introduction to quantum superconducting circuits - Rutgers Physics

INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: PART 1 QUAN

ELEC

UM – MECHANICAL

RONICS LAB

Applied Physics and Physics, Yale University R. VIJAY (U.C.Berkeley) I. SIDDIQI (U.C.Berkeley) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN

N. BERGEAL (ESPCI Paris) F. SCHACKERT A. KAMAL B. HUARD (ENS Paris) A. MARBLESTONE (MIT)

N. MASLUK M. BRINK K. GEERLINGS L. FRUNZIO M. DEVORET

Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve W.M. KECK

Rutgers Dec. 09

MESOSCOPIC ELECTRONS vs PHOTONS INDEPENDENT ELECTRON INTERFERENCES DIRECTLY REVEAL QUANTUM MECHANICS Schrödinger's equation

Example: Sharvin-Sharvin & A.B. effects Φ

2 ⎡ 2⎛ ⎤ eA ⎞ ∂ ∇ − + Ψ = Ψ i eU i ⎢ ⎜ ⎥ ⎟ ∂t ⎠ ⎢⎣ m ⎝ ⎥⎦

Interactions provide further quantum effects

INDEPENDENT PHOTON INTERFERENCES DO NOT DIRECTLY REVEAL QUANTUM MECHANICS Maxwell's equations

(

)

(

∂ c∇ × E + icB = i E + icB ∂t Planck's constant drops out! Need ph-ph interactions....

)

QUANTUM OPTICS

QUANTUM RF CIRCUITS

FIBERS, BEAMS

TRANSM. LINES, WIRES

BEAM-SPLITTERS

COUPLERS

MIRRORS

CAPACITORS

LASERS

~

GENERATORS

PHOTODETECTORS

AMPLIFIERS

ATOMS

JOSEPHSON JUNCTIONS

ADVANTAGES OF CIRCUITS: - PARALLEL FABRICATION METHODS - LEGO BLOCK CONSTRUCTION OF HAMILTONIAN - ARBITRARILY LARGE ATOM-FIELD COUPLING

DRAWBACKS OF CIRCUITS:

ARTIFICIAL ATOMS PRONE TO VARIATIONS

SELECTED BIBLIOGRAPHY ON QUANTUM CIRCUITS Michel Devoret, in "Quantum Fluctuations", S. Reynaud, E. Giacobino, J. Zinn-Justin, Eds. (Elsevier, Amsterdam, 1997) p. 351-385 Michel Devoret & John Martinis, Quant. Inf. Proc., 3, 351-380 (2004) Robert J. Schoelkopf & Steve M. Girvin, Nature 451, 664-669 (2008)

John Clarke & Frank K. Wilhelm, Nature 453, 1031-1042 (2008)

OUTLINE TODAY:

TOMORROW:

PRESENT BASIC CIRCUIT ELEMENTS

COMPARE QUBIT CIRCUITS

SUPERCONDUCTING CIRCUIT ELEMENTS

CAPACITANCE

INDUCTANCE

RESISTANCE

SUPERCONDUCTING CIRCUIT ELEMENTS e2 EC = 2C

CAPACITANCE

EL =

2 2

4e L

EJ =

INDUCTANCE

RESISTANCE

2

4e 2 L J

TRANSMISSION LINES ARE JUST L's AND C's

L

L C

L C

L C

L C

LIKEWISE, EVERY LINEAR PASSIVE RECIPROCAL RF COMPONENT IS BUILT FROM INDUCTANCES AND CAPACITANCES RESISTORS CAN BE THOUGHT OF AS INFINITE TRANSMISSION LINES (NYQUIST, CALDEIRA & LEGGETT)

DYNAMICAL VARIABLES OF CIRCUIT CIRCUIT : ARBITRARY NETWORK OF ELECTRICAL ELEMENTS node n

element

Vnp

branch np

loop

Inp node p

TWO DYNAMICAL VARIABLES CHARACTERIZE THE STATE OF EACH DIPOLE ELEMENT AT EVERY INSTANT: Voltage across the element:

Vnp ( t ) = ∫ E ⋅ d p

n

Current through the element:

I np ( t ) = ∫∫ j ⋅ dσ np

Signals: any linear combination of these variables

CONSTITUTIVE RELATIONS OF CIRCUIT ELEMENTS linear inductance:

V = L dI/dt non-linear inductance:

linear capacitance:

I = C dV/dt

E = LI2/2= Φ2/2L E = f(Φ) f non-quadratic

E = CV2/2= Q2/2L

Inductances and capacitances are "bottles" for electric and magnetic fields

KIRCHHOFF’S LAWS b

c a b

b d

∑ λ Vλ = 0

branches around loop

c

d a b

∑ν

branches tied to node

Iν = 0

CONSTITUTIVE RELATIONS + KIRCHHOFF'S LAWS

MAXWELL'S EQUATIONS ON A NETWORK

QUANTUM TREATMENT OF CIRCUITS Iβ rest of circuit

branch β

Vβ

Need to take branch flux and branch charge as basic variables:

φβ ( t ) = ∫ Vβ ( t ')dt ' t

−∞

Qβ ( t ) = ∫ I β ( t ')dt ' t

−∞

For every branch β in the circuit:

⎡φˆβ , Qˆ β ⎤ = i ⎣ ⎦

FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes

1) Choose a reference electrode (ground)


1) Choose a reference electrode (ground) 2) Choose a spanning tree (accesses every node, no loop)






φb

φc φa

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)

φd

φg

φf φe

FINDING A COMPLETE SET OF INDEPENDENT VARIABLES Φ2 Method of nodes

Φ3

Φ7

Φ1 Φ6

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 branch fluxes to ground (closure branch fluxes are expressed as differences between node fluxes)

Φ4

Φ5


Φ3

φh

Φ1 φf


φd

Φ4

example:

Φ7 φg Φ6 φe

Φ5

Φ 7 = φd + φe + φg φh = Φ 7 - Φ 6 + cst


Φ3

φh

Φ1 φf


φd

φg Φ6 φe

Φ4

example:

Φ7

Φ5

Φ 7 = φd + φe + φg φh = Φ 7 - Φ 6 + cst

Φn =

∑

tree branches β leading to n

φβ

φγ = Φ n (γ ) − Φ n (γ ) + cst +

−

TWO METHODS FOR DEFINING A COMPLETE SET OF INDEPENDENT VARIABLES Method of nodes

Method of loops

Defines loop charges

HAMILTONIAN OF CIRCUIT mechanical analog world

electrical world

φ

I

X +Q

-Q

V

M

Q (φ − φ0 ) H= + 2C 2L 2

φ=

∂H Q = ∂Q C

∂H − (φ − φ0 ) Q=− = L ∂φ

ωr =

f

2

1 LC

k

k ( X − X0 ) P H= + 2M 2 2

∂H P = X= ∂P M ∂H P=− = −k ( X − X 0 ) ∂X

2

ωr =

k M

A SUPERCONDUCTING CIRCUIT BEHAVING LIKE AN ATOM?

SIMPLEST EXAMPLE: SUPERCONDUCTING

MICROFABRICATION

LC OSCILLATOR

CIRCUIT

L ~ 3nH, C ~ 10pF, ωr /2π ~ 2GHz

ELECTRONIC FLUID SLOSHES BACK AND FORTH FROM ONE PLATE TO THE OTHER, INTERNAL MODES FROZEN BEHAVES AS A SINGLE CHARGE CARRIER

DEGREE OF FREEDOM IN ATOM vs CIRCUIT Superconducting LC oscillator

Example of Rydberg atom

L

velocity of electron → force on electron →

voltage across capacitor current through inductor

C

LC CIRCUIT AS A QUANTUM HARMONIC OSCILLATOR E

φ +Q -Q

hω r

φ

φˆ Qˆ φˆ Qˆ † aˆ = ; aˆ = +i −i φZPF QZPF φZPF QZPF φZPF = 2 ωr L

(

⎡⎣ aˆ , aˆ † ⎤⎦ = 1 trapped photons!

QZPF = 2 ωr C Hˆ = ωr aˆ † aˆ + 1

annihilation and creation operators

2

)

ALL TRANSITIONS BETWEEN QUANTUM LEVELS ARE DEGENERATE IN PURELY LINEAR CIRCUITS! E

φ

hω r

φ CANNOT STEER THE SYSTEM TO AN ARBITRARY STATE IF PERFECTLY LINEAR

NEED NON-LINEARITY TO FULLY REVEAL QUANTUM MECHANICS Potential energy

Position coordinate

JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION S

1nm

I S

Ι

Ι

LJ

superconductorinsulatorsuperconductor tunnel junction

Ι = φ / LJ

CJ

LJ =

φ0 I0

I0

φ

φ = ∫−∞ V ( t ')dt ' t

I = I 0 sin (φ / φ0 )

φ0 =

2e

JOSEPHSON TUNNEL JUNCTION PROVIDES A NON-LINEAR INDUCTOR WITH NO DISSIPATION S

1nm

I S

superconductorinsulatorsuperconductor tunnel junction

LJ

CJ

LJ =

U = − EJ cos (φ / φ0 )

Ι

φ = ∫−∞ V ( t ')dt ' t

φ02 EJ

φ

2 EJ

bare Josephson potential

φ0 =

2e

TRANSMISSION LINE AS 1D BOSON FIELD L

n-1

C

L Vn −1 I n −1 C

n

L

L

n+1

Vn I n C

n+2

L

Vn +1 I n +1 C

a Dynamical equations:

d Vn − Vn +1 = L I n dt d I n −1 − I n = C Vn dt

Continuum limit:

Field equations:

Vn +1 − Vn ∂V → ∂x a I n +1 − I n ∂I → ∂x a C L →C ; → L a a

∂V = −L ∂x ∂I = −C ∂x

∂I ∂t ∂V ∂t

CHARGE AND FLUX IN CONTINUUM LIMIT Φ ( x) Lδx

Π ( x)δ x

Φ ( x + δ x) Lδx

C δx

Lδx

C δx

Π ( x + δ x)δ x

δx ⎡⎣Φ ( x ) , Π ( x ) δ x ⎤⎦ = i ⎡⎣Φ ( x ) , Π ( x + δ x ) δ x ⎤⎦ = 0

δx→0

ˆ ( x ),Π ˆ ( x )⎤ = i δ ( x − x ) ⎡Φ 1 2 ⎦ 1 2 ⎣

Hamiltonian : energy density as a function of field and conjugate momentum: 2 2⎫ +∞ ⎧ 1 ˆ 1 ˆ ˆ ⎡ ⎤ ⎡ ⎤ H = ∫ dx ⎨ Π ( x )⎦ + ∇Φ ( x ) ⎦ ⎬ ⎣ ⎣ −∞ 2L ⎩ 2C ⎭

COMPATIBILITY WITH STANDARD QED For simplest geometry, consider stripline waveguide:

w

y

h

x

z B ( x1 , y1 , z1 )

E ( x2 , y2 , z2 )

Flux between strips: x

yb + h

−∞

yb

ˆ ( x ) = 1 dx Φ 1 ∫ ∫

Strip charge per unit length:

dy Bˆ z ( x, y, z1 )

ˆ (x ) = ε Π 2 0∫

z f +w

zf

dz Eˆ y ( x2 , y2 , z )

Commutation relations between field operators in standard QED:

i ∂ ⎡ Bˆ z ( x1 , y1 , z1 ) , Eˆ y ( x2 , y2 , z2 ) ⎤ = ⎣ ⎦ ε ∂x δ ( x1 − x2 ) δ ( y1 − y2 ) δ ( z1 − z2 ) 0 1 ˆ ( x ),Π ˆ ( x )⎤ = i δ ( x − x ) ⎡Φ 1 2 ⎦ 1 2 ⎣

OK!

END OF 1ST LECTURE

INTRODUCTION TO QUANTUM SUPERCONDUCTING CIRCUITS: PART 2 QUAN

ELEC

UM – MECHANICAL

RONICS LAB

Applied Physics and Physics, Yale University R. VIJAY (U.C.Berkeley) I. SIDDIQI (U.C.Berkeley) C. RIGETTI M. METCALFE (NIST) V. MANUCHARIAN

N. BERGEAL (ESPCI Paris) F. SCHACKERT A. KAMAL B. HUARD (ENS Paris) A. MARBLESTONE (MIT)

N. MASLUK M. BRINK K. GEERLINGS L. FRUNZIO M. DEVORET

Acknowledgements: R. Schoelkopf, S. Girvin, D. Prober & D. Esteve W.M. KECK

Rutgers Dec. 09

OUTLINE YESTERDAY:

TODAY:

PRESENT BASIC CIRCUIT ELEMENTS

COMPARE QUBIT CIRCUITS

QUANTUM TREATMENT OF CIRCUITS Iβ rest of circuit

branch β

Branch flux and branch charge are primary variables:

φβ ( t ) = ∫ Vβ ( t ')dt ' t

−∞

Vβ

Qβ ( t ) = ∫ I β ( t ')dt ' t

−∞

For every branch β in the circuit:

⎡φˆβ , Qˆ β ⎤ = i ⎣ ⎦

IRREVERSIBLE/REVERSIBLE CHARGE TRANSFER e tunneling quasiparticle energy

2Δ rate

before SIS TUNNEL JUNCTION

after

IRREVERSIBLE/REVERSIBLE CHARGE TRANSFER e tunneling quasiparticle energy

2Δ rate

before SIS TUNNEL JUNCTION

after

2e CP tunneling quasiparticle energy

matrix element

DYNAMICS OF JOSEPHSON ELEMENT FROM CHARGE POINT OF VIEW (IN ABSENCE OF Q.P. TUNNELING) integer Hopping

N0 − 1 N0 N0 + 1

Q =N 2e

1 h / e2 M = EJ = Gt Δ 2 16 Charge states:

Nˆ N = N N

Josephson tunneling hamiltonian in charge basis:

EJ ˆ HJ = − 2

∑( N N

N +1 + N +1 N

)

FROM NUMBER TO "PHASE" REPRESENTATION

ϕˆ = 2eφˆ /

⎡ϕˆ , Nˆ ⎤ = i ⎣ ⎦

eiϕˆ Nˆ e − iϕˆ = Nˆ − 1

EJ ˆ HJ = − N N +1 + N +1 N ( ∑ 2 N EJ iϕˆ (from expression of − iϕˆ = − ( e + e ) translation operator) 2 = − EJ cos ϕˆ

(

= − EJ cos φˆ / φ0 Josephson's

ϕ

)

) ϕ

Φ0 φ0 = = 2e 2π

runs on a line but tunnel hamiltonian is periodic

ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT

REST OF CIRCUIT

Z (ω ) CJ

EJ

U(t)

2

φ ( t ) = ∫−∞ ∫ E ( x,τ ) dxdτ t

1

N.B. The electric field E encompasses here all contributions of the force on the electrons doing work, including those usually called chemical potential effects.

ELECTRODYNAMICS OF JUNCTION IN ITS ENVRONMENT t

Qext = ∫ Idτ −∞

Q = 2eN REST OF CIRCUIT

EJ

QC

Z (ω )

CJ

U(t)

2

φ ( t ) = ∫−∞ ∫ E ( x,τ ) dxdτ t

1

Equation of motion:

∂ CJ φ + ∂φ

⎡ ⎛ φ ⎞⎤ ⎢ − EJ cos ⎜ ⎟ ⎥ = I φ , φ ,.... ⎝ φ0 ⎠ ⎦ ⎣

(

)

Can be obtained in general from a Lagrangian:

d ∂L ∂L − =0 dt ∂φ ∂φ

CJ 2 φ L = L J +L ext = φ + EJ cos + L ext φ , φ ,... 2 φ0

(

)

TWO CHARACTERISTIC ENERGIES OF ENVIRONMENT Total environment admittance:

Effective shunt capacitance of junction:

Effective shunt inductance of junction

Ytot (ω ) = iCJ ω + Z −1 (ω )

Im ⎡⎣Ytot (ω ) ⎤⎦

CΣ = lim

ω

ω →0

Leff

⎧⎪ ⎡ ⎤ ⎫⎪ 1 = lim ⎨Im ⎢ ⎥⎬ ω →0 ⎩⎪ ⎣ ωYtot (ω ) ⎦ ⎭⎪ (electron charge appears here instead of Cooper pair charge for convenience in some formulas)

2

Coulomb charging energy

Inductive energy

e EC = 2CΣ

EL

h / 2e ) ( = Leff

2 (form chosen for easy comparison with Josephson energy)

TWO BASIC SUPERCONDUCTING "ATOMS" flux

charge superconducting island

LJ

superconducting loop

ϕ∈

LJ

N∈

CJ

Φext

L > LJ

Cg

CJ U

I "HYSTERETIC RF SQUID"

E J > EL

EC

Δϕ 2π

Friedman et al. (2000)

"COOPER PAIR BOX"

1

EL = 0; EJ

EC

ΔN < 1

Bouchiat et al. (1998), Nakamura, Pashkin, Tsai (1999)

PHASE POTENTIALS OF SQUID AND BOX E J > EL

EL = 0; EJ

EC

EC

E SQUID

BOX

max. curvature

= EL − E J 2EJ −1

2eΦ ext /

0

ϕ 2π

+1

+2

−

ϕ 2π

1 2

e

+

1 2

iπ C gU / e

RICH LEVEL STRUCTURE, BUT EXTREME SENSITIVITY TO NOISE 09-III-10

PHASE-CHARGE DUALITY E J > EL

EL = 0; EJ

EC

EC

E SQUID

BOX

8 EC N g −

2π EL ϕext − π

−1

0

ϕ 2π

+1

+2

1 2

−1

+1

0

N

+2

CHARGE QUBIT FAMILY

THE SINGLE COOPER PAIR BOX ARTIFICIAL ATOM EJ

Bouchiat et al. (1998) Nakamura, Tsai and Pashkin (1999)

0.5EC U

Φ U

Q=2Ne Φ Cg

EJ

4 EC

QUANTRONIUM Vion et al. (2001)

U

Φ

Q=2Ne Φ

U

Cg

TRANSMON COOPER PAIR BOX J. Koch et al. (2007) A. Houck et al. (2007) J. Schreier et al. (2007)

U

Φ

EJ

50 EC

U

Q=2Ne

Φ Cg

Cottet et al. (2002)

Koch et al. (2007)

ANHARMONICITY vs OFFSET CHARGE INSENSITIVITY IN COOPER PAIR BOX

gap EJ

"C.P. BOX"

"QUANTRONIUM"

"TRANSMON"

FLUX QUBIT FAMILY

EXAMPLES OF QUANTUM CIRCUITS BELONGING TO THE RF-SQUID TYPE Readout Readout

Friedman, Patel, Chen, Tolpygo and J. E. Lukens, Nature 406, 43 (2000).

Chiorescu, Nakamura, Harmans & Mooij, Science 299, 1869 (2003).

"flux" qubit

"phase" qubit

Readout

Steffen et al., Phys. Rev. Lett. 97, 050502 (2006)

FLUXONS PLAY WITH FLUX QUBITS A ROLE SIMILAR TO WHAT COOPER PAIRS PLAY WITH CHARGE QUBITS Inductive energy

-1

0

1

Φ Φ0 ⎛ EJ ⎞ ~ exp ⎜⎜ −α ⎟⎟ E C ⎠ ⎝

OUR RECENT CHILD IN FLUX QUBIT FAMILY

NOISE IN THE TWO BASIC QUBITS

superconducting loop

flux

charge

LJ

LJ ϕ

ΦN Φext flux

superconducting island

CJ

L ∼ LJ

noise!

N

Cg

CJ

UN

U

charge noise!

I "RF SQUID"

Δϕ 2π

1

"COOPER PAIR BOX"

ΔN < 1

FLUXONIUM IDEA: SHUNT A COOPER PAIR BOX AT DC, LEAVE IT UNSHUNTED AT RF L

LJ LJ

Cg

CJ UN U

Manucharyan. et al., Science, 326, 113; arxiv 0906.0831 Koch. et al., Phys. Rev. Lett, to appear; arxiv 0902.2980 Manucharyan. et al., submitted to Nature, arxiv 0910.3039

PRACTICAL IMPLEMENTATION

write / d a e r

Small junction:

EJ/EC = 3.6 Array junctions (N =43):

EJA/ECA = 28 Readout f0 = 8 GHz; Q=400 Every island is shunted by at least one large junction: array performs as inductor

CHIP AND MODEL “Atom” EC = ½ e2/(CJ+Cc)=2.5 GHz EJ = ½ (Φ0/2π)2/LJ=8.9GHz EL = ½ (Φ0/2π)2/L=0.52GHz “Cavity” ωR=(LRCR)-1/2/2π = 8.2GHz ZR=(LR/CR)1/2 = 82Ω 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 72 hours w/o jumps or drifts! NO OFFSET CHARGE

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

A QUBIT OPERATING FOR ALL VALUES OF FLUX ~350MHz Inductive energy

9GHz

Potential energy

-1

1

0 Φext =1/4 Φ0

Φext =1/2 Φ0

Φ Φ0 Φext = 0

~2EJ

2π

2π2EL

2π

4π

ϕ

Φ0 TWO-TONE SPECTROSCOPY vs EXTERNAL FLUX Φ ≈ 2

A 370MHz ATOM MEASURED THROUGH A 8.2GHz CAVITY?

atom transition 0-1: store/manipulate atom transition 1-2: couple to readout

g12 = g ~ 100MHz ω12 - ωR ~ 1GHz χ= g122/(ω12 - ωR) ~10MHz At the same time ω01