CIRCUITS ET SIGNAUX QUANTIQUES (II) · 2017. 12. 28. · Implementation of protected qubits in...

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CIRCUITS ET SIGNAUX QUANTIQUES (II)

QUANTUM SIGNALS AND CIRCUITS (II)

Chaire de Physique MésoscopiqueMichel Devoret

Année 2009, 12 mai - 23 juin

Troisième leçon / Third Lecture

09-III-1

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write to "[email protected]"

09-III-2

PDF FILES OF ALL LECTURES WILL BE POSTED ON THIS WEBSITE

2

May 12: Daniel Esteve, (Quantronics group, SPEC-CEA Saclay)Faithful readout of a superconducting qubit

May 19: Christian Glattli (LPA/ENS)Statistique de Fermi dans les conducteurs balistiques : conséquences expéri-mentales et exploitation pour l'information quantique

June 2: Steve Girvin (Yale)Quantum Electrodynamics of Superconducting Circuits and Qubits

June 9: Charlie Marcus (Harvard)Electron Spin as a Holder of Quantum Information: Prospects and Challenges

June 16: Frédéric Pierre (LPN/CNRS)Energy exchange in quantum Hall edge channels

June 23: Lev Ioffe (Rutgers)Implementation of protected qubits in Josephson junction arrays

CALENDAR OF SEMINARS

09-III-3

CONTENT OF THIS YEAR'S LECTURES

NEXT YEAR: QUANTUM COMPUTATION WITH SOLID STATE CIRCUITS

09-III-4

1. Introduction and review of last year's course

2. Non-linearity of Josephson tunnel junctions

3. Readout of qubits

4. Amplifying quantum fluctuations

5. Dynamical cooling and quantum error correction

6. Defying the fine structure constant: Fluxonium qubit and

observation of Bloch oscillations.

OUT-OF-EQUILIBRIUM NON-LINEAR QUANTUM CIRCUITS

3

LECTURE III : USING THE NON-LINEARITY OF JOSEPHSON QUANTUM CIRCUITS FOR

QUBIT READOUT

1. Electrodynamics of the junction in its environment (ctn'd)

2. Artificial superconducting atoms

3. Semi-classical analysis

4. Readout of superconducting qubits

09-III-5

OUTLINE





09-III-5a

4

RESTOF

CIRCUIT CJEJ U(t)

( )Z ω

( ) ( )2

1

,t

t E x dxdτ τφ−∞

= ∫ ∫"gauge invariantphase difference"

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.

09-III-6

RESTOF

CIRCUIT CJEJ U(t)

( )Z ω

( ) ( )2

1

,t

t E x dxdτ τφ−∞

= ∫ ∫

Q

ext

tQ Idτ

−∞= ∫

2q eN=

Equation of motion:"gauge invariantphase difference"

( )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

09-III-7

5

TWO CHARACTERISTIC ENERGIES OF ENVIRONMENT

( )totImlim

YC

ω

ωωΣ →∞

⎡ ⎤⎣ ⎦=

Total environment admittance: ( ) ( )1tot JY jC Zω ω ω−= +

Effective shunt capacitance of junction:

Effective shunt inductance of junction ( ){ }1eff 0

lim Im totL Yω

ω ω−

→= ⎡ ⎤⎣ ⎦

Coulomb charging energy

2

2CeECΣ

=(electron chargeappear here insteadof Cooper pair chargefor convenience insome formulas)

Inductive energy( )2

ff

/ 2L

e

eE

L=

h(form chosen foreasy comparisonwith Josephsonenergy) 09-III-8a

TWO BASIC SUPERCONDUCTING "ATOMS"

Φext

flux

JL

JCJL L>

U

charge

JL

JCgC

I

supercon-ductingloop

island

"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=

∈∈

09-III-9

6

2ϕπ

12

− 12

+

/e gi C U eπ

2ϕπ

0 1+ 2+1−

2EJ

PHASE-CHARGE DUALITY

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−

BOX

09-III-11

E

182C gE N −

ext2 LE ϕπ π−

0 1+ 2+1−

7

OUTLINE





09-III-5b

Q=2Ne

THE SINGLE COOPER PAIR BOXARTIFICIAL ATOM

Φ

U

U

ΦgC

Bouchiat et al. (1998)Nakamura, Tsai and Pashkin (1999) J CE E<

09-III-12

8

U

QUANTRONIUM

Φ

U ΦgC

Q=2Ne

Vion et al. (2001)4J CE E

09-III-13

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

09-III-14

9

ANHARMONICITY vs OFFSET CHARGEINSENSITIVITY IN COOPER PAIR BOX

Koch et al. (2007)

"QUANTRONIUM"

"TRANSMON"

Cottet et al. (2002)

"C.P. BOX"

09-III-15

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)

09-III-16

10

"FLUXONIUM" QUBITV. Manucharyan et al. (2009)

array junctionsbehaving

as a large shuntinginductance

small junction, as in quantronium

09-III-17

THE LANDSCAPE OF SUPERCONDUCTING ARTIFICIAL ATOMS

/L JE E

/J CE E

0 1

1

10

100

1000

10000

"Cooper Pair Box"

"Quantronium"

"Transmon"

"Fluxonium"

100000 "Phase qubit"

"Flux qubit"

analogousto atomic

A (!?)

analogous to atomic Z (!?)09-III-18

11

HARMONIC APPROXIMATION

( ) ( )x

22

e tˆ

ˆ cos8ˆ

ˆ2 2J LJ

extC E

NH E E

N ϕϕ

ϕ− −= − +

( )2

,

2ˆˆ

28ˆ

2e

J h JCxtN

EN

H E ϕ−= + ′ J CE E′

offsetˆ ˆϕ ϕ ϕ− →

09-III-19


( ) ( )x

22

e tˆ

ˆ cos8ˆ

ˆ2 2J LJ

extC E

NH E E

N ϕϕ

ϕ− −= − +

( )2

,

2ˆˆ

28ˆ

2e

J h JCxtN

EN

H E ϕ−= + ′

8 C JP

EEω =

′Josephson plasma frequency

,1ˆ ˆ2J h PH nω ⎛ ⎞= +⎜ ⎟

⎝ ⎠photon

representation

† †ˆ ; , 1n c c c c⎡ ⎤= =⎣ ⎦

J CE E′offsetˆ ˆϕ ϕ ϕ− →

09-III-19a

12


( ) ( )x

22

e tˆ

ˆ cos8ˆ

ˆ2 2J LJ

extC E

NH E E

N ϕϕ

ϕ− −= − +

( )2

,

2ˆˆ

28ˆ

2e

J h JCxtN

EN

H E ϕ−= + ′

Josephson plasma frequency

Spectrum independent of DC value of Next

,1ˆ ˆ2J h PH nω ⎛ ⎞= +⎜ ⎟

⎝ ⎠photon

representation

† †ˆ ; , 1n c c c c⎡ ⎤= =⎣ ⎦

4 4

†4 4

4 ˆˆ16

4 ˆˆ16

J C

C J

J C

C J

E Ec i NE E

E Ec i NE E

ϕ

ϕ

= +

= −

J CE E′offsetˆ ˆϕ ϕ ϕ− →

09-III-19b

8 C JP

EEω =

′

CAN WE GO 1 STEP BEYONDTHE HARMONIC APPROXIMATION

AND OBTAIN MEANINGFUL RESULTS?

09-III-20

13

YES, BUT WE NEED TO CONSIDERREGIMES WHICH LEND

THEMSELVES TO A SEMI-CLASSICALAPPROXIMATION

09-III-21

OUTLINE





09-III-5c

14

ATOM IN SEMI-CLASSICAL REGIME:CIRCULAR RYDBERG ATOM

p+

2r

ω

2

20

2

1 3

1

Ry

Ry2

2

e

n

n

n n

nn n

m r n

r a n

En

nerd

ω

ω → ±

→ ±

=

=

= −

=

=

old Bohr theory essentially “exact”!

constraint

( )2

, ,n x y zΨ

09-III-22


p+

2 nr

nω

2

20

2

1 3

1

Ry

Ry2

2

e

n

n

n n

nn n

m r n

r a n

En

nerd

ω

ω → ±

→ ±

=

=

= −

=

=


constraint

09-III-22a

15


p+

2 nr

nω

2

20

2

1 3

1

Ry

Ry2

2

e

n

n

n n

nn n

m r n

r a n

En

nerd

ω

ω → ±

→ ±

=

=

= −

=

=Coulombpotential


valid when: ( ) 1EE

ω∂∂

09-III-22b


p+

2 nr

nω

2

20

2

1 3

1

Ry

Ry2

2

e

n

n

n n

nn n

m r n

r a n

En

nerd

ω

ω → ±

→ ±

=

=

= −

=

=Coulombpotential


valid when: ( ) 1EE

ω∂∂

, 1 1, 2 , 1n n n n n nω ω ω+ + + +−09-III-22c

16

Josephsonpotential energy

Junction phase ϕ

TRANSMON AS ANALOGOF CIRCULAR RYDBERG ATOMS

2EJ

+π−π

J CE E

09-III-23


2EJ

ω

Absorption lines at high T:

+π−π

0

01ω12ω , 1 1, 2 , 1n n n n n nω ω ω+ + + +−

Junction phase ϕ


09-III-23a

17


2EJ

ω

Absorption lines at low T

+π−πJunction phase ϕ


09-III-23b


2EJ

ω

Absorption lines after π pulse on 0-1:

+π−πJunction phase ϕ


09-III-23c

18

E2EJ

ω(0)

( )Eω

JUNCTION NON-LINEAR INDUCTANCE

( ) 10 PJ JL C

ω ω= =

00

09-III-24

E2EJ

ω(0)


( ) 10 PJ JL C

ω ω= =

00

ωΔ ≥ Γ Classical condition for strong non-linearity,implies drive

( )Eω

09-III-24a

19

E2EJ

ω(0)

, 1n nω +


( ) 10 PJ JL C

ω ω= =

00

( )EωQUANTUM-MECHANICALLY, DISCRETE ENERGY LEVELS

09-III-24b

E2EJ

ω(E)= ω(0)(1-E/8EJ)

ω(0)

, 1n nω +


quartic correction of cosine potential

00

( )Eω

09-III-24c

20

E2EJ

ω(E)= ω(0)(1-E/8EJ)

ω(0)

, 1n nω +


JEJL

( )

2

2

2

20

2

4 8

JJ

JJ JJ

J

Le E

IL LELE I

δ

=

= =

( )JL Eδ

( ) ( )1

J

EL E C

ω =

00

( )Eω

09-III-24d

QUANTUM NON-LINEARITY ENERGY SCALE

( )

( ) ( )( ) ( )

( )

01, 1 01

01, 1 1, 2

, 1 1, 2

22

, 1 1, 2 2 2

2

, 1 1, 2

11 ; 08

18

18

18 / 2

2

n nJ

n n n nJ

Pn n n n

J

Jn n n n

J J

n n n n CJ

n nE

E

E

LL C e

e EC

ωω ω

ωω ω

ωω ω

ω ω

ω ω

+

+ + +

+ + +

+ + +

+ + +

⎛ ⎞= − ≥⎜ ⎟

⎝ ⎠

− =

− =

⎛ ⎞− = ⎜ ⎟⎜ ⎟

⎝ ⎠

− = =

weak quantum non-linearity

strong quantum non-linearity

/CE Γ

/CE Γ

decay rateof quantumlevel

09-III-25e

neglect zero-point correction

21

OUTLINE





09-III-5d

THE QUBIT MEMORY READOUT PROBLEM

QUBIT READOUTON

OFF0 1

QUBIT READOUTON

OFF0 1

QUBIT READOUTON

OFF0 1 or

09-III-26

22


QUBIT READOUTON

OFF0 1

QUBIT READOUTON

OFF0 1

1

0

QUBIT READOUTON

OFF0 1

0 1

0 10 11F ε ε= − −

1ε0ε

FIDELITY:

or

09-III-26a


QUBIT READOUTON

OFF0 1

QUBIT READOUTON

OFF

1) SWITCH WITH ON/OFF RATIO AS LARGE AS POSSIBLE2) READOUT WITH F AS CLOSE TO 1 AS POSSIBLEWANT:

0 1

1

0

QUBIT READOUTON

OFF0 1

0 1

0 10 11F ε ε= − −

1ε0ε

FIDELITY:

or

09-III-26b

23

TWO MAIN STRATEGIES

09-III-27

STATE DECAY STRATEGY

1

0

1

0

Martinis, Devoret and Clarke, PRL 55 (1985)Martinis, Nam, Aumentado and Urbina, PRL 89 (2002)

09-III-28

24

QUBITCIRCUIT

10

or

or

rf signal in rf signal out

DISPERSIVE READOUT STRATEGY

01ω ω≠

Blais et al. PRA 2004, Walraff et al., Nature 2004

FILT

ER

FILTER

09-III-29

QUBITCIRCUIT

10

or

or



01ω ω≠

QUBIT STATEENCODED IN PHASE

OF OUTGOING SIGNAL,NO ENERGY

DISSIPATED ON-CHIP


FILT

ER

FILTER

09-III-29a

25

QUBITCIRCUIT

10

or

or


A) SHELTER QUBIT FROM ALL RADIATION EXCEPT READOUT RFB) AMPLIFY OUTGOING SIGNAL WITH LOWEST NOISE POSSIBLEC) SEND ENOUGH PHOTONS TO BEAT NOISE


01ω ω≠

QUBIT STATEENCODED IN PHASE

OF OUTGOING SIGNAL,NO ENERGY

DISSIPATED ON-CHIP


FILT

ER

FILTER

09-III-29b

END OF LECTURE

CIRCUITS ET SIGNAUX QUANTIQUES (II) · 2017. 12. 28. · Implementation of protected qubits in...

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