APS 2010 March Meeting, Tutorial #3

Advances in Josephson Quantum Circuits

Portland Convention Center Sunday, March 148:30 a.m. - 12:30 p.m.

Instructors:

Michel Devoret, Yale University"Introduction to superconducting quantum circuits"

Yasunobu Nakamura, NEC Japan"Superconducting qubits coupled to a transmission line "

John Martinis, University of California, Santa Barbara"Precision Control of Josephson Qubits"

Leo DiCarlo, Yale University"Production and detection of entanglement in cQED processors"

Introduction to superconducting quantum circuits

Outline• Motivation: quantum information

• Why Josephson junctions?

• Main flavors of Josephson qubits

• Readout of qubits

• 1-qubit qnd 2-qubit gates

Final version of this presentation available athttp://qulab.eng.yale.edu/archives.htm (talks)

J. Clarke and F. Wilhelm, Nature 453, 1031-1042 (2008)

RECENT REVIEWS ON JOSEPHSONQUANTUM CIRCUITS

R. Schoelkopf and S. M. Girvin, Nature 451, 664-669 (2008)

M.H. Devoret and J.M. Martinis, Quantum Information Processing 3,163 (2004)

J.M. Martinis, Quantum Information Processing 8, 81 (2009)

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

CLASSICAL BIT = SWITCH

1

0

Mechanical switch Electrical switch

CMOSTransistor

0 1

Bit state is either 0 or 1: 1) strong dissipation and 2) kTnoise<< B

0 1

EnergyCoordinate

B kTnoise

QUANTUM BIT: 2 LEVELSFORMING EFFECTIVE SPIN 1/2

ENERGY

}

MOLECULE, ATOM, PARTICLE...

10α β+

0

1

2

34

Qubit state can be 0 and 1: 1) no dissipation and 2) kTnoise<< hω01

0

1

spin up

spin down

y

z

x

θ

φ2

2

2

2

cos e

sin e

i

i

φ

φ

θ

θ

α

β

+

−

=

=

Bloch sphererepresentation

RELAXATION TIMES OF QUANTUM MEMORY

T1 PROCESS

random fieldsin x,y plane

Tφ PROCESS

random fieldalong z

2

1

11 1

2

T

T Tφ

=+

DECOHERENCE TIME 01 2Tω DECOHERENCEQUALITY FACTOR

2N = 1024 POSSIBLE CONFIGURATIONS

REGISTER WITH N=10 BITS:

THE POWER OF QUANTUM SUPERPOSITION

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 1 0

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 0

classically, can store and work only onone number between 0 et 1023

2N = 1024 POSSIBLE CONFIGURATIONS

classically, can store and work only onone number between 0 et 1023

REGISTER WITH N=10 BITS:

THE POWER OF QUANTUM SUPERPOSITION

0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 1 0

1 1 1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 0“quantally”, can store and work on an

arbitrary superposition of these numbers!

0 1 2 2 10 1 2 ..... 2 1N

Nα α α α−

Ψ = + + + + −

QUANTUM PARALLELISM

suppose a function f { } ( ) { }0,1023 0,1023j n f j∈ → = ∈

Classically, need 1000 ×10-bit registers (10,000 bits) to storeinformation about this function and to work on it.

Quantum-mechanically, a 20-qubit register can suffice!

( )2 1

/ 20

12

N

Nj

j f j−

=

Ψ = ∑Function encoded in a superposition of states of register

MICROFABRICATION L ~ 3nH, C ~ 1pF, ωr /2π ~ 4GHz

SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT

HOW CAN A SUPERCONDUCTING CIRCUITBEHAVE LIKE AN ATOM?

ELECTRONIC FLUID FLOWS BACK AND FORTH BETWEEN PLATES:ALL ELECTRONS BEHAVE AS A SINGLE CHARGED ENTITY

see practical LC superconducting resonators: Lindström et al., PRB 80, 132501 (2009)Paik & Osborn, APL 96, 072505 (2010)

+Qφ

-Q

QUANTUM CIRCUITS IN A NUTSHELL: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 excitation quanta of circuit(standing photons)

φ

φ

WAVEFUNCTIONS OF LC CIRCUIT

rωhI

φ

E

Ψ(φ)Ψ0

0Ψ1

In every energy eigenstate,(standing photon state)

current flows in opposite directions simultaneously!

2 rφ

01

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 Tω5 GHz 15 mK

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

Emissionspectrum

frequency01ω12ω23ω34ω

JOSEPHSON TUNNEL JUNCTIONPROVIDES A NON-LINEAR INDUCTOR

WITH NO DISSIPATION

1nm SI

S

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

φ

( )0cos /JU E φ φ= −

CJLJ

Ι

( )' 't

V t dtφ−∞

= ∫

0 2eφ =

20 0

0J

J

LE Iφ φ

= =

Bare Josephson potential

1nm SI

S

ordinary inductance

ENERGY SCALESOF THE JOSEPHSON JUNCTION "ATOM"

RESTOF

CIRCUIT

extq( )ˆ ' '

tQ I t dt

−∞= ∫

ˆˆ

ˆˆ

2

2QN

e

e

φϕ =

=

ˆˆ, iNϕ⎡ ⎤ =⎣ ⎦

( )2

ˆ co8 ˆ2

ˆs

exCJ

tJEH

NNE ϕ

−= −

2

2Cj

E eC

=

2ext

extqN

e=

Hamiltonian:

Coulomb charging energy for 1e

18JE = ΔNT

Josephson energy

gap

# condion channels

barrier transpcy

reduced offset charge

valid foropaque barrier

φ

HARMONIC APPROXIMATION

( )2

ˆ co8 ˆs2J

ex

J

t

C

NE E

NH ϕ

−= −

( )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:

JCE E , low energy

3 TYPES OF BIASES

U

chargeJL

JC 2 gC2 gC

( )2/ 2ˆ

ˆ8 cos2C JgN

E EC U

ϕ−

−

"Cooper pair box"ϕ̂ lives on circle

N̂ integer

3 TYPES OF BIASES

U

Φb

charge fluxJL JL

JC JC2 gC2 gC L

( )2/ 2ˆ

ˆ8 cos2C JgN

E EC U

ϕ−

−

"Cooper pair box"ϕ̂ lives on circle

N̂ integer

b2

2 ˆˆˆ8 co

2

s2 2C L J

eNE E E

ϕϕ

⎛ ⎞−⎜ ⎟⎝ ⎠+ −

Φ

"RF-Squid", lives on line, real numberϕ̂ N̂

3 TYPES OF BIASES

UIb

Φb

charge flux currentJL JL JL

JC JCJC2 gC2 gC L

( )2/ 2ˆ

ˆ8 cos2C JgN

E EC U

ϕ−

−2

b

0

ˆˆ ˆ8 cos

2C J IN E IE ϕ ϕ

⎛ ⎞− −⎜ ⎟

⎝ ⎠

"Cooper pair box"ϕ̂ lives on circle

N̂ integerin the limit

b

LΦ → ∞

→ ∞b

bIL

Φ→

b2

2 ˆˆˆ8 co

2

s2 2C L J

eNE E E

ϕϕ

⎛ ⎞−⎜ ⎟⎝ ⎠+ −

Φ

"RF-Squid", lives on line, real numberϕ̂ N̂

EFFECTIVE POTENTIALOF 3 MAIN BIAS SCHEMES

"phase" bias

"flux" bias

"charge" bias

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)

a few levels here.quasi-continuum

there.

0b 2

ΦΦ

=

SUPERCONDUCTING ARTIFICIAL ATOMS"MENDELEEV" TABLE

/L JE E

/J CE E

0 1

1

10

100

1000

10000

Cooper Pair Box

Quantronium

Transmon

Fluxonium

100000Phase qubit

Flux Qubit

inverse ofnumber of wells

in potential

charge fluctuationsrelative to phase fluctuations

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 POSSIBLE3) FAST, 4) PRESERVE STATE (QND)

WANT:

0 1

1

0

QUBIT READOUTON

OFF0 1

0 1

0 10 11F ε ε= − −

1ε0ε

FIDELITY:

or

pointervariable

10 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 session V26 )

"Circuit QED": Review by Blais et al., Phys. Rev. A 75, 032329 (2007)

SCHEMATIC OF COOPER PAIR BOXESIN A MICROWAVE RESONATOR (CAVITY)

Roles of cavity: 1) Filter, 2) Dispersive measurement, 3) Quantum bus

IN

OUT

PAULI SPIN MATRICES AND ROTATIONS

1 00 1

0 11 0

00

1 00 1

z

x

y

w

Z

X

iY

i

I

σ

σ

σ

σ

⎡ ⎤⎢ ⎥= =⎢ ⎥−⎣ ⎦⎡ ⎤⎢ ⎥= =⎢ ⎥⎣ ⎦⎡ ⎤−⎢ ⎥= =⎢ ⎥⎣ ⎦⎡ ⎤⎢ ⎥= =⎢ ⎥⎣ ⎦

[ ] ( )

[ ] ( )

[ ] ( )

00

00

0 11 0

1 00 1

z z

x x

y y

w

iZ i R

i

iX i R

i

Y i R

I Identity

σ π

σ π

σ π

σ

⎡ ⎤−⎢ ⎥=− = →⎢ ⎥⎣ ⎦⎡ ⎤−⎢ ⎥=− = →⎢ ⎥−⎣ ⎦⎡ ⎤−⎢ ⎥=− = →⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥= = →⎢ ⎥⎣ ⎦

usefulnotationof Paulispinmatrices

HERMITIAN (MEASUREMENT) UNITARY (GATE)

ELEMENTARY GATES ARE π/2 ROTATIONS

( ) ( ) ( ) ( )ˆ

cos sin2 2

yxx x

H t t t tσσ

ω φ ω φ⎡ ⎤ ⎡ ⎤= +⎣ ⎦ ⎣ ⎦

( ) ( )ˆ

cos2

zz x z x

H t t tωσ ω ω φ σ⎡ ⎤= + +⎣ ⎦In lab frame:

In rotating frameat Larmor freq.:

Do rotating wave approximation

Pulse 90°taround x

( )x tω area π/2

transverse osc.field amplitude

φ = 0

Pulse 90°taround y

( )x tω area π/2

φ = π/2

[ ]1/ 2X [ ]1/ 2Y[ ]1/ 2Z : shift Zeeman field

NATURAL ENTANGLING OPERATIONS

[ ]1/ 2ZZSecular interaction: 21intˆ

z zH g σ σ=

( ) ( )intˆ ˆexp /U iHτ τ= −

Flip-flop interaction:

( )1

1 1

nt 2i

2 2

ˆ . .

x x y y

H g h c

g

σ σ

σ σ σ σ⊥ + −

⊥

= +

= +

[ ] [ ]1/ 2 1/ 2X YX Y

adjustment of gate duration time:4gπ

τ =

1 2

PAIRWISE COUPLING v.s. BUS COUPLING

qubit

microwave transmission line resonator

coupling element

qubit coupling element:capacitor, inductor,auxiliary qubit

TWO-QUBIT QUANTUM PROCESSORslide courtesy of

Leo DiCarlo & Rob Schoelkopf

see 1 qubit and 2 cavities: B. Johnson et al. , 3 qubits and 1 cavity L. DiCarlo et al.: T26, W6

1Q 2Q

1f

2V2f

T29, V26, Y26

1V

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

P.I.'s

Grads

Post-Docs

Res. Sc.

Collab.

M. D.R. VIJAY (UCB)M. METCALFE (NIST)V. MANUCHARIANF. SCHAKERTN. MASLUKA. KAMALI. SIDDIQI (UCB)C. WILSON (Chalmers)E. BOAKNIN (McK)N. BERGEAL (ESPCI)C. RIGETTIM. BRINK

D. 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. SCHOELKOPFB. TUREK (MIT)J. CHOWB. JOHNSONA. SEARSM. READA. WALRAFF (ETH)H. MAJER (Vienna)A. HOUCK (Princeton)D. SCHUSTERL. DiCARLOL. SUNH. PAIKL. FRUNZIO

P. ZOLLER(Innsbruck)

W.M. KECK