introduction to quantum superconducting circuits: … · beam-splitters mirrors lasers...
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INTRODUCTION TO QUANTUMSUPERCONDUCTING CIRCUITS:
PART 1
Applied Physics and Physics, Yale University
N. BERGEAL (ESPCI Paris)F. SCHACKERT A. KAMAL B. HUARD (ENS Paris)A. MARBLESTONE (MIT)
R. VIJAY (U.C.Berkeley)I. SIDDIQI (U.C.Berkeley)C. RIGETTIM. METCALFE (NIST)V. MANUCHARIAN
N. MASLUKM. BRINKK. GEERLINGSL. FRUNZIOM. DEVORET
RutgersDec. 09
Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve
W.M. KECK
LABQUAN RONICSUM – MECHANICAL
ELEC
MESOSCOPIC ELECTRONS vs PHOTONSINDEPENDENT ELECTRON INTERFERENCES DIRECTLY REVEAL QUANTUM MECHANICS
INDEPENDENT PHOTON INTERFERENCESDO NOT DIRECTLY REVEAL QUANTUM MECHANICS
22 eAi eU im t
⎡ ⎤ ∂⎛ ⎞∇ − + Ψ = Ψ⎢ ⎥⎜ ⎟ ∂⎝ ⎠⎢ ⎥⎣ ⎦
Schrödinger's equation
( ) ( )c E icB i E icBt
∂∇× + = +
∂
Maxwell's equations
Interactions provide further quantum effects
Planck's constant drops out!Need ph-ph interactions....
Φ
Example: Sharvin-Sharvin & A.B. effects
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
~
John Clarke & Frank K. Wilhelm, Nature 453, 1031-1042 (2008)
Robert J. Schoelkopf & Steve M. Girvin, Nature 451, 664-669 (2008)
Michel Devoret & John Martinis, Quant. Inf. Proc., 3, 351-380 (2004)
Michel Devoret, in "Quantum Fluctuations", S. Reynaud, E. Giacobino,J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p. 351-385
SELECTED BIBLIOGRAPHYON QUANTUM CIRCUITS
OUTLINE
TODAY: PRESENT BASIC CIRCUIT ELEMENTS
TOMORROW: COMPARE QUBIT CIRCUITS
SUPERCONDUCTING CIRCUIT ELEMENTS
CAPACITANCE INDUCTANCE
RESISTANCE
CAPACITANCE INDUCTANCE
RESISTANCE
2
2CeEC
=2
24LEe L
=2
24JJ
Ee L
=
SUPERCONDUCTING CIRCUIT ELEMENTS
L LL
C C C C
TRANSMISSION LINES ARE JUSTL's AND C's
L
LIKEWISE, EVERY LINEAR PASSIVE RECIPROCAL RF COMPONENT IS BUILTFROM INDUCTANCES AND CAPACITANCES
RESISTORS CAN BE THOUGHT OF AS INFINITE TRANSMISSION LINES(NYQUIST, CALDEIRA & LEGGETT)
L
DYNAMICAL VARIABLES OF CIRCUITCIRCUIT : ARBITRARY 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
linear inductance:
linear capacitance:
V = L dI/dt
I = C dV/dt
CONSTITUTIVE RELATIONSOF CIRCUIT ELEMENTS
E = LI2/2= Φ2/2L
E = CV2/2= Q2/2L
non-linear inductance: E = f(Φ)f non-quadratic
Inductances and capacitances are "bottles" for electric and magnetic fields
KIRCHHOFF’S LAWS
ba
c
b
d ba
c
b
d
branchesaround loop
0Vλ
λ =∑branchestied to node
0Iν
ν =∑
CONSTITUTIVE RELATIONS+
KIRCHHOFF'S LAWS
MAXWELL'SEQUATIONS
ON A NETWORK
QUANTUM TREATMENT OF CIRCUITS
Vβ
Iβ
,ˆ ˆ iQβ βφ⎡ ⎤⎦ =⎣
For every branch β in the circuit:
restof
circuit
branch β
Need to take branch fluxand branch charge as basicvariables:
( ) ( )
( ) ( )
' '
' '
t
tQ t
t dt
t dI
V
t
t
ββ
ββφ−∞
−∞
=
=
∫∫
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
Method of nodes
1) Choose a reference electrode(ground)
Method of nodes
1) Choose a reference electrode(ground)
2) Choose a spanning tree(accesses every node, no loop)
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
Method of nodes
1) Choose a reference electrode(ground)
2) Choose a spanning tree(accesses every node, no loop)
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
Method of nodes
1) Choose a reference electrode(ground)
2) Choose a spanning tree(accesses every node, no loop)
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
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)
φa
φb φc
φe
φgφf
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
φd
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)
4) Node flux is sum of branchfluxes to ground (closure branchfluxes are expressed as differencesbetween node fluxes)
Φ1
Φ2 Φ3
Φ4
Φ6
Φ5
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
Φ7
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 is sum of branchfluxes to ground (closure branchfluxes are expressed as differencesbetween node fluxes)
φh
φh = Φ 7 - Φ 6 + cst
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
φd
φf
Φ4
Φ6
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 is sum of branchfluxes to ground (closure branchfluxes are expressed as differencesbetween node fluxes)
φh
φh = Φ 7 - Φ 6 + cst
FINDING A COMPLETESET OF INDEPENDENT VARIABLES
φd
φf
( ) ( ) cstn nγγ γφ+ −
Φ Φ= − +
tree branchesleading to
n
nβ
βφΦ = ∑
Φ4
Φ6
TWO METHODS FOR DEFINING A COMPLETESET OF INDEPENDENT VARIABLES
Method of nodes
Method of loops
Defines loop charges
+Qφ
-Q
HAMILTONIAN OF CIRCUIT
V
X
Mk
electrical world mechanical analog world
I
f
( )0
H QQ C
HQL
φ
φ φφ
∂= =
∂
− −∂= − =
∂( )0
H PXP MHP k X XX
∂= =
∂∂
= − = − −∂
1r LC
ω = rkM
ω =
( )220
2 2QHC L
φ φ−= +
( )220
2 2k X XPH
M−
= +
MICROFABRICATION L ~ 3nH, C ~ 10pF, ωr /2π ~ 2GHz
SIMPLEST EXAMPLE: SUPERCONDUCTING LC OSCILLATOR CIRCUIT
A SUPERCONDUCTING CIRCUITBEHAVING 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 Rydberg atom SuperconductingLC oscillator
L C
velocity of electron → voltage across capacitorforce on electron → current through inductor
DEGREE OF FREEDOM IN ATOM vs CIRCUIT
+Qφ
-Q
φ
E
LC CIRCUIT AS A QUANTUMHARMONIC OSCILLATOR
rωh
( )
†
ZPF ZPF ZPF ZPF
ZPF
ZPF
†
ˆ ˆ ˆ ˆˆ ˆ;
2
2
ˆ 1ˆ ˆ 2
r
r
r
Q Qa i a iQ Q
L
Q C
H a a
φ φφ φ
φ ω
ω
ω
= + = −
=
=
= +
annihilation and creation operators†ˆ ˆ, 1a a⎡ ⎤ =⎣ ⎦
trapped photons!
φ
φ
E
ALL TRANSITIONS BETWEEN QUANTUM LEVELS ARE DEGENERATE
IN PURELY LINEAR CIRCUITS!
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 =
0
0JL
Iφ
=
0I
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
JJ
LEφ
=
bare Josephson potential
2 JE
TRANSMISSION LINE AS 1D BOSON FIELD
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
CHARGE AND FLUX IN CONTINUUM LIMIT
xδ
( )xΦ ( )x xδΦ +
C xδ( )x xδΠ ( )x x xδ δΠ +
( ) ( ),x x x iδ⎡ ⎤Φ Π =⎣ ⎦
( ) ( ), 0x x x xδ δ⎡ ⎤Φ Π + =⎣ ⎦
L xδ
C xδ
L xδL xδ
( ) ( )2 21 1ˆ ˆ ˆ
2 2H dx x x
C L+∞
−∞
⎧ ⎫⎡ ⎤ ⎡ ⎤= Π + ∇Φ⎨ ⎬⎣ ⎦ ⎣ ⎦⎩ ⎭∫
Hamiltonian : energy density as a function of field and conjugate momentum:
( ) ( ) ( )1 2 1 2ˆ ˆ,x x i x xδ⎡ ⎤Φ Π = −⎣ ⎦0xδ →
COMPATIBILITY WITH STANDARD QED
( )1 1 1, ,B x y z ( )2 2 2, ,E x y z
xz
y
( ) ( )1
1 1ˆˆ , ,b
b
x y h
zyx dx dy B x y z
+
−∞Φ = ∫ ∫ ( ) ( )2 0 2 2
ˆˆ , ,f
f
z w
yzx dz E x y zε
+Π = ∫
h
w
( ) ( ) ( ) ( ) ( )1 1 1 2 2 2 1 2 1 2 1 20 1
ˆ ˆ, , , , ,z yiB x y z E x y z x x y y z z
xδ δ δ
ε∂⎡ ⎤ = − − −⎣ ⎦ ∂
Commutation relations between field operators in standard QED:
Flux between strips: Strip charge per unit length:
For simplest geometry, consider stripline waveguide:
( ) ( ) ( )1 2 1 2ˆ ˆ,x x i x xδ⎡ ⎤Φ Π = −⎣ ⎦ OK!
END OF 1ST LECTURE
INTRODUCTION TO QUANTUMSUPERCONDUCTING CIRCUITS:
PART 2
Applied Physics and Physics, Yale University
N. BERGEAL (ESPCI Paris)F. SCHACKERT A. KAMAL B. HUARD (ENS Paris)A. MARBLESTONE (MIT)
R. VIJAY (U.C.Berkeley)I. SIDDIQI (U.C.Berkeley)C. RIGETTIM. METCALFE (NIST)V. MANUCHARIAN
N. MASLUKM. BRINKK. GEERLINGSL. FRUNZIOM. DEVORET
RutgersDec. 09
Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve
W.M. KECK
LABQUAN RONICSUM – MECHANICAL
ELEC
OUTLINE
YESTERDAY: PRESENT BASIC CIRCUIT ELEMENTS
TODAY: COMPARE QUBIT CIRCUITS
QUANTUM TREATMENT OF CIRCUITS
Vβ
Iβ
,ˆ ˆ iQβ βφ⎡ ⎤⎦ =⎣
For every branch β in the circuit:
restof
circuit
branch β
Branch flux and branch chargeare primary variables:
( ) ( )
( ) ( )
' '
' '
t
tQ t
t dt
t dI
V
t
t
ββ
ββφ−∞
−∞
=
=
∫∫
IRREVERSIBLE/REVERSIBLECHARGE TRANSFER
S I STUNNEL
JUNCTION
e tunnelingquasi-particleenergy
before after
2Δrate
IRREVERSIBLE/REVERSIBLECHARGE TRANSFER
S I STUNNEL
JUNCTION
e tunneling
2e CP tunneling
quasi-particleenergy
before after
quasi-particleenergy
2Δrate
matrixelement
DYNAMICS OF JOSEPHSON ELEMENTFROM CHARGE POINT OF VIEW
(IN ABSENCE OF Q.P. TUNNELING)
2Q Ne
=0N 0 1N +0 1N −
21 /2 16J t
h eM E G= = Δ
integer
Charge states:
N̂ N N N=
Josephson tunneling hamiltonian in charge basis:
( )ˆ 1 12
JJ
N
EH N N N N= − + + +∑
Hopping
FROM NUMBER TO "PHASE" REPRESENTATION
( )
( )
( )0
ˆ ˆ
ˆ 1 12
e e2
cos
cos /
ˆˆ
JJ
N
i iJ
J
J
EH N N N N
E
E
E
ϕ ϕ
ϕ
φ φ
−
= − + + +
= − +
= −
= −
∑
ϕ(from expression oftranslation operator)
ˆ /ˆ2eϕ φ= ˆˆ, N iϕ⎡ ⎤ =⎣ ⎦ˆ ˆˆ ˆe e 1i iN Nϕ ϕ− = −
00 2 2e
φπ
Φ= =
Josephson's runs on a line but tunnel hamiltonian is periodicϕ
RESTOF
CIRCUIT CJEJ U(t)
( )Z ω
( ) ( )2
1
,t
t E x dxdτ τφ−∞
= ∫ ∫
ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT
N.B. The electric field Eencompasses here allcontributions of the forceon the electrons doing work, including those usuallycalled chemical potentialeffects.
RESTOF
CIRCUIT CJEJ U(t)
( )Z ω
( ) ( )2
1
,t
t E x dxdτ τφ−∞
= ∫ ∫
CQ
ext
tQ Idτ
−∞= ∫
2Q eN=
Equation of motion:
( )0
cos , ,....J JC E Iφφ φ φφ φ
⎡ ⎤⎛ ⎞∂+ − =⎢ ⎥⎜ ⎟∂ ⎝ ⎠⎣ ⎦
Can be obtained in general from a Lagrangian:
0ddt φφ
∂ ∂− =
∂∂L L ( )2
ext e0
xt , ,..c .os2
JJ J
C E φ φφφφ
= + = + + LL L L
ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT
TWO CHARACTERISTIC ENERGIES OF ENVIRONMENT
( )tot
0
Imlim
YC
ω
ωωΣ →
⎡ ⎤⎣ ⎦=
Total environment admittance: ( ) ( )1tot JY iC Zω ω ω−= +
Effective shunt capacitance of junction:
Effective shunt inductance of junction ( )eff 0
1lim Imtot
LYω ω ω→
⎧ ⎫⎡ ⎤⎪ ⎪= ⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭
Coulomb charging energy2
2CeECΣ
=(electron chargeappears here insteadof Cooper pair chargefor convenience insome formulas)
Inductive energy( )2
ff
/ 2L
e
eE
L=
h(form chosen foreasy comparisonwith Josephsonenergy)
TWO BASIC SUPERCONDUCTING "ATOMS"
Φext
flux
JL
JCJL L>
U
charge
JL
JCgC
I
supercon-ductingloop
supercon-ductingisland
"HYSTERETIC RF SQUID" "COOPER PAIR BOX"
ϕ
N
12
ϕπ
Δ 1NΔ <
Friedman et al. (2000) Bouchiat et al. (1998), Nakamura, Pashkin, Tsai (1999)
J L CE E E> 0;L J CE E E=
∈∈
2ϕπ
12
− 12
+
/e gi C U eπ
2ϕπ
0 1+ 2+1−
2EJ
PHASE POTENTIALS OF SQUID AND BOX
ext2 /eΦ
J L CE E E> 0;L J CE E E=
RICH LEVEL STRUCTURE, BUT EXTREME SENSITIVITY TO NOISE
SQUID BOX
max. curvature
L JE E= −
09-III-10
E
2ϕπ
PHASE-CHARGE DUALITY
J L CE E E> 0;L J CE E E=
SQUID
N0 1+ 2+1−
BOXE
182C gE N −
ext2 LE ϕπ π−
0 1+ 2+1−
CHARGE QUBIT FAMILY
Q=2Ne
THE SINGLE COOPER PAIR BOXARTIFICIAL ATOM
Φ
U
U
ΦgC
Bouchiat et al. (1998)Nakamura, Tsai and Pashkin (1999) 0.5J CE E
U
QUANTRONIUM
Φ
U ΦgC
Q=2Ne
Vion et al. (2001)4J CE E
Q=2Ne
TRANSMON COOPER PAIR BOX
U
Φ
U
gC
Φ
J. Koch et al. (2007)A. Houck et al. (2007)J. Schreier et al. (2007)
50J CE E
ANHARMONICITY vs OFFSET CHARGEINSENSITIVITY IN COOPER PAIR BOX
Koch et al. (2007)
"QUANTRONIUM"
"TRANSMON"
Cottet et al. (2002)
"C.P. BOX"gapEJ
FLUX QUBIT FAMILY
EXAMPLES OF QUANTUM CIRCUITSBELONGING TO THE RF-SQUID TYPE
Chiorescu, Nakamura, Harmans & Mooij,Science 299, 1869 (2003).
Friedman, Patel, Chen, Tolpygo and J. E. Lukens,Nature 406, 43 (2000).
"phase" qubit
Steffen et al., Phys. Rev. Lett. 97, 050502 (2006)
Readout
Readout
"flux" qubit
Readout
FLUXONS PLAY WITH FLUX QUBITSA ROLE SIMILAR TO WHAT
COOPER PAIRS PLAY WITH CHARGE QUBITS
Inductive energy
0
ΦΦ0 1-1
~ exp J
C
EE
α⎛ ⎞
−⎜ ⎟⎜ ⎟⎝ ⎠
OUR RECENT CHILD INFLUX QUBIT FAMILY
Φext
flux
JL
JCJL L∼
charge
JL
JCgC
I
supercon-ductingloop
"RF SQUID" "COOPER PAIR BOX"
ϕ N
12
ϕπ
Δ 1NΔ <
ΦN
U
UNfluxnoise! charge
noise!
NOISE IN THE TWO BASIC QUBITS
supercon-ductingisland
U
JL
JCgC
JL L
FLUXONIUM IDEA: SHUNT A COOPER PAIR BOXAT DC, LEAVE IT UNSHUNTED AT RF
UN
Manucharyan. et al., Science, 326, 113; arxiv 0906.0831Koch. et al., Phys. Rev. Lett, to appear; arxiv 0902.2980Manucharyan. et al., submitted to Nature, arxiv 0910.3039
Readout f0 = 8 GHz; Q=400
Small junction:EJ/EC = 3.6Array junctions (N =43):EJA/ECA = 28
PRACTICAL IMPLEMENTATION
Every island is shunted by at least one large junction: array performs as inductor
read/write
EC = ½ e2/(CJ+Cc)=2.5 GHz
EJ = ½ (Φ0/2π)2/LJ=8.9GHz
EL = ½ (Φ0/2π)2/L=0.52GHz
ωR=(LRCR)-1/2/2π = 8.2GHz
ZR=(LR/CR)1/2 = 82Ω
CHIP AND MODEL
“Atom”
“Cavity”
Coupling constant
g = ωR(ZR/2RQ)1/2 /(1+CJ/Cc)= 135MHz (RQ=1 kΩ)
CJ/Cc=11
PARAMETERS OF THE ARRAY
EJA=22.5GHz
ECA=0.8GHz
Cg~CJ/2000
N =43 <(CJ/Cg)1/2
TWO-TONE SPECTROSCOPY vs EXTERNAL FLUX
0-3 transition
0-2 transition
0-1 transition
resonator & vacuum Rabi
Inset: 0-2 transition symmetry-forbiddenat zero flux
More than 106 points,taken over >72 hoursw/o jumps or drifts!
Parasitic resonance in the array(accounted by a minor model correction)
NO OFFSET CHARGE
0
2Φ
Φ≈
THY vs EXP CONFIRMS SINGLE COOPER PAIR REGIME
V.Manucharyan, J.Koch, L.Glazman, M.Devoret,Science, 326, 113; arxiv 0906.0831
LOCATING FLUXONIUM ON THE SUPERCONDUCTING QUBIT MAP
ϕπ2 π4π2
Φext =1/2 Φ0
Pot
entia
l ene
rgy
Φext =1/4 Φ0 Φext = 0
~2EJ 2π2EL
Inductive energy
0
ΦΦ0 1-1
~350MHz
A QUBIT OPERATING FOR ALLVALUES OF FLUX
9GHz
TWO-TONE SPECTROSCOPY vs EXTERNAL FLUX 0
2Φ
Φ≈
A 370MHz ATOM MEASURED THROUGH A 8.2GHz CAVITY?
g12 = g <1|N|2> ~ 100MHzω12 - ωR ~ 1GHzχ= g12
2/(ω12 - ωR) ~10MHz
atom transition 0-1: store/manipulateatom transition 1-2: couple to readout
At the same time ω01 << ωR
refle
cted
pha
se (d
eg)
time (μs)
TIME DOMAIN COHERENCE
ext 0/Φ ΦExternal flux
Tim
e, n
sFr
eque
ncy,
GH
z
COHERENCE AS A FUNCTION OF FLUX BIAS
quasiparticles, quantum phase slips?
Q1=130,000
1/ 012 6
010fT
Aω−
∂Φ∂
=⎡ ⎤= Φ⎣ ⎦
ext 0/Φ ΦExternal flux
COHERENCE AS A FUNCTION OF FLUX BIASFr
eque
ncy,
GH
zTi
me,
ns
CONCLUSIONS AND PERSPECTIVES
QUANTUM CIRCUITS OFFER A RICH PLAYGROUND FOR EMULATING EXISTING MANY-BODY QUANTUMSYSTEMS AND CREATING NEW ONES
PRESENT CHALLENGE IS QUBIT WITH BUILT-INERROR-CORRECTION, AND MORE GENERALLYQUANTUM FEEDBACK
STRONG COUPLING, NON-PERTURBATIVE REGIME IS EASILY ACCESSIBLE
ϕ
MECHANICAL ANALOG OF RF SQUID
spring : loop inductance
moment of inertia of pendulum : junction capacitance
torque due gravity : Josephson current
potential energy of pendulum : Josephson energy
angle of pendulum : gauge invariant phase difference
ϕ
U( )ϕ
MECHANICAL ANALOG OF COOPER PAIR BOX
rotatingmagnet
angular velocity:
0
gCC
UφΣ
Ω =
compass needle
needleangular momentum:
Nh
torque on needle due to magnetvelocity of needle in magnet frame
current through junctionvoltage across junction
ϕ̂
Introduction to quantum superconducting circuits
1) Usual 2deg physics: circuit with normal electrons.quantum circuits means that the electron obeys S. equation over the whole scalof circuit. Electron propagate as Fermi waves
2) What about electromagnetic Bose wave propagation?
3) Need non-linearity. But usually, non-linear element are alsodissipative. Dissipation not good for QM
4) Josephson element: non-linear, non-dissipative, point-like
5) Lumped element electromagnetism: L and C's
6) L bottle for B field. C bottle for E field.
( ) ( ) ( )2
, , ,U x t x t i x tm t
⎡ ⎤− ∂Δ + Ψ = Ψ⎢ ⎥ ∂⎣ ⎦
( ) ( )c E icB i E icBt
∂∇× + = +
∂
Constitutive equations+ Kirchhoff equations for circuits are equivalent to Maxwell's equation
Define flux and charge for one element
Flux and charge are conjugate variablesHow do we understand it?
2 choices flux is position, charge is positionConnections between elements
Example of harmonic oscillator
Always in the correspondence limit
The Josephson junction non linear inductance
Two superconducting electrodes: islands
1sole degree of freedom in an island : its number of Cooper pair
Charge hopping between two islands (zero external field)
Gives back Josephson formula
Arrays of Josephson junctions tends toward inductancesQuantum phase slips
Conclusions: finite networks are caricature of distributed systemsinfinite networks are more powerful than distributed systemssingle junction is non-linear inductance. Infinite junction array is linear inductance
Artificial Josephson junction atoms
3 types of energies
Capacitive environem: easyInductive environment: more sutble