H

eigh

t (e

)

Electrometer Operating Point (Vg)

370

185

740

01Re ( )Z

0

0.6 1 1.3 sT m

Cg

2e SET 2e

Vg

Cc Cg

2e

Cc 01Re ( )Z

1Peak Height T

0.3

Conclusions

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

•Box Hamiltonian determined with spectroscopy

•Long excited-state lifetime while continuously measured.

•Box measures electrometer input impedance

*2 01 100Q T

1 1 01 100,000Q T

0.895 0.002CE 0.313 0.004JE

2ggC V

e

n

Box State Depends on Electrometer Bias

Vds (mV)

250290

1200

0

420470760

•Long Excited-state lifetime >1 ms :

•Electrometer affects T1

• RF-SET measures charge states of box

• Spectroscopic determination of Hamiltonian of box

• Dephasing time ~ 1 ns :

Conclusions

*2 01 100Q T

1 1 01 100,000Q T

(w/ continuous measurement)

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

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 broadenedkT/Ec

-1 1

200 mK 16 mK

0

-1

1

First demonstrated by Lafarge et al, ’91

(Saclay)

B 1 TeslaEc/4

n=-1 n=0 n=1Ec

E

2ggC V

e0-0.5 0.5

n

ˆ ˆ ˆel JH H H

Cooper-pair Box Spectrum: Electrostatic and Josephson

2e

2e

EJ /4Ec

Condition:Two level System

4c

c J

E

E E kT

EJ

01 Larmor frequency

10-40 GHz

T1 Excited state lifetime 0.1-10 ms

R Rabi frequency

T2* Ensemble decoherence time

Cooper-pair Box as Spin 1/2

2JE

effB

elE

ZS

Time scales

/ E

z

x

0

( )VS

The Quantum Spectrum Analyzer

( )VS

( )VS

pCmeas

?

Vbias

Measures all Noise Classical (symmetric) Quantum (asymmetric)

2e

2e

kT/4Ec

n=-0.5

n=0 n=1Ec

E

2ggC V

e0-0.5 0.5

n

ˆ ˆ ˆel qpH H H

Cooper-pair Box Spectrum: Electrostatic and Quasi-particle

n=0.5

n=1.5

n=-1

( ) cT E

2e-periodic Cooper-pair Staircase

Odd: single q.p.

Even: no q.p.

( ) ln effT kT N

1

0 1

0E

2ggC V

e

n

Cooper-Pair Resonance Spectroscopy

2ggC V

e

Vapp

0 1 0.5

38 GHz

35 GHz

Cg

Vapp=Vg+Vacsint1

0