Jan 08, 2016
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)