What NEC Did

• Engineered a two-state system

• Initialized the system

• Formed a superposition state

• Allowed the system to evolve

• Stopped the evolution

• Read out the final state

The Box

The box is 700x50x15nm; It has about 108 conduction electrons.

e2/2C = 117V; T=30mK, so kT = 3V; for aluminum, =230V

Scope of Talk

• Tunneling

• The wave function

• Josephson’s equations and the squid

• The Cooper pair box

• Schrodinger equation solution

• Josephson quasiparticle current

Tunneling

Josephson’s Equations

dtrAtrtrtr

2

10

21 ,2

,,,

eV

dt

d

II c

2

sin

Energy in a Junction

tIVdtE

0

dtdt

dIE

t

c

2

sin 0

0

dI

E c

0

0 sin2

cos1 JEE

.

20 cJ IE Finally, where

Squid

C

B d c

a b

I

I1 I2

I

Figure 6.1: Schematic of a two-junction squid. Current enters at the top, splits into two branches, and recombines at the bottom. The junctions are at a-b and d-c. A magnetic field B threads the loop formed by the two branches. The integration path is denoted C.

Energy of a Squid

22

11

sin

sin

c

c

II

II

2

sin2

cos2 212121

cIIII

dacdbcabCnd 2

d

ccd

b

aab

dA

dA

02

01

2

2

a

d

a

d

a

dda

c

b

c

b

c

bbc

dAdJne

md

dAdJne

md

02

0

02

0

2

2

2

2

2

2

C

dAn 0

12

22

012

22

n

0

10

sincos2

cII

0cos0

1

0

max cos2

CII

0

0 cos

C

J

IE

Cooper Pair Box

Vg Box

Capacitor

Junction

Rj

Cg

Box,

Cj

Vg

Energy of the Box

i

ii VCne

C

neVC ii

i

22

1 ii

i VCU

C

neVCC

CU ggj 22

1 22

C

CneVW g

gg

C

VneC

C

neVCC

CWUE gg

ggjg 22

1 22

C

CVCneVC

CE ggg

gg

21

22

12

2

Hamiltonian

2cos gCJ QnEEH

To arrive at the quantum mechanical Hamiltonian, note that

These two exponents are just raising and lowering operators.

2/cos ii ee

1 niini eee

2222 gC

n

J QnEnnnnE

H

H can be rewritten:

Hamiltonian

• For our system, we can safely consider only the two lowest energy states,

• In that case,

20 and

2

2

22

2

gCJ

Jgc

QEE

EQE

H

Josephson Quasiparticle Current

Figure 9.1: The double junction Cooper pair box. Junction 2 has very high impedance, has very low critical current, and is voltage biased.

Capacitor Junctions

Vg VbBox

1

2

2

2Re

EeV Cbqp

Vb=650V, EC=117V, and R2=30M

qp=1.6x108sec-1, the lifetime is about 6ns.

(Averin and Aleshkin)