TRAITEMENT DESSIGNAUX QUANTIQUES

PAR LESCIRCUITS SUPRACONDUCTEURS

Department of Applied Physics, Yale University

Michel DEVORET

Juin

2015, Centre Jouvence, Mont Orford

ECOLE

DE PHYSIQUE DU

MONT ORFORD

PLAN DU COURS

Leçon

I: Introduction, questions et concepts de base

Leçon

II: Modes temporels

et fonction

d'onde

d'un photon

Leçon

III: Un amplificateur

calculable préservant

la phase:

Leçon

IV: Limite

quantique

des amplificateurs

SAMPLE rV t

t

pI t 1 2

tTm

Measurement value :

212

ei tr r w

p

M w t V t dt V

Z I w t dt

Probesignal

Responsesignal

ELECTRICAL TRANSPORT MEASUREMENT

12cos Re i tr pV t Z e I processing

Variations of M are

acquired as a functionof external drive and bias fields, and

thermodynamic parameters like

temperature

w(t) : window function

M M iM

if linear &1mT

in-phase andquadrature

compnents

cosp pI t I t w t cosr rV t V t w t

A FEW EXAMPLESFROM MESOSCOPIC

PHYSICSSAMPLEprobe response

tunneljunction

atomicpoint contact

qubit

w resonator

nano-mechanicalresonator

w LCresonator

nanomagnetic

system

2D-gas point contact, nanowire

NOISE REDUCES THE INFORMATIONEXTRACTED FROM MEASUREMENT

2 r NV t V t V t 2 +

= r Nw w

M V V

deterministic stochastic

signal energy:

t

221 2S r r w

E V t dt VR R

SAMPLE

22ReR Z

noise energy: 22N N w

E VR

signal to noise: S

N

ESNRE

Shannon information: 2ln 1 S

N

EE

(strict validity requires noise be gaussian)

1SNR 1 bit of information in value of M

3 origins for noise: a) probe, b) sample, c) processing.

A I t CURRENT

V V t VOLTAGE

,cZ v WAVE AMPLITUDE

22

c

c

ZZ

IA

V ttt

dimension: (watts)1/2

1A t

2 1dA t A tv

d

2A t

DIFFERENT DESCRIPTIONS OF A SIGNAL

2 2,xA A t : total energy flux through a section of line

SPATIAL MODE DESCRIPTION:SIGNAL PORTS

portport a port b

port c

.....

..... .....

port d

port l

inAoutA

inBoutB

notation: port index vs

signal amplitude symbol

GEOMETRIC REPRESENTATION OF SIGNAL MODE

N noise disc(diam.= 2

)

FRESNEL VECTOR

N = signal mode energy in photon number = signal mode phase

FRESNEL "LOLLYPOP"

in-phase amplitude compnent

out-phase amplitude compnent

Im a a

Re a a

Classical → Quantum: N1 coth4 2 Bk T

: mode index(spatial and temporal)

AThermalequilibrium1/2 ˆ

/ 2

Aa a

N

GN

OUTIN

b

b

a

a

QUANTUM LIMITED AMPLIFICATION WITH ALINEAR, PHASE-PRESERVING AMPLIFIER

STANDARD

QUANTUM LIMIT: AMPLIFIER ADDS ONLYANOTHER ½

PHOTON OF NOISE !MINIMUM REQUIRED BY HEISENBERG PRINCIPLEFOR A MEASUREMENT OF BOTH QUADRATURES

Shimoda, Takahasi

and Townes, J. Phys. Soc. Jpn. 12, 686 (1957); Haus

and Mullen, Phys. Rev. 128, 2407 (1962); Caves, Phys. Rev. D 26, 1817 (1982)

min 1 112 G

HEMT's 20 JPA's 1

14

12 2G

N

OUTIN

b

ba

AMPLIFICATION AT THE QUANTUM LIMIT WITH ALINEAR, PHASE-SENSITIVE AMPLIFIER

(Caves, 1982)

de-amplification

amplification

QUANTUM LIMIT : NO ADDED NOISE (!)+ SQUEEZING OF QUANTUM FLUCTUATIONS

'N N

2min min 1/ 2 1/ 21 116

G G

G

G

14

a

RELATIONSHIP BETWEEN PHASE-PRESERVINGAND PHASE-SENSITIVE AMPLIFICATION

+

UNVOIDABLE ADDED NOISE OF PHASE-PRESERVINGAMPLIFIER CAN BE UNDERSTOOD AS AMPLIFIED QUANTUM NOISEOF HIDDEN PORT OF BEAM SPLITTER. HOWEVER A MORE PRECISE

MODEL OF NOISE IS NEEDED TO OPTIMIZE THE AMPLIFIER.

phase sensitiveamplifiers withtwo # phasereferences

HOW CAN WE MAKE PRACTICAL SIGNAL PROCESSING DEVICESWHOSE NOISE PERFORMANCE

REACH THEQUANTUM LIMIT?

LINK WITH QUANTUM OPTICS

LASER

atom(s)

photo-multiplier

photon counting

LASER

intensity msmt

DIRECT DETECTION HOMODYNE DETECTION

= ?

+Q

-Q

HAMILTONIAN FORMALISM

V

X

Mk

electrical world mechanical world

I

f

2 2

,2 2QH QC L

charge on capacitor

time-integralof voltage on

capacitor

conjugate variables

2 2

,2 2P kXH X PM

momentum of mass

positionof massw/r

wall

conjugate variables

+Q

-Q

HAMILTON'S EQUATION OF MOTION

V

X

Mk

electrical world mechanical world

I

f

H QQ C

HQL

H PXP MHP kXX

1r LC

rkM

FROM CLASSICAL TO QUANTUM PHYSICS:CORRESPONDENCE PRINCIPLE

ˆ

ˆX X

P P

ˆ ˆ ˆ, ,H X P H X P

position operator

momentum operator

hamiltonian

operator

ˆ ˆ, , /PB

A B A B A B A B iX P P X

Poisson bracket

commutator

ˆ ˆ, 1 ,PB

X P X P i position andmomentum

operators do

not commute

+Q

-Q

E

LC CIRCUIT AS QUANTUM HARMONIC OSCILLATOR

r

†

†

ˆ 1ˆ ˆ 2ˆ ˆ ˆ ˆ

ˆ ˆ;

2 2 /

2

r

r r r r

r r r

r r

H a a

Q Qa i a iQ Q

L C

Q C

annihilation and creation operators

†,ˆ ˆ, 1i a aQ

1r LC

TRANSMISSION LINES

L L LLL

C C C C

1

1n n n

n nn

I

I I

dLdt

V V

C Vddt

nI1nV 1nV 1nI 1nI

a

nV

Dynamical equations: Continuum limit:

1

1

;

n

n

n

n

I I I

V

C LC L

a

Va

x

Vx

a a

L It

I VC

V

x

x

t

Field equations:

n n+1 n+2n-1

PROPAGATING WAVE AMPLITUDES

0 pA x v t

2 WATTSA

0, pA x t A x v t solution:

, ,

, ,1c

c

c

Z

Z

LZ

A x t A x t

A x t A x t

V V V

VC

I I

IV

I

I

1pv

L C

x

HOW DO WE QUANTIZE THIS DISTRIBUTED

ELEMENT SYSTEM?

0 pA x v t

characteristicimpedance

propagationvelocity

1

px vA

tA

LAGRANGIAN OF FREE SCALAR FIELD

x

1ground1 1 1, ,t x

dx t dt E t xx

2 21

2 2Cx

t L x

Lagrangian

density:

Momentum density: / lLx C

t t

= CHARGE

DENSITY ON

WIRE

Commutation relations: 1 2 1 2ˆ ˆ,x x i x x

1 2 1 2ˆ ˆ ˆ ˆ, , 0x x x x

For transmission

line, field is a "node flux":

WHAT DOES THIS MEAN?

x

x x x

C x x x x x x

,x x x i

, 0x x x x

L x

C x

L xL x

2 21 1ˆ ˆ ˆ

2 2H dx x x

C L

Hamiltonian gives energy density as a function of field and conjugate momentum:

1 2 1 2ˆ ˆ,x x i x x 0x

FOURIER DOMAIN REPRESENTATION

-1ˆ ˆ e21ˆ ˆ e2

ikx

ikx

k dx x

x dk k

-1ˆ ˆ e21ˆ ˆ e2

ikx

ikx

k dx x

x dk k

†ˆ ˆk k †ˆ ˆk k

Commutation relations:

1 2 1 2

1 2 1 2

ˆ ˆ ˆ ˆ, , 0

ˆ ˆ,

k k k k

k k i k k

NEW

OPERATORS

ARE NOT

HERMITIAN !

Hamiltonian has now independent mode form:

2 21ˆ ˆ ˆ ˆ ˆ2

H dk k k k C k kC

ˆ ˆk k &

with 2

2

2 2p

kkC L

v k

FIELD LADDER OPERATORS

†

1 ˆ ˆˆ2

1 ˆ ˆˆ2

ia k k C k kk C

ia k k C k kk C

†

†

ˆ ˆ ˆ2

1ˆ ˆ ˆ2

k a k a kk C

k Ck a k a k

i

SIMILAR BUT NOT IDENTICAL TO STANDING MODE LADDER OPERATORS

k

0a k

k0

0a k

†0a k †

0a k

DispersionRelation:

IT IS CONVENIENTTO INTRODUCE ASORT OF "MIRROR"DISPSION RELTION

pv k

FIELD LADDER OPERATORS IN FREQUENCY DOMAIN

k

0a

0

0a

0a 0a

†0 0

†0 0

ˆ ˆ 0

ˆ ˆ 0

a a k

a a k

† †ˆ ˆ

ˆ ˆ

di a adtdi a adt

Makes sense since:

Introduce new notationextending frequencies to negative

values, for both positive

and negative k:

Right moversLeft movers

Creation

Annihilation

Commutation relations: 1 2 1 2 1 2ˆ ˆ, sga a

1 2 1 2ˆ ˆ ˆ ˆ, , 0a a a a

Hamiltonian:

0ˆˆ ˆˆ ˆH aad aa

ˆ vac 0H

[Courty, Grassia

and Reynaud, Europhys.Lett. 46 (1999) 31]

- 4 - 2 0 2 4

- 1

- 0.5

0

0.5

1

- 10 - 5 0 5 10

- 1

- 0.5

0

0.5

1

WAVELET BASISChoose a complete orthonormal

set of functions of time, with both time and frequency

locality, forming a discrete basis (wavelet):

1 1 2 2 1 2 1 2m p m p m m p pW t W t dt

For instance, Shannon "bandpass" wavelets based on sinc function:

sin / 2exp

/ 2mp

b t pb imbt

b pt

tW

71

max

ReW tW

t b

e 1 1for ,2 2

0 elsewhere

mp

m

ip

p

m b m bb

W

W

2b

71

max

ReWW

WAVELETS HAVE

TWO INDICES

m specifies carrier frequencyp specifies instant of time

01

max

W tW

70

max

WW

- 4 - 2 0 2 4

- 1

- 0.5

0

0.5

1

- 10 - 5 0 5 10

- 1

- 0.5

0

0.5

1 71

max

ReW tW

t b

71

max

ReWW

01

max

W tW

70

max

WW

SHANNON WAVELET

exp sinc / 2mp b imbt bW t t p

TEMPORAL MODES

time

angularfrequency

bm

Tiling of

t-

plane:

a signal "atom": pairof wavelets (m,p)&(-m,p)

-m

t

0 p

/ 2 Remp mp mpW t W t W t

Each pair of conjugate basis functions correspond to a pair of real basis functions

in-phase basis component (position)

/ 2 Immp mp mpW t W t i W t quadrature

basis component (momentum)

THE RF PHOTON FINALLY APPEARS!

time

angularfrequency

bm

p

Tiling of

t-

plane:

must fill these

boxes with an amplitude

ˆ ˆmp mpa d W a

The flying photon ladder operator:

1 21 1 2 2 1 2 1 2sgˆ ˆ,m p m p m m p pm ma a where 1 2 1 2, , ,m m p p

-m

THE "AMPLITUDE"OF THE CONTENT

OF THE BOX IS THE

PHOTON OPERATOR

t

0

†ˆ ˆmp mpa a

2b

OTHER POSSIBLE MODE BASES

time

angularfrequency

'

'bm

-localized tiling: signal "atom"

-m

p

t

OTHER POSSIBLE MODE BASES

time

angularfrequency

''

"bm

t-localized tiling:

signal "atom"

p

t

SCALING TILINGangularfrequency

time

t

Mallat, S. M., "A Wavelet Tour of Signal Processing" (Academic Press, San Diego, 1999)

TILING OFTIME-

FREQUENCYPLANE

IN MUSIC

A musical scorehas similarities

with tiling of time-frequency plane.

However, shape, rather

than width, codefor duration.

08-III-25

PARAMETRIC AMPLIFICATION COMBINESANHARMONICITY, DRIVE AND DISSIPATION

† † ††3

† h.c.a b ccH ga ab ca cb b

g3 "3-wave mixing process"

000

001

bD a

drive resonance condition:

110

010

100drive,AKApump

signal

idler

populationinversion

3 a b cg

A minimal, calculable amplifier!

3 standing modes

VARIATIONS ON AMPLIFIER TYPE

Non-degenerate:

Degenerate:

Doubly-degenerate:

(Yale, Paris)

† †† †3

† h.c.ba cc cb b b ca a ag

†3

† †2 h.c.a c g ca a c c a

† †2 24 h.c.a aga a a

bD a

2D a

*4 4;a aD aa g ng

(Boulder, Berkeley, ETH)

(Tsukuba, Göteborg,Saclay, UCSB,....)

1) All above can be used also as bifurcation amplifier (Yale, Saclay, Berkeley,...)2) Drive can be DC voltage (Berkeley, Madison, Grenoble,...)

THE LIST OF PARTS(1)

a b

c

stand † ††lin

cbaH a ca cb b

stand3

†† h.c.nlH b cg a

3g

stand stand standlin nlH H H

non-linearelement

A PURELY INDUCTIVE MIXER

0

2ext

3g

JOSEPHSON

RING MODULATOR: WHEATSTONE BRIDGE CONFIGURATION

DIODE RING MODULATOR PERFORMSPRODUCT OF TWO SIGNALS (WITH DAMPING)

inut inoC A B;in no t iu AB C ;in no t iu BA CA B

C

DIODE RING MODULATOR PERFORMSPRODUCT OF TWO SIGNALS (WITH DAMPING)

DIODE RING MODULATOR PERFORMSPRODUCT OF TWO SIGNALS (WITH DAMPING)

DIODE RING MODULATOR PERFORMSPRODUCT OF TWO SIGNALS (WITH DAMPING)

0

2ext

2 2 2 higher order terms2 8

eff effJ J

r bi g c ca banL LE I IIII I

JOSEPHSON MODULATOR: REVERSIBLE MIXING

0

20

cos ext

effJ

J

LE

2

04

effJL

baI I3aI2

baI I

Wheatstonebridge symmetryeliminates many

undesirable terms

aI bIcI

0 2e

QUANTUM OP-AMP LIST OF PARTS

a b

c

stand † ††lin

cbaH a ca cb b

stand3

†† h.c.nlH b cg a

3g

stand stand standlin nlH H H stand coupl envH H H H

ab

c

A completelycalculableamplifier!

THE LIST OF PARTS

a b

c

stand † ††lin

cbaH a ca cb b

stand3

†† h.c.nlH b cg a

3g

stand stand standlin nlH H H †

t termerm

acoupl

AH d

b

a

cCB

a

term term

env A Ad

B

H

CCB

stand coupl envH H H H

ab

c

A completelycalculableamplifier!

ωpump

= ωsignal

+ ωidler

Amplification mode:

Bergeal

et al., Nature 2008Abdo

et al., APL 2011, PRB

2013

SP

∆∑

180°

hybrid

Signal

Idler

JRM

(Josephson

ParametricConverter)

I

Pure conversion mode:ωpump

= ωsignal

-

ωidler

MICROWAVE DESIGN OF THE JOSEPHSON OP-AMP

QUANTUM-LIMITED PARAMP: 8-JUNCTION JPC

•

Eliminates hysteresis•

Tunability

over ~400MHz 25

20

15

10

5

0

Gai

n (d

B)

7.577.567.557.547.53 Frequency (GHz)

See also Roch, et. al.

(PRL2011)

AMPLIFIER PERFORMANCE

8

6

4

2

0

NR

(dB

)

7.507.487.467.447.42

x109

G=20 dBNVR= 9 dBBW= 8 MHzD.R. ~ 10 photons @ 4.5 MHzTunability

~400 MHz

7.42 7.46 7.50

Frequency (GHz)

NV

R (d

B)

0

4

8

~88% of fluctuations of signal at 300K are quantum noise!

Idler

Signal

Quantum limited Josephson

amplifiers

:

an overview

RF PoweredDC PoweredDegenerate Non-Degenerate

Phase sensitive:amplifyonly onequadrature

Boulder (JPA), Yale (JBA), Berkeley (JBA), NEC, Caltech, ...

Phase preserving:amplifybothquadratures

Yale (JPC), ENS-Paris (JPC)

Berkeley, LLNL, NIST, Wisconsin, Syracuse, Princeton, etc....(MSA)

Amplifier TypeO

pera

tion

of A

mpl

ifier

MSA: Microwave Squid Amplifier; Muck et al., APL (2001) -> SLUG amplifier, McDermott et al. (2013)JBA: Josephson

Bifurcation Amplifier; Siddiqi

et al., PRL

(2004)JPA: Josephson

Parametric Amplifier; Castellanos-Beltran et al., Nature Physics (2008)JPC: Josephson

Parametric Converter; Bergeal

et al., Nature (2008)

CHARACTERISTIC FREQUENCIESAND ENERGY SCALES

3g

1MHz 100MHz 10GHz 1THz

ab c

a

b

D ca b

3 cg g n

f

0

2

eff effeff JJ

I Ee

4a b a bp p Q Q Staying below critical photon number:

participation ratio ( )eff effi J i Jp L L L quality factor /i i

THE QUANTUM LANGEVIN EQUATION

Operator equation extending Heisenberg' equation

Contains damping + drive + noise

Written here in RWA for single field and single operator,but can be more refined

Implements Fluctuation Dissipation Theorem

Easily transformed to form giving input-output relations:

in outa a a a

2 2out ina a

a ad di a i adt dt

stand ,2

inaa

d ia H a a adt

AMPLIFIER EQUATIONS

3

3

*

†

†

3

2

2

2

inaa a

i

in

nb

c

b b

cc

a a a a

a

d i idtd i idtd i

g

g

g

c

c

b

b b b b

i a cbc cdt

c

Abdo, Kamal

& MHD, Phys. Rev. B 87, 014508 (2013)

AMPLIFIER EQUATIONS

Crucial remark: Quantum fluctuations in the c field are negligible

compared with deterministic amplitude determined by the pump.

22

Di tc

c t c t c t

c t n e

c c

Input field = coherent state (Glauber)

22

ee

!

cc Dc

c

nin tn

cinc c

n c

nt n

n

3

3

*

†

†

3

2

2

2

inaa a

i

in

nb

c

b b

cc

a a a a

a

d i idtd i idtd i

g

g

g

c

c

b

b b b b

i a cbc cdt

c

†

† †

†e e

e e

a a

b b

b b

a a

c c

c c

i t i ta a a a

i t

out in

out i tb b

out in

out ib b

in n

d di i i idt dt

d

a a

a

b b

b b

g

di i i i

g

gdt

g adt

3 ecig ng e Di t

cc t c t n

FLUCTUATION-LESS PUMP APPROXIMATION

After a Fourier transform, in frequency domain:

*

* * *

a out a ina s b s a s b s

b out b ina b I I a b I I

h ig a h ig aig h b ig h b

Where: S I D , , ,a b a b a bh i i

,, 2

a ba b

a ab

b

g g

SCATTERING MATRIX

S S

I I

ab

bout in

bb

out inaa

a rbs a

bsra

1 S aa

a

i

3 e i

c

ba

ng

For zero detuning, i.e. 1a b 0baa b Grr

2 22 amp

0 2 amp

11

a

a

b

b

G

2(3)ampb

c

b

g n

Idler port appears as negative impedanceport to the signal!!

formula for attenuator, with a <0 for aux. port

2* *

2*aa

ab

a b

r

2

2*a

bbb

a b

r

*

2 2* *

2 2a

a b a bb bas si i

1 I bb

b

i

,, 2

a ba b

PHASE CONJUGATION

outa

ina outb

inbG G

e 1i G

e 1i G

in

out

S ISI 0

S ISI 0

Reflection amplifierIdler port brings in

minimum added noise †e 1iut io in na bGaG

BANDWIDTHS a b I

1 1

21 1

22

021 1

1/ 20

1

11

a b

a b

G

i GGi

1 1 1a b Bandwidth decreases as gain increases:

General result validfor most parametric

amplifiers

- 2 - 1 0 1 2

0

10

20

30

40

1/ 20

1

2 a b

G a b

B G

10 L

og10

G

2

pumpstrength

(some exceptionsexist, seeMetelmann

et al. 2014)

DYNAMIC RANGE LIMITATIONS

Log G0

0

good:

gainindependent

of input power

~1dB/dB

23

( )~ Log a b c

g

23Log

b

a

a c

gP

0bad:

gainfalls with

input power

NON-DEMOLITION MEASUREMENT OF TRANSMON QUBIT

Readout phase

f

90º

-90ºfr

resonator width

Readout amplitude

ffr

dispersive shift

qubit

&readoutpulses

JPC

e

e

g

g

HEMT

I

Q

phase-preserving,non-degenerate

amplifier

2 2I Q

1tan /Q I

Circuit QED: Walraff

et al. (2004),..., Gambetta et al. (2008)

ref

readoutfrequency

QUANTUM JUMPS

0

5

-5

0 50 100 150Time (s)

MEET THE TEAM MEMBERS

K. GeerlingsM. Hatridge

B. AbdoA. Kamal

K. SliwaA. Narla

S. Shankar

L. Frunzio

I. Pop

Y. Liu

Z. Leghtas

M. Mirrahimi

Z. Mineev

R. SchoelkopfS. GirvinMHD

L. Glazman

THANK YOU TO THEORGANIZERS AND THE

STUDENTS