Chang-Pu Sun

(孙昌璞)

Institute of Theoretical Physics,

Chinese Academy of Sciences

宏观量子态:量子比特

Related Publications and References therein

1.P. Zhang, Y. D. Wang, and C. P. Sun，

Phys. Rev. Lett. 95, 097204 (2005)2. Y.-x. Liu, J. Q. You, L. F. Wei, C. P. Sun, and F. Nori，Phys. Rev. Lett. 95, 087001 (2005)3. C. P. Sun, L. F. Wei, Y.-x. Liu, and F. Nori, Phys. Rev. A 73, 022318 (2006)3.Y. B. Gao, Y. D. Wang, and C. P. Sun，Phys. Rev. A 71, 032302 (2005)4. Y. D. Wang, Z. D. Wang, and C. P. Sun，Phys. Rev. B 72, 172507 (2005) 5. Y. D. Wang, P. Zhang, D. L. Zhou, and C. P. Sun, Phys. Rev. B 70, 224515 (2004) 6.Y. D. Wang, Y. B. Gao and C. P. Eur. Phys. Jour. B 40, 321-326 (2004).7. P. Zhang, Z. D. Wang, J. D. Sun, and C. P. Sun, Phys. Rev. A 71, 042301 (2005)

3 Review Articles:1. G. Wendin, V.S. Shumeiko, cond-mat/0508729

Superconducting Quantum Circuits, Qubits and Computing,

2. Y. Makhlin, G. Schön, and A. Shnirman, Rev. Mod. Phys. 73, 357 (2001)Quantum-state engineering with Josephson-junction devices

3. J.Q. You & F. Nori, Phys. Today, 58(11), 42 (2005)Superconducting Circuits and Quantum Information

Circuit QED

Nature 431 162 (2004)

Blais, A., et al., Phys. Rev.A 69, 062320 (2004)

Outlines

1. Microscopic Theory for Josephson

Effects

2. Macroscopic Qubit

based on Superconducting Circuits

A. Charge Qubit; B: Phase Qubit

; C: Flux Qubit

3. Superconducting Quantum Network: Circuit QED

A. Direct Coupling ; B: with LC Oscillator

C . With Transmission Line ; D: With Large Junction:

1. Microscopic Theory for Josephson

Effects

(J.Q. You & F. Nori, Phys. Today, 2005)

Macroscopic Qubits with Josephson Junction (JJ Qubit)

0 1

N electrons (N+2) electrons

‘

Field Field0 1

Charge Qubit

Flux Qubit

Phase Qubit

φ

Fast Progresses in Experiments

SQUID loopSQUID loop

ProbeProbeBox Box GateGate

Tunnel junctionTunnel junction

SingleSingleCooperCooper--pair pair tunnelingtunneling ReservoirReservoir

0 200 400 6000

2

4

6

8

10

Q0/e=0.51Td=16ns Tcoh=h/EJ

T=30mK

Pul

se-in

duce

d cu

rrent

(pA

)

Pulse width (ps)

Quality Factor

Q=Td X w01 ~ 5

700x50x15nm; 108 conduction electrons. e2/2C = 117mV;

T=30mK, kT = 3mV; for aluminum, D=230mV

NEC Box:

φ

E

0Δ

0ex 21φφ −=

φ0 φ

1

Ec < EJΦ

Single junction SQUID Cooper Box

φ

0

1ω

Ec > EJn

E

0

JE

q

( )eQ 221

0 −−=

22002: Q~ 2X 105

Han, SCIENCE 296 ,2002

2003: Q ~ 6 X 103

Mooij,Science ,2003

0.0 0.1 0.2 0.3 0.4 0.5 0.6

30

35

40

45

switc

hing

pro

babi

lity

(%)

time between pulses Δt (µs)

Δν

= 19.84 MHzTϕ

= 500 +/- 50 ns

2002: Q~ 2.5 X 104Vion, SCIENCE 296 ,2002

Superconductivity

1.Macroscopic ground stat2.Elimination of low-energy excitations3.High-density electrons ⇒

short screening length

Normal Metal Formed QubitSuperconductor

Anharmonicity

⇒

Two Level System (Qubit)

・・・

・・・

・・・

Harmonic Oscillator anharmonicity

Josephson Equation

Josephson Effects

I

s i n ,

2 ( , ) ,

2

cI IdI e n r td t

d e Vd t

ϕ

ϕ

=

= −

=

( ) ( ) ( ) ( ) dtrAtrtrtr ⋅Φ

−−= ∫2

10

21 ,2,,, πθθϕ

n

Cooper-pair tunneling -2e

V

AC Currents: 0V ≠

DC Currents: 0V = Aharonov –Bohm Effects

phase of Superconductor = Order Parameter ϕ =

Josephson (1962)

Tunneling EnergiesTunneling Energies

∫=t

dtIVE0

)(

( ) dtdt

dIt

c ⎟⎠⎞

⎜⎝⎛ ′Φ′= ∫

ϕπ

ϕ2

sin 00

ϕϕπ

ϕ′′Φ

= ∫ dI c0

0 sin2

( )ϕcos1−= JEE

Small Junction

CJ <10-15F

0.5 KJE ≈π20 cJ IE Φ=

““Potential Energy”:

Microscopic Theory for Josephson Effects

Anderson Pseudo –Spin For JJ Tunneling

∑ += +

qkqkkqT chCCTH

,

).( ∑ += +

qkqkkqT chgH

,

).( σσ

,

(1 )k k kzk k k k k

C C

C C C C

σ

σ

+ +−

+ +− −

=

= − −

LN RN

SD 1SD 1

q k

SD 2SD 2 Left L Phaseφ =

RightR Phaseφ =

( ) 0LiL k k k

kS u v e φ σ= ∏ + ( ) 0Ri

R q q qq

S u v e φ σ= ∏ +

1 1

2

, S

2[ , ] 0

1 ( )2

L R R LN N

z

z zz q k

q k

S b b b b

S S SN

S σ σ

+ ++ −

+ −

= =

≅ =

= −∑ ∑

1212

NL k L

k

NR q R

q

S bN

S bN

σ

σ

→∞ +

→∞ +

= ⎯⎯⎯→

= ⎯⎯⎯→

∑

∑

,[ , ] ,

i i

z

S e S eS S S

φ φ−+ +

+ +

= =

=

[ , ]zS iφ =

Macroscopic Quasi-Spin Representation

[ , ]n iφ =N →∞

Microscopic Description for Macroscopic Quantum Tunneling

0)(

0)(

qi

qqq

R

ki

kkk

L

RL

R

L

evuS

evuS

SSBCS

σ

σ

φ

φ

+∏=

+∏=

⊗=

Super –current ：22 [ , ]

1( )2

L T L

zL k

k

eJ e N H Ni

N σ

•

= − =

= −∑2 ( . )k q

kq

iJ e h cσ σ+= − −∑

in Left SCL Phaseφ =

in Right SCR Phaseφ =

BCS S

tates

Josephson Current at

( )

1

2 .

sin( )

L Rik q k q

kq

L R

J BCS J BCSeg u v u v e h ciJ

φ φ

φ φ

−

=

= +

≡ −

∑

gg k =

1 ( )4

22 [ , ]

T

z z T

JH L Lie

eJ e L L Hi

+ −

•

= +

−= − =

11

ˆ ˆ( ) sin( ) 2 L RJJ L L Ji

φ φ+ −= − = −

, , , ,ˆ L R L R L R L Rφ ϕ φ ϕ=

Interaction with Cavity Field

12 ( )a z J x yH a a E i a a gω ω σ σ σ+ += + + + +

Charge Qubit

V

Y. Makhlin, et.al Rev. Mod. Phys. 73 (2001) 357

Coulomb Blocked Mechanism

for Formation of Charge Qubit

kμ

( )212 g gE C V neC

= +

0=n

1=n

Spin 1/2

Small C Junction : CJ <10-15F, E~100

2(2 )4 4 K2c

eECΣ

= ≈

“Kinetic Energy”

Cooper Pair Box (CPB)

( )∑ −=i

ii VCne φ

1i i

i

CV neC

φ= +∑ ( ) ( )22 21 1

2 2 2i i j g gi

neU C V C C V

C Cφ= − = +∑

( ) ( )2 2

2 221 12 2 2

g g gg g g

C V C eE U W C V ne n nC C C

⎛ ⎞⎟⎜= + = + + − ⎟≈ −⎜ ⎟⎜ ⎟⎝ ⎠

Vg

CP Box,n electrons

Vj

Cg

CjC1

V1 φ

( ) ij j i

i T

CW e V VC

= −∑

Effective Hamiltonian

number n ⇔

phase position ⇔

momentum

in =],[ ϕ

EJ

n=-2 -1 0 1 2 3

Tight-binding model in 1d latticeof charge number states

-1 0 1

E

/π

2EJ

ϕ

ϕ

∑ += nnei 1ϕ

( )1 1c o s = ( 1 . )2 2

i ie e n n h cϕ ϕϕ −= + + +∑

( ) ϕcos4 2JgC EnnEH −−=

Two Level Approximation

( 1 1 0 0 ) ( 1 0 0 1 )2

Jq

EH E≈ − − +

31 1ˆ ˆ2 2z x xB Bσ σΗ = − −

+ -

3

σ =|e g|,σ =|g e|,

σ =|e e|-|g g|,

1 12 2gn = +Δ≈ ( ) ( )2 2

1 0g gn n− ≈ −

|e =|1

|g |0=

2 1),z g x JB n B E∝ − ∝（

( ) ( )∑∑ ++−−=n

J

ngC chnnEnnnnEH .1

24 2

Controllable Charge Qubit

( )

( )

0

x

0 0

0

0

0

Manupulation Via Flu

cos cos

2 cos cos

x x J x

x xJ J J

xJ

B E Controllable

E E E

E

θ π θ π

π θ

= Φ −

⎛ ⎞ ⎛ ⎞Φ Φ− = − + − −⎜ ⎟ ⎜ ⎟Φ Φ⎝ ⎠ ⎝ ⎠

⎛ ⎞Φ= − ⎜

Φ

⎟Φ⎝ ⎠

B

S

xy0

0

2 1) via gate voltage

( ) 2cos cos via Flux

z g

xx J x J

B n

B E E πθ

∝ −

Φ∝ Φ = −

Φ

（

Rabi Coupling Controlled via SQUID

J

xΦ

( )A r

ab

( ) s in ( )x

b

a

I

A d

θ

θ

Φ ≈

= ⋅∫

Controllable Rabi Coupling via By AB effect

22

11

sinsin

ϕϕ

c

c

IIII

=

=

⎟⎠⎞

⎜⎝⎛ +

⎟⎠⎞

⎜⎝⎛ −

=+=2

sin2

cos2 212121

ϕϕϕϕcIIIIC

B dc

ab

II1 I2

I

( )

( ) ∫

∫

⋅Φ

−−=−

⋅Φ

−−=−

d

ccd

b

aab

dA

dA

02

01

2

2

πϕθθ

πϕθθ

( ) ( ) ( ) ( )2 b a c b d c a dCd nθ π θ θ θ θ θ θ θ θ∇ ⋅ = = − + − + − + −∫

∫∫∫

∫∫∫

⋅Φ

−⋅Φ

−=⋅∇=−

⋅Φ

−⋅Φ

−=⋅∇=−

a

d

a

d

a

dda

c

b

c

b

c

bbc

dAdJnemd

dAdJnemd

02

0

02

0

22

2

22

2

ππθθθ

ππθθθ

∫ ⋅Φ

+=−C

dAn0

1222 ππϕϕ

012

22ΦΦ

+=−ππϕϕ n

⎟⎟⎠

⎞⎜⎜⎝

⎛ΦΦ

+⎟⎟⎠

⎞⎜⎜⎝

⎛ΦΦ

=0

10

sincos2 πϕπcII

0

0

cosC eJ

IE ππ

⎛ ⎞Φ Φ= ⎜ ⎟Φ⎝ ⎠

Controllable

Inside A Superconductor: J=0

Measurement of Charge States

2e

Cooper-pair box

JE

Resonance probe

t

t

quantum oscillations

measurementprobe

Dissipation

Flux Qubit

and Phase Qubit

2 1ϕ ϕ ϕ≡ −[ , ]N iϕ =

2

cos2 J exQH E UC

ϕ= − +

Flux qubitJ cE E

J cE E

xΦϕI

C o o p e r p a irsN

11

ine ϕψ =

22

ine ϕψ =

, R CZ

Phase qubitJ cE E

ϕ

2Q en=

depends on the circuitexU

[ , ]n iϕ =

Classical Dynamics : a Current Biased J Junction

RSJ model

sI

C

R

bI

Current bias: 0

2ex bU IϕπΦ

= −

s bdV VC I Idt R

+ + =

Conservation of current:

21

20

sin 2b cC R I It t

π− ⎛ ⎞∂ Φ ∂Φ Φ+ = − ⎜ ⎟∂ ∂ Φ⎝ ⎠0( / 2 )ϕ πΦ ≡ Φ

Mechanical Analog of a Josephson junction

1 ,UC R− ∂Φ + Φ = −

∂Φ

( cos ), / .

J b

b b c

U E ii I I

ϕ ϕ= − +≡

0 sins cU I I ϕ∂

= ⇒ =∂Φ

-10

-5

Ene

rgy

(EJ)

3210Junction phase [2π]

2 1/4cp b

0

2pIUw = = (1- i )m CF′′

- a “particle” in a tilted washboard potential

Quantum mechanical description: Phase Qubit

Two-level approximation:

0 00 01

10 1 11

ˆ ˆ,

ˆ ˆE I I

HI E I

δ δ

δ δ

⎛ ⎞+Φ Φ= ⎜ ⎟⎜ ⎟Φ +Φ⎝ ⎠

ˆ ˆ| |ij i jΦ ≡ ⟨ Φ ⟩

( ) ( )b dcI t I I tδ= +

2

0

ˆ ˆ ˆcos(2 ) ( )2 J bQH E I tC

π Φ= − − Φ

Φ

JJ Hamiltonian:

400

200

0

I(μA)

1050V (mV)

/ Bk Tqp nR R eΔ∼

Nb: / 16 KNbN: / 30 K

B

B

kk

Δ ≈Δ ≈

nR

qpRrI

3

4.2 K/ 10qp n

TR R=

≈

cI

Flux Qubit: Single-JJ rf

SQUID

•Quantum state can be engineered and controlled accurately.•Immune to the background charge noise (in charge qubit)

easily isolated from environment • Coupling between qubits is straightforward.

Why flux qubits?

1/ 2ex =

| 0⟩|1⟩

cIJ cE E

xΦϕI

22

0

( ) 2cos2 2

eJ

QH EC L

π⎛ ⎞Φ −Φ Φ= + − ⎜ ⎟Φ⎝ ⎠

-6

-4

-2

0

2

4

6

Ene

rgy

-6 -4 -2 0 2 4 6

ε

| 0⟩|1⟩

Quantum Mechanics from Lagrangian

( ) [ ]2 22

20 01

1 1, 1 cos 22 2 2 2J J

q

L C E fL

ϕ ϕ ϕ ϕ ϕ ππ π

⋅ ⋅Φ Φ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞= − − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

0

extf Φ=Φ

202

2JLP C ϕ

πϕ

⋅

⋅

∂ Φ⎛ ⎞= = ⎜ ⎟⎝ ⎠∂

( )2

2 20

11 cos2 2JPH E M IM

ϕ ϕ ϕ= + − + Ω −

202

2\

JM CπΦ⎛ ⎞= ⎜ ⎟

⎝ ⎠

[ ]0 0 22 2q extIL fϕ ϕ ππ πΦ Φ

= − Φ = − +

0E

1E

2E

30

20

10

0

-10

Ener

gy (G

Hz)

0.70.60.50.40.3φ

11T−

| 0⟩ |1⟩

0 0I =

xΦϕI

0 0I ≠

( )2

2 20

11 cos2 2JPH E M IM

ϕ ϕ ϕ= + − + Ω −

Real qubits are multi-level

0.496 0.497 0.498 0.499 0.500 0.501 0.502 0.503 0.504 0.505

Applied Flux

-3

-2

-1

0

1

2

3

4

5

Ener

gy (K

)

ω

1 ( )2 z xH εσ σ= + Δ

-6

-4

-2

0

2

4

6

Ene

rgy

-6 -4 -2 0 2 4 6

ε

Two level Approximation

(J.E. Mooij

et al., Science, 1999)

Three Junction Flux Qubit

•Small Size : Rf SQUID = large Loop..•Controllable Coupling.

Why Three Junction flux qubits?

JEα

JEJEEX ⊗ Φ

/ 1, 0.8J CE E α =

Quantum Mechanics with Hamiltonian

JEα

JEJE ex ⊗ Φ1 2 3

0 0 0

2 2 2sin sin sinJ J JE E EI π π α πϕ ϕ ϕ= = =Φ Φ Φ

( )01 3 22q extIL ϕ ϕ ϕ

πΦ

+Φ = − − +

( ) ( )1 2 1 21 1, 2 2p mϕ ϕ ϕ ϕ ϕ ϕ+ = − =

( )

2 220

2

[ (1 2 ) ]2

11 cos cos cos 2 22

J p m

J p m m q

L C

E f L I

ϕ α ϕπ

ϕ ϕ α π ϕ

⋅ ⋅Φ⎛ ⎞= + +⎜ ⎟⎝ ⎠

′⎡ ⎤− − − + +⎣ ⎦

( )

22

2

22 2

12 cos cos cos 2 22

pmJ J

m p

J p m J m q

PPH E EM M

E E f L I

α

ϕ ϕ α π ϕ

= + + +

′− − + + m m mP M ϕ⋅

=

p p pP M ϕ⋅

=

20(1 2 ) (1 2 )2

2m p JM M Cα απΦ⎛ ⎞= + = + ⎜ ⎟

⎝ ⎠

1ϕ 2ϕ

3ϕ

qL

I

Flux Qubit: Two-Level System

“Dipole Transition ‘

Induced by Microwave

JEJEex ⊗ Φ

JEα ext ext fΦ ⇒Φ +Φ

fΦ

30

2 sin( 2 )J fH H E fπ α ϕ π⇒ + + ΦΦ

0 0

2 2sin 2I J ext m fH Eπ πα ϕ⎡ ⎤

= − Φ + Φ⎢ ⎥Φ Φ⎣ ⎦

Symmetry to depends on External Flux m extϕ Φ

Liu, You, Wei , Sun, and Nori, PRL, (2005) Symmetry Breaking

Δ–Artificial Macroscopic Atoms

(Liu, You, Wei, Sun & Nori, Phys. Rev. Lett., 2005)

ϕ3

ϕ4

ϕ1 ϕ2

Φs

Φe

EJCJ

γEJ

γEJ

γCJ

γCJ

EJCJ

0

2

P

T

M

(a) (b)

1

Tunable Flux Qubit: Artificial atom for a Micromaser

The symmetric SQUID provides an effective, tunable Josephson junction that is controlled by the flux through the SQUID loop.

(You, Liu, Sun & Nori, quant-ph/0512145,,PRB 2007)

Superconducting Quantum Network: Circuit QED

A: Direct Coupling:

Charge Qubit

B: Indirect Coupling:

Charge Qubit

Coulomb Interaction Via Data Bus

⊗⊗

Why

Need Two Qubit

Coupling?

A. Barenco

et al Phys.Rev. A52 (1995) 3457

Universal quantum logical gates:

A universal set of quantum gates:any unitary U can be decomposed into a product of successive gates of the basic ones

Universality Theorem :

A general U can be decomposed into single- qubit rotations and a nontrivial two-qubit gate

1 111 12

H ⎛ ⎞= ⎜ ⎟−⎝ ⎠

HH

1 00 ie Φ

⎛ ⎞Φ=⎜ ⎟

⎝ ⎠ Φ |y>

|x>

Φ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=Φ

Φie

B

000010000100001

)(yx ixye x yΦ

Coupled Charge Qubits

NEC-RIKEN Experiments

|e

|g

|e

|g

Nature,2003, 241,843

⊗⊗

Capacitive Coupling Of Charge Qubits

⊗⊗C

V

2 21 21 1 [ ]( )2 2 2CH CV C

eϕ ϕ−

= =

21( )2 2

np Ce e

ϕ= =

2

1 22

CeH n nC

=

2 2( )gn n−

21 1 1 1 1

22 2 2 2 2

1 1 2 2

4 ( ) cos

4 ( ) cos

( )( )

c g J

c g J

m g g

H E n n E

E n n E

E n n n n

θ

θ

= − −

+ − −

+ − −

1 11 1

2 212 22

1 2

a z J x

a z J x

z z

H E

E

ω σ σ

ω σ σ

χσ σ

= +

+ +

+

Models of NEC Capacitive Coupling

NEC Experiment: Multi-frequency

Science, 2003

Capacitive Coupling Of Phase Qubits

Two Charge Qubit

Logic Gates

Yamamoto et al

Nature, 425,941

Experiment Results

Vion

,et al , Nature,2002，Averin,2003;Gleland and Geller 2003Wang, Zhang, Zhou, Sun, 2002

V

Flux

Small Junction

Island

V

Flux

Island

Large Junction

inductance

Nano-Mech R

Superconducting Quantum Network: Circuit QED

|e

|g

|e

|g

Analog of Cavity QED: Virtual Photon Exchange

[1] [1] [2] [2]I 1 + - 2 + -H ( σ σ ) ( σ σ )g a a g a a+ +∝ + + +

[1] [2]eff z zg σ σ

1 2

[1] [2] [2] [1]eff 1 - 2 + 2 - 1 +

[1] [2] [1] [2]- + + -

H σ σ σ σ

(σ σ +σ σ )G G

G a G a G a G a

a a

+ +

+Δ

∝ × + ×

≈

Revisit Cavity QED by LC Circuit + SQUID

|e =|1

|g |0=

12H= a a+ +g(a + a )zω ωσ σ σ+ +

+ −

12 ( )a z yH a a i a a gω ω σ σ+ += + + −

Rotation Wave Approximation (RWA) ：JC-Model

Without RWA

+[a,a ]=1

Cavity = Quantization LC Circuit

2 2

2 2qHe L

φ= + [ , ]q iφ =

124

124

( ) ( ),4

( ) ( )4

Cq a aLLi a a

Cφ

+

+

= +

= −φ q

L C

H a aω +=

LC Circuit With J Junctions = Cavity QED

φ

LCg

q

n

)cos()()(422

222

φηθφφ ′−−−++= xJgc EnnELe

qH

2

, [ , ]2 ( )c

J g

eE n iC C

θ= =+

1)4

( 412

<<′= ηηCL

c o s ( ) c o s s inθ η φ θ η φ θ′ ′− ≈ −

2( )sin 4 ( ) cosc g JH a a ig a a E n n Eω θ θ+ +≈ + − + − −

CJ

12sin ( )i i

yi e eθ θθ σ−= − ≈(| 1 | |1 0 |ie n nθ = + ≈∑

Two-Level Approximation .12cos ( )i i

xe eθ θθ σ−= + ≈

12 ( )a z J x yH a a E i a a gω ω σ σ σ+ += + + + −

Effective Hamiltonian without RWA

| 0 1| a | 0 1| a+

|e

|g

|e

|g

|e

|g

|e

|g

|1 0 | a+ |1 0 | a

4 Basic Processes

Rabi Rotating Representation

|0 c o s | 0 s i n | 1 ,2 2

|1 s i n | 0 c o s | 12 2

c c

c c

θ θ

θ θ

> = > + >

> = − > + >

21 / 4

0

2 2 2

1 , ( )4

1 1 6 (1 2 )

J

J

a J c g

E C LgL C C C

E E n

πω

φ

ω

= =

= + −

12 ( )a z yH a a i a a gω ω σ σ+ += + + −

y yi ie eθσ θσα ασ σ −→

t a n / 4 (1 2 )J c gE E nθ = −

| 0 1| a | 0 1| a+

|e

|g

|e

|g

|e

|g

|e

|g

Rotation Wave Approximation

|1 0 | a+ |1 0 | a

( )ai te ω ω+ ( )ai te ω ω− + ( )ai te ω ω− ( )ai te ω ω− −

Fast oscillations

12 [ ]a zH a a g a aω ω σ σ σ+ +

− += + + +

Indirect Coupling By Large Junction

Wang, Zhang, Zhou, C.P. Sun (2002), published in PRB,2004

20

2 20

(1 cos )1 2

L J

J

H E N E

E N E a a

θ

θ ω +

= + −

≈ + =

2 20( ) cos cosc g j JH E n n E E N Eϕ θ= − − + −

Cgq

nCJ

0,,

EE θ

,,

c

j

EE ϕ

Mooij ‘s group: Nature 431, 159–162 (2004).

( ) x zH g a a f a aσ σ+ += + + +Ω

Coherent Approximation

2 2( ) cos 2 cos( )cos ,2c g c J JH E n n E N E E θθ ϕ= − + − − −Θ

( )2 2( ) cos ' cos ' cos ,c g J c JH E n n E E N Eϕ ϕ θ= − − + + −

† †( ) ,z x xH g a a f a aωσ σ σ⎡ ⎤= + + + +Ω⎣ ⎦

( ) / 2ϕ ϕ ϕ′ ′′

= +

⊗⊗Θ

'ϕ

θ

''ϕ

'' ' 2θ ϕ ϕ+ − = Θ

2 cos H n T ϕ= − 12 ( )2effH T a a T+ + −

0 2 4 6 8 10 12 14 16 18

-100

-80

-60

-40

-20

0

20

40

60

80

100

120

140

Eig

en E

nerg

y

Level Number

2[| 1| | 1 |] | |2

⎧ ⎫= − >< + + + >< + ><⎨ ⎬⎩ ⎭

∑n

TH n n n n n n n

21 2| exp[ ] ( ) | n n ni

N i H i iT T

ψ∞

=−∞

>= − >∑

1/ 21/ 2( )

2 !n nNn

απ

=

Shi, Chen, Song, Comm. Theor. Phys. (2005).

Numerical Test for Coherent Approximation

14

J

c

ETE

=

|e

|g

|e

|g

Recall :

(2) (1) (1) (2)( )f effH J σ σ σ σ+ − + −+

2| |eff

a

gJω ω

=−

2

2 1| |a

gω ω−

Virtual Photon Exchange=Slow Process

Qubit Qubit

Data Bus: A Quantize Filed

A Slow Process

|e

|g

chaFtFttH .ˆ]ˆ)(ˆ)([)( 2211 ++= ββ

1 2ˆ ˆ 0F F⎡ ⎤ =⎣ ⎦，

+∗

⎥⎦

⎤⎢⎣

⎡= +∑ atat

jkjk

jk eeFFitU )(ˆ)(ˆ

,,

ˆˆexp)( ααμ

( ') 0t tα = =

Our dynamical Scheme for Fast Operation

“

Lie algebraic solution of linear differential equations”, J. Math. Phys. 4, 575 (1963)

( )

( ) ( ) ( )

1N

k

ˆ ( ) , ˆ ˆ ( ) , then solution of

[ 0 ] excan be expressed s pa

m

i

k

i

k

iIf H t dU H t Udt

U I U t t

t

H

c H

g

=

= =

= = ⎡ ⎤⎣ ⎦

∑

∏

Mathematics for our Protocol: Wei-Noman

Method

2 ( ) *( )1ˆexp( . ) exp( | | )2

t a t aFa h c F e eα α+

− = −

Island 1Island 2

2 2

1,2

( ) (cos cos ) cosck k gk Jk k k c Jk

H E n n E E N Eϕ ϕ θ=

′ ′′= − − + + −∑

2gk gk

gk

C Vn

e=

Superconducting Circuit for Two Quibit

Coupling

Large Junction

∑=

++ Ω+++=2,1

)(k

xkkxkk aafaagH σσ

field. magnetic external by tuned be can 0φ

π xkk

Φ=Θ

0110ˆ2

ˆ8

ˆ41

41

kkkkxkJ

c

c

J NEEi

EEa +=⎟⎟

⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛= σθ

kJkJ

ck E

EEg Θ⎟⎟

⎠

⎞⎜⎜⎝

⎛−= sin

32

41

JcEE2=ΩkJkk Ef Θ−= cos

Low Energy Effective Theory

( ) ( )[ ]( )[ ] ( )[ ]xkk

kxkk

xxI

atiBatiB

tiAtiCtU

σσ

σσ+

=

−−×

−−=

∏ ˆexpˆexp

exp)(*

2,1

21

( ) ( ) ( ) ( ) ( ) ( )2 2

1 2 1 22 1 11 , 1 , 1i t i t i tkk

g g g g gB t e A t e t C t e ti i i

− Ω − Ω − Ω+⎛ ⎞ ⎛ ⎞= − = − − = − −⎜ ⎟ ⎜ ⎟− Ω Ω Ω Ω Ω⎝ ⎠ ⎝ ⎠

Time Evolution

[ ]21exp)( xxxx iU σθσθ =

( ) ⎥⎦⎤

⎢⎣⎡

Ω−= 212

214exp xxnIggnitU σσπ

22W hen ( ) ( 1)ni

kn k n

n gt B t ei

ππ − ΩΩ= = − −

Ω Ω，

can be tuned torealize the required2-bit operation

ig

Two Qubit

Logic Gate Operation : Method 1

( )kk fgHH ,−=′

( ) ⎟⎠⎞

⎜⎝⎛ −

ΩΩΩ

= TTggTM2

sin22 21

( ) ( )[ ]21exp22

~xxIII TiMTUTUTU σσ−=⎟

⎠⎞

⎜⎝⎛

⎟⎠⎞

⎜⎝⎛′=

( )[ ] ( )[ ] ( )[ ]xkkxkkk

xxI atiBatiBtAitU σσσσ ˆexpˆexpexp)( *

2,121

+

=∏′−=′

( ) ,112 21 ⎟⎠⎞

⎜⎝⎛ −−

Ω−ΩΩ− te

igg ti

Control T to realize a required 2-bit

operation

Two Qubit

Logic Gate Operation : Method 2

•

Fast : 10-10s. (stronger effective coupling strength)

•

Manipulate easily（Φ）

•

Scalable.

What is good?

c JE ECoherent

limitc JE E

Can not be too fast !Contradict! !

What is question ?

Need to optimal !!!!

fastOperation

Experiment For Flux Qubits

Coupling

Delft group

Repot in March Meeting 2006

Another Scheme for Coupling:

J.Q. You, J.S. Tsai & F. Nori, Phys. Rev. Lett., 2002

Multiple qubits are connected in parallel with a common inductance.

σx σx -type interaction

Coupling of Two Qubits

2

1,2

22 1

{ ( ) cos }

( ) cos

c k gk Jk kk

c J

H E n n E

E N n n E

ϕ

ξ ξ η θ=

= − −

+ − − + −

∑

Iθ 1

1Iϕ2

2Iϕ

1Φ2Φ

22 1( ) cosF c JH E N n n Eξ ξ η θ= − − + −

1 2| ( , ) |FH E V n n E=

(1) (2)1 2( , ) z zV n n κσ σ=

Circuit QED : Superconducting Transmission line Resonator

5 μmDC +6 GHz in

out

Cross-section of mode:

E

10 μm

+ + --B

Transmission line “cavity”

Dressed Artificial Atoms :

01 Rω ω=

“vacuum Rabi splitting”

2g

/ Rω ω

T

2γ κ+

1

Resonance

140 dBmprobeP = − 1710 W −=

/ 2rn ω κ=

1n ≤

/ 2 12 MHz/ 2 0.6 MHz/ 2 1 MHz

2g πππ

κγ

===

Experimental Observation

Cavity Field in Superconducting Transmission Line Resonator

~1cm

22

2 2

1( , ) ( , )E x t E x tu t

∂∇ =

∂

(0, ) 0 ( , )E t E L t= =

0 L

2( ) sin n xn L Lv x π=

( , ) ( ) ( )n nE x t v x V t= ∑

Boundary Condition for Cavity Field

AJm

= −Λ

(0, ) 0 ( , )

(0, ) 0 ( , )

J t J L t

A t A L t

= =

= =

0 L

From the theory of superconductivity

BCS -> Ginzburg-Landau Theory

The London Pippard's Equation

22* ** *( )

2 * 2 *x x x xie eJ A

m m c= − Ψ ∂ Ψ−Ψ∂ Ψ − Ψ

( ) ( )A x J x∝

( )| 0boundary tE r l ∂∂∇ = − − In n I⋅ ⋅ =

( )2

22 2

1E EE rgV rc glt v t

∂ ∂∇ = + + +

∂ ∂

( ) ( )† ,2

nn n nV t a a

cω

= +

Quantization of The Cavity Field In TLR

22

2 ( ) ( ) 0n n nd V t V tdt

ω+ = 22 2n

nuLπω ⎛ ⎞= ⎜ ⎟

⎝ ⎠

General Case : with dissipationsn

Standard Quantum Limit (SQL):

1 2d x x= −

2 2 21 2

21 2

22

( ) ( ) ( )

2( )( ) ( )

( )

px xm

p xm

m

d

xm

x

τ

τ

τ τ

Δ= Δ + Δ +

Δ≥ Δ + Δ

= + Δ

Δ

∼

1x 2xp

mτ Δ

1 2p xΔ Δ ≥Free Particle

Measurement of length Limited by Uncertain Relation

Harmonic Oscillator (0)( ) (0)cos( ) sin( )px t x t tm

ω ωω

= +

22 2 2

2 2

(( ) )( ) cos ( ) sin ( )px t tm

x tm

ω ωω ω

Δ=Δ Δ + ≥

TLR : Standard Quantum Limit

( )2( ) ( ) ( )n n n n nV t V t V t f tγ ω⋅⋅ ⋅

+ + =

( ) ( ) ( )jn n njV t Q t tξ≡ + ∑

( ) ( )2 2 21 22 [ ]

( ) coth .2 2

nj nj nj njn

j B nj nj

b t b tt

k T mω ω

σω

+⎛ ⎞= ⎜ ⎟

⎝ ⎠∑

Y.D. Wang C. P. Sun, e-print , 2004

2 2 2( ) ( )SQLn n nV Q σΔ = Δ +

Electronic Coupling of Charge Qubit

to TLR

† †cos [ ( ) ( )] ( )rmsxV a t a t V a a

Lc Lω π⎛ ⎞= + ≡ +⎜ ⎟

⎝ ⎠

2 (1 2 ) 2 (1 2 )

( )(1 2 ).

z zQ C g C g

g zg

H E n E n

Ce a a N

C Lc

σ σ

ω σ+

Σ

= − − = − −

− + − −

dc

r

⊗

( )2

c gE E n n= −

( ).gg g

Cn V V

e= +

J C-Model

Y Rotate

RWA

Magnetic Coupling of Charge Qubit

to TLR

†( ),q k k kk

i a aφΦ = −∑3/ 20 1( )

2kS L c

r v kμφπ π

=⊗ †( ).q i a aφΦ = −

( )00

0 0 0

cos / cos

cos sin

( )

c q

qc c

y X a a

π π

ππ π

ξσ ησ

ΦΦ

+

Φ +Φ⎛ ⎞= ⎜ ⎟Φ⎝ ⎠

Φ⎛ ⎞ ⎛ ⎞Φ Φ= +⎜ ⎟ ⎜ ⎟Φ Φ Φ⎝ ⎠ ⎝ ⎠= + +

J C-ModelRWA

Qubit coupling via virtual photon exchange:

212 ~ /J g Δ

( )2NRNumber of Ops ~ max , / 40 1200op gγ κ⎡ ⎤Γ Δ −⎣ ⎦ ∼

Operation rate:

2~ /op gΓ Δ

(top

~10-100 ns)~1cm

Quantum Entanglement of 2 JJ Qubits

2

2

22 1( ) ( ) .2

( )z z z zq i j i j i j i j

g gH a gaω σ σ σ σ σσ σ σ+ − − ++⎡ ⎤ ⎡ ⎤≈ + + + Ω + + +⎢ ⎥ ⎢ ⎥Δ Δ⎣

+⎦ ⎣ ⎦ Δr

2 Qubit

Logic Operations

U2qt exp −i g2

Δ t a†a 12 i

z jz

1

cos g2

Δ t i sin g2

Δ t

i sin g2

Δ t cos g2

Δ t1

⊗ r,

t Δ/4g2 50 ns iSWAP

Decoherence-Free Subspace : { , }DFS = ↑↓ ↓↑

U2q exp−iΔ/3g2x1x2 t Δ/3g2