Michel DEVORET, Applied Physics and Physics, Yale University

JOSEPHSON QUBIT CIRCUITSAND THEIR READOUT

2009 APS March Meeting Pittsburgh, PA

W.M. KECK

Final version of this presentation available at http://qulab.eng.yale.edu/archives.htm

Sponsors:

• OPTICAL PHOTONS

• NUCLEAR SPINS

• IONS

• ATOMS

• MOLECULES

• QUANTUM DOTS

• SUPERCONDUCTING JOSEPHSONTUNNEL CIRCUITS

QUANTUM INFORMATION PROCESSING

micro

MACRO

couplingwith envt.

# of atoms

10µm

50µm

from A. O. Niskanen et al. Science 316, 723 (2007) courtesy of J. Martinis, 2009 from Metcalfe et al., 2007

103 106 109 1012 1015 1018 1020100

106 103 100 10-3 10-6 10-9 10-12109

IR UV RXμW γRF /

ELECTROMAGNETIC SPECTRUM

VIDEOAUDIO

ELECTRICITY OPTICS

f (Hz)

λ (m)

10 GHz ~ 0.5K

1 bit = RF signal with 0/1 photon?

MICROFABRICATION L ~ 3nH, C ~ 10pF, ωr /2π ~ 2GHz

SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT

HOW CAN A SUPERCONDUCTING CIRCUITBEHAVE LIKE AN ATOM?

ELECTRONIC FLUID SLOSHES BACK AND FORTHFROM ONE PLATE TO THE OTHER, INTERNAL MODES FROZEN,

BEHAVES AS A SINGLE CHARGE CARRIER

Example of H atom with largeprincipal quantum number

SuperconductingLC oscillator

L C

velocity of electron → voltage across capacitorforce on electron → current through inductor

DEGREE OF FREEDOM IN ATOM vs CIRCUIT

+Qφ

-Q

FLUX AND CHARGE DO NOT COMMUTE

V

I

ˆ ˆ, iQφ⎡ ⎤⎣ =⎦

LIφ = Q VC=

+Qφ

-Q

φ

E

LC CIRCUIT AS QUANTUMHARMONIC OSCILLATOR

rωh

( )†

†

ˆ 1ˆ ˆ 2ˆ ˆ ˆ ˆ

ˆ ˆ;

2

2

r

r r r r

r r

r r

H a a

Q Qa i a iQ Q

L

Q C

ω

φ φφ φ

φ ω

ω

= +

= + = −

=

=

annihilation and creation operatorsfor mesoscopic excitation of circuit

φ

φ

WAVEFUNCTIONS OF LC CIRCUIT

rωhI

φ

E

Ψ(φ)Ψ0

0Ψ1

In every energy eigenstate,(photon state)

current flows in opposite directions simultaneously!

2 rφ

φ

φ

EFFECT OF DAMPING

important: as littledissipation as possible dissipation broadens energy levels

E

112 2n r

r

iE n

RC

ω

ω

⎡ ⎤⎛ ⎞= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=

QQ

φ

CAN PLACE CIRCUIT IN ITS GROUND STATE

Br k Tω10 GHz 25mK

rωh

E

residual dissipationprovides

reset of circuit

φ

φ

E

PB: ALL TRANSITIONS ARE DEGENERATE!

CANNOT STEER THE SYSTEM TO AN ARBITRARY STATEIF PERFECTLY LINEAR

rωh

Potential energy

Position coordinate

NEED NON-LINEARITY TO FULLYREVEAL QUANTUM MECHANICS

JOSEPHSON TUNNEL JUNCTIONPROVIDES A NON-LINEAR INDUCTOR

WITH NO DISSIPATION

1nm SI

S

superconductor-insulator-

superconductortunnel junction

φ

ΙΙ = φ / LJ

( )0 0sin /I I φ φ=

CJLJ

Ι

( )' 't

V t dtφ−∞

= ∫

0 2eφ =

20 0

0J

J

LE Iφ φ

= =

JOSEPHSON TUNNEL JUNCTIONPROVIDES A NON-LINEAR INDUCTOR

WITH NO DISSIPATION

1nm SI

S

superconductor-insulator-

superconductortunnel junction

φ

( )0cos /JU E φ φ= −

CJLJ

Ι

( )' 't

V t dtφ−∞

= ∫

0 2eφ =

20 0

0J

J

LE Iφ φ

= =

bare Josephson potential

0

2 2ˆ

ˆ2ˆ 1

41

e

er

pm

AH eπε

⎛ ⎞= − −⎜ ⎟⎜ ⎟

⎝ ⎠

RESTOF

CIRCUIT(CONTROL

& READOUT)

ˆextq

THE Hamiltonian:(we mean it!)

( )21 2coˆ s2 ext

JJJ

CH EQ q eφ

= − −

COUPLING PARAMETERSOF THE JOSEPHSON "ATOM"

Comparable with lowest ordermodel for hydrogen atom

Josephson tunnel junctioncontribution to total circuit

Hamiltonian

TWO ENERGY SCALESTwo dimensionlessvariables:

ˆˆ

ˆˆ

2

2QN

e

e

φϕ =

=

ˆˆ, iNϕ⎡ ⎤ =⎣ ⎦

( )2

co8 ˆs2

ext

JCJ

NE EH

Nϕ

−= −

2

2CJ

E eC

=

2x

exte tqNe

=

Hamiltonian becomes :

Coulomb charging energy for 1e

18JE = ΔNT

Josephson energy

gap

# condion channels

barrier transpcy

reduced offset charge

valid foropaque barrier

HARMONIC APPROXIMATION

( )2

co8 ˆs2

ext

JCJ

NE EH

Nϕ

−= −

( )22

,2ˆ

82

e

CJ h J

xtNE

NEH ϕ−

= +

8 CP

JE Eω =

Josephson"plasma" frequency:

Spectrum independent of DC value of Next

( )28

2J

J

CZe

EE

=

JosephsonRF impedance:

SOME JOSEPHSONTUNNEL JUNCTIONS

IN REAL LIFE

100nm

EJ ~ 50K

EJ ~ 0.5K

ωp ~ 30-40GHz

credit L. Frunzio and D. Schuster

credit I. Siddiqi and F.Pierre

RF CONTROL & BIAS vs SENSITIVITY TO NOISE

"charge" qubit "flux" qubit

+_

"phase" qubit

off 0 / 2Φ < Φ

0off 2

ΦΦ =

0off 2

ΦΦ >

"phase"

"flux"

"charge"

ΔΦoff ΔEJΔQoffflavors

noises

- +-

charged impurityunstable tunnel channel

loose elec. dipoles

see R. McDermot's tutorial

island

a.k.a. Cooper Pair Box

Devoret and Martinis, Quant. Inf. Proc. 2004

THE MONSTERS WE FEAR:

freak electron spin

EFFECTIVE POTENTIALOF 3 MAIN BIAS SCHEMES

"phase" bias

"flux" bias

"charge" bias

UC Berkeley, NIST, UCSB,U. Maryland, I. Neel Grenoble...

TU Delft, NEC, NTT, IBM,MIT, UC Berkeley, SUNY, IPHT Jena ....

CEA Saclay, NEC, YaleChalmers, JPL, ...

22

eh

ϕπ

φ=

2ehφ

2ehφ

12

− 12

+

e giN

see also proposals fortopologically protected

qubits, for exampleFeigelman et al. PRL 92,

098301 (2004)

HOW DO WE FIND THE HAMILTONIANOF AN ARBITRARY CIRCUIT?

Yurke B. and Denker J.S., Phys. Rev. A 29, 1419 (1984)

Devoret M. H. in "Quantum Fluctuations", S. Reynaud, E. Giacobino,J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p. 351-385

G. Burkard, R. H. Koch, and D. P. DiVincenzo, Phys. Rev. B 69, 064503 (2004)

L CCJLJ

Cc

Example: 2 Josephson junctions capacitively coupled to 1 resonator mode

C'JL'J

Cc

U

Φ U

ΦgCisland

COOPER PAIR "BOX"

Bouchiat PhD Thesis 97, et al. Physica Scripta '98Nakamura, Pashkin & Tsai, Nature '99

qbit space = Hilbert space of 0 or 1quanta in this non-linear oscillator

Φ controls EJ, CgU/2e controls Ng

Review by Blais et al., Phys. Rev. A 75, 032329 (2007)

SCHEMATIC OF COOPER PAIR BOXESIN A MICROWAVE CAVITY

TWO-QUBIT QUANTUM PROCESSORslide courtesy of

L. DiCarlo & Rob Schoelkopf

V17.6 Thu. March 19

see also 1 qubit and 2 cavities: B. Johnson et al. V17.4

small junction

array junctions

50 Ω port 1

50 Ω port 2resonator

"FLUXONIUM QUBIT"V. Manucharyan et al. Q17.5 Wednesday

A FEW USEFUL IDEAS FOR CIRCUITHAMILTONIANS.....

BRANCH VARIABLES

Vβ

Iβ

restof

circuit

banch βIntroduce branch flux and charge

position variable: φ Xmomentum variable: Q Pgeneralized force : Ι fgeneralized velocity: V V

( ) ( )

( ) ( )

' '

' '

t

tQ t

t dt

t dI

V

t

t

ββ

ββφ−∞

−∞

=

=

∫∫

Vβ

Iβ

,ˆ ˆ iQβ βφ⎡ ⎤⎦ =⎣

For every branch β in the circuit:

restof

circuit

banch βIntroduce branch flux and charge

( ) ( )

( ) ( )

' '

' '

t

tQ t

t dt

t dI

V

t

t

ββ

ββφ−∞

−∞

=

=

∫∫

BRANCH VARIABLES

PROBLEM: NOT ALL BRANCHVARIABLES ARE INDEPENDENT

b bd

branchesaround loop

0Vλ

λ =∑branchestied to node

0Iν

ν =∑

KIRCHHOFF'S LAWS:

b

c

b

a

d

a

c

IMPOSE CONSTRAINTS ON BRANCH VARIABLES

TWO METHODS FOR DEFINING A COMPLETESET OF INDEPENDENT VARIABLES

Method of nodes

Method of loops

08-II-11a

Defines loop charges

Defines node fluxes

Method of nodes

1) Choose a reference electrode(ground)

2) Choose a spanning tree(accesses every node, no loop)

3) Select tree branch fluxes(closure branches left out)

Φ1

Φ2 Φ3 Φ7

Φ5

φg

φe

Φ 7 = φd + φe + φgexample:

4) Node flux Φ n is sum of branchfluxes to ground (closure branchfluxes are expressed as differencesbetween node fluxes)

φh

φh = Φ 7 - Φ 6 + cst

EXAMPLE OF A COMPLETESET OF INDEPENDENT VARIABLES

φd

φf

( ) ( ) cstn nγγ γφ+ −

Φ Φ= − +

tree branchesleading to

n

nβ

βφΦ = ∑

Φ4

Φ6

INDUCTIVE vs CAPACITIVE ELEMENTS

( )0

' ' 't

E I Vdt I dφ

φ φ−∞

= ⋅ =∫ ∫I

φ

Inductance : current I function only of flux φ

Electrical equivalent of spring: φ ↔ X ; I ↔ f

( )0

' ' 't Q

E V Idt V Q dQ−∞

= ⋅ =∫ ∫Q

V

Capacitance : voltage V function only of charge Q

Electrical equivalent of mass: Q ↔ P ; V ↔ V

NODE CHARGES

The conjugate coordinates of node fluxesare node charges: they are the sum of all the charges

going into capacitances linked to this node.

This can be demonstrated by writing the Lagrangianof the circuit from its dynamical equations

and performing a Legendre transform to obtain the Hamiltonian

Dynamicalequations

Lagrangian conjugatecoordinates Hamiltonian

capacitive branchesexiting from

n

n

Q Qβ= ∑ n

( ), ,L x x t ( ), ,H x p tLp x∂= ∂

[ ], nn iQΦ =for eachnode:

HAMILTONIAN OF TWO CAPACITIVELYCOUPLED RESONATOR MODES

LbCaLa

Cc

Cb

Φ1 Φ2

( )2 21 2

1 1 2 2

2 21 2 1 2

1 2 3

ˆ ˆˆ ˆˆ ˆ ˆ, ; ,2 2

ˆ ˆ ˆ ˆ

2 2

a b

H Q QL L

Q Q Q QC C C

Φ ΦΦ Φ = +

+ + +

Reduced capacitance matrix:

a c c

c b c

C C CC C C+ −⎡ ⎤

= ⎢ ⎥− +⎣ ⎦C

Inverse of reduced capacitance matrix:

1 1 b c c

c a ca b a c b c

C C CC C CC C C C C C

− +⎡ ⎤= ⎢ ⎥++ + ⎣ ⎦

C

ANHARMONICITY vs CHARGE SENSITIVITYCooper pair box levels are "exactly soluble" (A. Cottet, PhD thesis, Orsay, 2002)

290 μm

ANHARMONICITY vs CHARGE SENSITIVITYIN THE LIMIT EJ /EC >> 1

peak-to-peak charge modulation amplitude of level m:

anharmonicity:

TRANSMON: SHUNT CPB JUNCTION WITH CAPACITANCE

Courtesy of J. Schreier and R. Schoelkopf

J. Koch et al. Phys. Rev. A '07( )

12 01

12 01 / 2 8C

J

EE

ω ωω ω

−→

+

ω01

ω12ω02 2

Sufficient to control the artificial atom as a two level system: Qubit

SPECTROSCOPY OF A JOSEPHSON ATOM

Anharmonicity:

01 12 455MHz CEω ω− =

ω01

ω02 2

ω12

J. Schreier et al., Phys. Rev. B '08

Slide courtesy of J. Schreier and R. Schoelkopf

THE MEMORY READOUT PROBLEM

QUBIT READOUTON

OFF0 1

QUBIT READOUTON

OFF

1) SWITCH WITH ON/OFF RATIO AS LARGE AS POSSIBLE2) READOUT WITH F AS CLOSE TO 1 AS POSSIBLEWANT:

0 1

1

0

QUBIT READOUTON

OFF0 1

0 1

0 10 11F ε ε= − −

1ε0ε

FIDELITY:

or

STATE DECAY STRATEGY

1

0

1

0

Martinis, Devoret and Clarke, PRL 55 (1985)Martinis, Nam, Aumentado and Urbina, PRL 89 (2002)

QUBITCIRCUIT

10

or

or

rf signal in rf signal out

A) FILTER OUT EVERYTHING ELSE THAN READOUT RFB) REPEAT WITH ENOUGH PHOTONS TO BEAT

NOISE : USE THE BEST AMPLIFIER AS POSSIBLE

DISPERSIVE READOUT STRATEGY

01ω ω≠

QUBIT STATEENCODED IN PHASE

OF OUTGOING SIGNAL,NO ENERGY DISSIPATED

ON-CHIP

Blais et al. PRA 2004, Walraff et al., Nature 2004

(see whole sessions J3 & L17 + Q17.6& Y34.4)

Si or Al2O3

Al or Nb

SUPERCONDUCTING MICROWAVE RESONATOR: ANALOG OF FABRY-PEROT CAVITY FILTER

~

can get Qup to 106

( in bulk:Q ~ 1010)

ADC

length ~ 1cm, but photon decay length ~ 10km!

IN OUT

1/2 photon: ~1nA~100nV

ν ~ 10GHz

CHIP

w~20μm

VERY SMALL MODE VOLUME

3mm

10mm

100μm

Nb

SUPERCONDUCTING CAVITY = FABRY-PEROT

qubit goes here (see slide 2, lower right, in this talk)

OFF-RESONANT NMR-TYPEPULSE SEQUENCE

FOR QUBIT MANIPULATION

EXAMPLE OF QUANTRONIUM IN MICROWAVE CAVITY

|0>

|1>

READOUTPROBING

PULSE

QUBIT STATEENCODED IN PHASE

OF TRANSMITTED PULSE

or

QUANTRONIUM IN MICROWAVE CAVITY

|0>

|1>

QUANTRONIUM IN MICROWAVE CAVITY

~ ADCIN OUT

AWG

40ns40ns200ns

t t

Metcalfe et al.Phys. Rev. B6174516 (2007)

latching effectdue to bifurcation

of cavity mode

50Ω 50Ω

Cin~200aF Cout~20fFCcouple~6fF

qubit cavity readout cavity

qubitreadoutjunction

IN-LINE TRANSMON WITH BIFURCATING MICROWAVE CAVITY READOUT

See V17.6 M. Brink et al. Thursday morning, and also recent Saclay group results(in preparation)

Acknowledgements: Circuit Quantum Electrodynamics GroupsDepts. Applied Physics and Physics, Yale

P.I.'s

Grads

Post-Docs

Res. Sc.Undrgrds

Collab.

M. D.R. VIJAY (UCB)C. RIGETTI M. METCALFE (NIST)V. MANUCHARIANF. SCHAKERTN. MASLUKA. KAMALI. SIDDIQI (UCB)C. WILSON (Chalmers)E. BOAKNIN (McK)N. BERGEAL (ESPCI)M. BRINKA. MARBLESTONED. ESTEVE et coll.(Saclay)B. HUARD (LPA/ENS)

S. GIRVINT. YUL. BISHOP

J. KOCHJ. GAMBETTA(U. Waterloo)E. GINOSSARA. NUNNENKAMP

F. MARQUARDT(Munich)A. BLAIS(Sherbrooke)A. CLERK (McGill)

R. SCHOELKOPFJ. CHOWB. TUREK (MIT)J. SCHREIERB. JOHNSONA. SEARSM. READA. WALRAFF (ETH)H. MAJER (Vienna)A. HOUCK (Princeton)D. SCHUSTERL. DiCARLOL. FRUNZIOJ. SCHWEDEP. ZOLLER(Innsbruck)

W.M. KECK

Acknowledgements: Circuit Quantum Electrodynamics GroupsDepts. Applied Physics and Physics, Yale

P.I.'s

Grads

Post-Docs

Res. Sc.Undrgrds

Collab.

M. D.R. VIJAY (UCB)C. RIGETTI M. METCALFE (NIST)V. MANUCHARIANF. SCHAKERTN. MASLUKA. KAMALI. SIDDIQI (UCB)C. WILSON (Chalmers)E. BOAKNIN (McK)N. BERGEAL (ESPCI)M. BRINKA. MARBLESTONED. ESTEVE et coll.(Saclay)B. HUARD (LPA/ENS)

S. GIRVINT. YUL. BISHOP

J. KOCHJ. GAMBETTA(U. Waterloo)E. GINOSSARA. NUNNENKAMP

F. MARQUARDT(Munich)A. BLAIS(Sherbrooke)A. CLERK (McGill)

R. SCHOELKOPF(Keithley Award, T3)J. CHOWB. TUREK (MIT)J. SCHREIERB. JOHNSONA. SEARSM. READA. WALRAFF (ETH)H. MAJER (Vienna)A. HOUCK (Princeton)D. SCHUSTERL. DiCARLOL. FRUNZIOJ. SCHWEDEP. ZOLLER(Innsbruck)

W.M. KECK