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