INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: … · beam-splitters mirrors lasers photodetectors atoms ... resistors can be thought of as infinite transmission lines (nyquist,

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


2) Choose a spanning tree(accesses every node, no loop)


Method of nodes




Method of nodes




Method of nodes



3) Select tree branch fluxes(closure branches left out)

φa

φb φc

φe

φgφf


φd

Method of nodes




4) Node flux is sum of branchfluxes to ground (closure branchfluxes are expressed as differencesbetween node fluxes)

Φ1

Φ2 Φ3

Φ4

Φ6

Φ5


Φ7

Method of nodes




Φ1

Φ2 Φ3 Φ7

Φ5

φg

φe

Φ 7 = φd + φe + φgexample:


φh

φh = Φ 7 - Φ 6 + cst


φd

φf

Φ4

Φ6

Method of nodes




Φ1

Φ2 Φ3 Φ7

Φ5

φg

φe

Φ 7 = φd + φe + φgexample:


φh

φh = Φ 7 - Φ 6 + cst


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


RICH LEVEL STRUCTURE, BUT EXTREME SENSITIVITY TO NOISE

SQUID BOX

max. curvature

L JE E= −

09-III-10

E

2ϕπ

PHASE-CHARGE DUALITY


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

Documents

INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: … · beam-splitters mirrors lasers photodetectors atoms ... resistors can be thought of as infinite transmission lines (nyquist,