1 CIRCUITS ET SIGNAUX QUANTIQUES QUANTUM SIGNALS AND CIRCUITS Chaire de Physique Mésoscopique Michel Devoret Année 2008, 13 mai - 24 juin Première leçon / First Lecture 08-I-1 This College de France document is for consultation only. Reproduction rights are reserved. VISIT THE WEBSITE OF THE CHAIR OF MESOSCOPIC PHYSICS http://www.college-de-france.fr Questions, comments and corrections are welcome! 08-I-2 PDF FILES OF ALL LECTURES WILL BE POSTED ON THIS WEBSITE http://www.physinfo.fr/lectures.html and follow links to:
Transcript
1
CIRCUITS ET SIGNAUX QUANTIQUES
QUANTUM SIGNALS AND CIRCUITS
Chaire de Physique MésoscopiqueMichel Devoret
Année 2008, 13 mai - 24 juin
Première leçon / First Lecture
08-I-1
This College de France document is for consultation only. Reproduction rights are reserved.
VISIT THE WEBSITE OF THE CHAIROF MESOSCOPIC PHYSICS
http://www.college-de-france.fr
Questions, comments and corrections are welcome!
08-I-2
PDF FILES OF ALL LECTURES WILL BE POSTED ON THIS WEBSITE
http://www.physinfo.fr/lectures.html
and follow links to:
2
May 13: Denis Vion, (Quantronics group, SPEC-CEA Saclay)Continuous dispersive quantum measurement of an electrical circuit
May 20: Bertrand Reulet (LPS Orsay)Current fluctuations : beyond noise
June 3: Gilles Montambaux (LPS Orsay)Quantum interferences in disordered systems
June 10: Patrice Roche (SPEC-CEA Saclay)Determination of the coherence length in the Integer Quantum Hall Regime
June 17: Olivier Buisson, (CRTBT-Grenoble)A quantum circuit with several energy levels
June 24: Jérôme Lesueur (ESPCI)High Tc Josephson Nanojunctions: Physics and Applications
CALENDAR OF SEMINARS
NOTE THAT THERE IS NO LECTURE AND NO SEMINAR ON MAY 27 ! 08-I-3
PROGRAM OF THIS YEAR'S LECTURES
NEXT YEAR: STRONGLY NON-LINEAR AND/OR DISSIPATIVE CIRCUITS
08-I-4
Lecture I: Introduction and overview
Lecture II: Modes of a circuit and propagation of signals
Lecture III: The "atoms" of signal
Lecture IV: Quantum fluctuations in transmission lines
Lecture V: Introduction to non-linear active circuits
Lecture VI: Amplifying quantum signals with dispersive circuits
3
LECTURE I : INTRODUCTIONAND OVERVIEW
1. Review of classical radio-frequency circuits
2. Quantum information processing
3. Quantum-mechanical LC oscillator
4. A non-dissipative, non-linear element: the Josephson junction
5. Energy levels and transitions of the Cooper Pair Box
6. Summary of questions addressed by this course
08-I-5
CLASSICAL RADIO-FREQUENCY CIRCUITS
Hergé, Moulinsart
1940's : MHz 2000's : GHz
Communication
Computation
US Army Photo
Communications
Sony
08-I-6
4
SIMPLEST EXAMPLE :
THE LC OSCILLATOR
La radio? Mais c'est très simple!Eisberg, 1960
L C
08-I-7
WHAT IS A RADIO-FREQUENCY CIRCUIT? A RF CIRCUIT IS A NETWORK OF ELECTRICAL ELEMENTS
element
node p
loopbranch
np
node n
Vnp
Inp
TWO DYNAMICAL VARIABLES CHARACTERIZE THE STATEOF EACH DIPOLE ELEMENT AT EVERY INSTANT:
Voltage across the element:
Current through the element:
( )p
nnpV t E d= ⋅∫( ) nnp pj dI t σ= ⋅∫∫
Signals:any linear
combinationof these variables
08-I-8
5
KIRCHHOFF’S LAWS
ba
c
b
d ba
c
b
d
branchesaround loop
0Vλ
λ =∑branchestied to node
0Iν
ν =∑08-I-9
EACH ELEMENT IS TAKEN FROM A FINITE SET OF ELEMENT TYPES
inductance:
capacitance:
V = L dI/dt
I = C dV/dt
resistance: V = R IIds = G(Igs,Vds,) VdsIgs = 0
transistor: g
s d
diode: I=G(V)
CONSTITUTIVE RELATIONS
EACH ELEMENT TYPE IS CHARACTERIZED BY A RELATION BETWEENVOLTAGE AND CURRENT
LINEAR NON-LINEAR
08-I-10
6
GENERALIZATIONS: MULTIPOLES, PORTS
dipole quadrupole hexapole etc....
n polesn-1 currentsn-1 voltages
WILL DEAL WITH THESE HIGHER DEGREES OF COMPLEXITY LATER
ALSO, CONSTITUTIVE RELATION OF ELEMENT NEED NOT BE LOCAL IN TIME
AN ELEMENT CAN BE CONNECTED TO MORE THAN TWO NODES
08-I-11
COMPUTING THE DYNAMICAL STATE OF A CIRCUIT IN CLASSICAL PHYSICS
A CIRCUIT BEHAVING QUANTUM-MECHANICALLYAT THE LEVEL OF CURRENTS AND VOLTAGES ?
INCOMPRESSIBLE ELECTRONIC FLUID SLOSHES BETWEEN PLATES.NO INTERNAL DEGREES OF FREEDOM.
d ~ 0.5 mm
d << λ: LUMPED ELEMENT REGIME
SUPERCONDUCTIVITY ONLY ONE COLLECTIVE VARIABLE
08-I-15c
Rydberg atom SuperconductingLC oscillator
L C
DEGREE OF FREEDOM IN ATOM vs CIRCUIT:SEMI-CLASSICAL DESCRIPTION
08-I-16
10
Rydberg atom SuperconductingLC oscillator
velocity of electron → voltage across capacitorforce on electron → current through inductor
DEGREE OF FREEDOM IN ATOM vs CIRCUIT
2 possible correspondences: velocity of electron → current through inductor force on electron → voltage across capacitor
charge dynamics
field dynamics
φL C
semiclassical picture
08-I-16b
+Qφ
-Q
FLUX AND CHARGE IN LC OSCILLATOR
LIφ = Q VC=
V
-I
X
Mk
electrical world mechanical world
I
f
position variable: φ xmomentum variable: Q Pgeneralized force on mass : Ι fgeneralized velocity: V V
1xk
f= − P VM=
generalized mass : C Mgeneralized spring constant: 1/L k
08-I-17a
0x X X= −
equilibriumposition of
spring
11
+Qφ
-Q
HAMILTONIAN FORMALISM
V
X
Mk
electrical world mechanical world
I
f
( )2 2
,2 2QH QC L
φφ = +
charge on capacitor
time-integralof voltage oninductor =flux in inductor
conjugate variables
( )2 2
,2 2P kxH x PM
= +
momentum of mass
elongationof spring
conjugate variables 08-I-18
0x X X= −
+Qφ
-Q
HAMILTON'S EQUATION OF MOTION
V
X
Mk
electrical world mechanical world
I
f
H QQ C
HQL
φ
φφ
∂= =
∂∂
= − = −∂
H PxP M
HP kxx
∂= =
∂∂
= − = −∂
1r LC
ω = rkM
ω =
08-I-19
0x X X= −
12
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 andmomentumoperators donot commute
08-I-20
X and P are conjugate variables
+Qφ
-Q
FLUX AND CHARGE DO NOT COMMUTE
V
I ˆ ˆ, iQφ⎡ ⎤⎣ =⎦
This fundamental resultarises from the
correspondence principleapplied to the LC oscillator
We can also obtain this result from quantum field theory through the commutationrelations of the electric and magnetic field. See third lecture in this course.
08-I-21
Not every pairs of variables can satisfy such commutation relation. As A.J. Leggett once remarked in a seminar presenting his seminal work on macroscopic quantummechanics: " We cannot quantize the equations of the stock market!"
13
+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 operators
08-I-22
φ
φ
THERMAL EXCITATION OF LC CIRCUIT
Br k Tω5 GHz 10mK
rωhI
E
Can place the circuitin its ground state
08-I-22a
14
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
08-I-23
φ
φ
EFFECT OF DAMPING
important:negligible dissipation dissipation broadens energy levels
E
112 2n r
r
iE n
RC
ω
ω
⎡ ⎤⎛ ⎞= + +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦=
QQ
seewhylater
08-I-24
15
φ
φ
E
ALL TRANSITIONS ARE DEGENERATE
CANNOT STEER THE SYSTEM TO AN ARBITRARY STATEIF PERFECTLY LINEAR
rωh
08-I-25
Potential energy
Position coordinate
NEED NON-LINEARITY TO FULLYREVEAL QUANTUM MECHANICS
08-I-26
16
1. Review of classical radio-frequency circuits
2. Quantum information processing
3. Quantum-mechanical LC oscillator
4. A non-dissipative, non-linear element: the Josephson junction
5. Energy levels of the Cooper Pair Box
6. Summary of questions addressed by this course
08-I-5d
OUTLINE
JOSEPHSON JUNCTIONPROVIDES A NON-LINEAR INDUCTOR
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
= =
08-I-27b
17
2
0
212 ˆ
ˆ 14e
erm
AH epπε
⎛ ⎞= − −⎜ ⎟⎝ ⎠
RESTOF
CIRCUIT
extq
THE hamiltonian:(we mean it!)
( )21 2cos2 ext
jJj
CH EQ q eφ
= − −
COUPLING PARAMETERSOF THE JOSEPHSON JUNCTION "ATOM"
Comparable with lowest ordermodel for hydrogen atom
08-I-28a
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
08-I-29
reduced offset charge
valid foropaque barrier
18
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 Next08-I-30
TUNNEL JUNCTIONSIN REAL LIFE
100nm
EJ ~ 50K
EJ ~ 0.5K
ωp ~ 30-40GHz
credit L. Frunzio and D. Schuster
credit I. Siddiqi and F.Pierre
08-I-31
19
1. Review of classical radio-frequency circuits
2. Quantum information processing
3. Quantum-mechanical LC oscillator
4. A non-dissipative, non-linear element: the Josephson junction
5. Energy levels and transitions of the Cooper Pair Box
TU Delft, NEC, NTT, MITUC Berkeley, IBM, SUNYIPHT Jena ....
CEA Saclay, YaleNEC, Chalmers, JPL, ...
2ehφ
2ehφ
2ehφ
08-I-32a
20
U
Φ
island
COOPER PAIR"BOX"
Bouchiat et al. 97Nakamura, Pashkin & Tsai 99Vion et al. 2002
2 control parameters CgU/2e and Φ/Φ0
U
ΦCg
08-I-33
ω01
ω12ω02 2
Sufficient to control the junction as a two level system
SPECTROSCOPY OF A COOPER PAIR BOX
Anharmonicity:
01 12 455MHz CEω ω− =
ω01
ω02 2
ω12
J. Schreier et al. 08
Slide courtesy of J. Schreier and R. Schoelkopf 08-I-34
21
ANHARMONICITY vs CHARGE SENSITIVITY
Cooper pair box soluble in terms of Mathieu functions (A. Cottet, PhD thesis, Orsay, 2002)
08-I-35
290 μm
ANHARMONICITY vs CHARGE SENSITIVITYIN THE LIMIT EC/EJ << 1
peak-to-peak charge modulation amplitude of level m:
anharmonicity:
TRANSMON: SHUNT JUNCTION WITH CAPACITANCE
Courtesy of J. Schreier and R. Schoelkopf
J. Koch et al. 07
( )12 01
12 01 / 2 8C
J
EE
ω ωω ω
−→
+
08-I-36
22
1. Review of classical radio-frequency circuits
2. Quantum information processing
3. Quantum-mechanical LC oscillator
4. A non-dissipative, non-linear element: the Josephson junction
5. Energy levels and transitions of the Cooper Pair Box
6. Summary of questions addressed by this course
08-I-5f
OUTLINE
CAN WE BUILD ALL THE QUANTUM INFORMATIONPROCESSING PRIMITIVES OUT OF SIMPLE CIRCUITSAND CONNECT THESE CIRCUITS TO PERFORM ANY
DESIRED FUNCTION?
08-I-37
23
TRANSM. LINES, WIRES
COUPLERS
CAPACITORS
GENERATORS
AMPLIFIERS
JOSEPHSON JUNCTIONS
FIBERS, BEAMS
BEAM-SPLITTERS
MIRRORS
LASERS
PHOTODETECTORS
ATOMS
DRAWBACKS OF CIRCUITS:
ADVANTAGES OF CIRCUITS: - PARALLEL FABRICATION METHODS- LEGO BLOCK CONSTRUCTION OF HAMILTONIAN- ARBITRARILY LARGE ATOM-FIELD COUPLING
ARTIFICIAL ATOMS PRONE TO VARIATIONS
QUANTUM OPTICS QUANTUM RF CIRCUITS
~
08-I-38
HOW DO WE TRANSLATE FROM THE LANGUAGEOF ELECTRICAL SIGNALS COUPLED
IN A NON-LINEAR CIRCUITINTO THE LANGUAGE OF
PHOTONS INTERACTING WITH ATOMS(EMISSION, ABSORPTION, SCATTERING, DETECTION)?
HOW DO WE DO TREAT QUANTUM-MECHANICALLYA DISSIPATIVE, NON-LINEAR, OUT-OF-EQUILIBRIUM
ENGINEERED SYSTEM?
08-I-39
24
SELECTED BIBLIOGRAPHYBooksBraginsky, V. B., and F. Y. Khalili, "Quantum Measurements" (Cambridge University Press,
Cambridge, 1992) Cohen-Tannoudji, C., Dupont-Roc, J. and Grynberg, G., "Atom-Photon Interactions"
(Wiley, New York, 1992)Haroche, S. and Raimond, J-M., "Exploring the Quantum" (Oxford University Press, 2006)Mallat, S. M., "A Wavelet Tour of Signal Processing" (Academic Press, San Diego, 1999) Nielsen, M. and Chuang, I., “Quantum Information and Quantum Computation” (Cambridge, 2001)Pozar, D. M., "Microwave Engineering" (Wiley, Hoboken, 2005)Tinkham, M., "Introduction to Superconductivity" (2nd edition, Dover, New York, 2004)
Review articlesCourty J. and Reynaud S., Phys. Rev. A 46, 2766-2777 (1992)Devoret M. H. in "Quantum Fluctuations", S. Reynaud, E. Giacobino, J. Zinn-Justin, Eds.
(Elsevier, Amsterdam, 1997) p. 351-385Makhlin Y., Schön G., and Shnirman A., Nature (London) 398, 305 (1999).Devoret M. H., Wallraff A., and Martinis J. M., e-print cond-mat/0411174Blais A., Gambetta J., Wallraff A., Schuster D. I., Girvin S., Devoret M.H., Schoelkopf R.J.
Phys. Rev. (2007) A 75, 032329
ArticlesKoch J, et al. Phys. Rev. A 76, 042319 (2007)Schreier J., et al, .arXiv:0712.3581, J. A. Schreier, et al., Phys. Rev. B 77, 180502(R) (2008) Houck A. A., et al. arXiv:0803.4490, A. A. Houck, et al., Phys. Rev. Lett. 101, 080502 (2008)
