The rf-SQUID Quantum Bit
(the superconducting flux qubit)
C. E. Wu(吳承恩), C. C. Chi(齊正中)
Materials Science Center and Department of Physics, National Tsing Hua University, Hsinchu ,
Taiwan, R.O.C.
A rf-SQUID qubit is a superconducting loop
interrupted by a small Josephson junction
x -- external flux applied to loop
0 -- flux quanta (=2.07*10-15 wb)
 -- total flux in the loop
L -- loop inductance
C -- junction capacitance
i -- supercurrent in the loop
Ic -- junction critical current
EJ -- junction coupling energy(=Ic 0/2π )
L
EJ
C
(x, )
i
(/2)0 +  = n 0
 = x + iL
(n = integer)
BCS: i = Ic sin()
 -- phase difference across junction
The Hamiltonian of the rf-SQUID qubit
L
EJ
C
(x, )
i
Q2 1 2
H
 Li  E J cos(Δ )
2C 2
Q 2 (Φ  Φ x ) 2
Φ


 E J cos(2π )
2C
2L
Φ0
ˆ  
; Q
ˆ
i 
2 2
(Φ  Φ x ) 2
Φ


E
cos(2π
)
J
2
2C 
2L
Φ0
2 2
 V(  )
2
2C 
A double well potential
(Φ  Φx ) 2
Φ
V ( ) 
 E J cos(2π )
2L
Φ0
2
2
Φ
Φ Φ
Φ
 02 2π 2 (  x ) 2  E J cos(2π )
4π L
Φ0 Φ0
Φ0
Φ
Φ
Φ
;   ;  x  x ; U 0  02
Φ0
Φ0
4π L
 U 0 [2π 2 (   x ) 2   J cos(2π  )]
; J 
E J 2π I c L

U0
Φ0
1.5
x=0.5
J=0.8
x=0.5
J=1
V()/U0
V()/U0
1.5
1.0
1.0
0.5
0.2
0.4
0.6
0.8
0.2
 / 
1.45
0.4
0.6
0.8
 / 
1.5
1.40
1.4
x=0.5
J=1.2
1.30
x=0.51
J=1.2
1.3
V()/U0
V()/U0
1.35
1.25
1.20
1.2
1.1
1.15
1.0
0.2
0.4
0.6
 / 
0.8
0.2
0.4
0.6
 / 
0.8
Dimensionless Hamiltonian
2
H2C
1
2C
2
 V(  )
2

2
2
 V(  )
2
2

(

/

)
0
0
1 (2e) 2  2
- 2
 V(  )
2
4 2C 
2
 - 2 Ec
 U 0 [2 2 (   x ) 2   J cos(2 )]
2
4

1
H
1
2
h
 - 2 c
 [2 2 (   x ) 2   J cos(2 )]
2
U0
4

βc 
Ec
U0
; E c  superconduct ingchargingenergy
Energy levels quantization and lowest two states
wave function in a symmetric double well
potential
βJ=1.10, βc=7.55*10-4 , x=0.5
V()
U0
ε3
ε2

-1/2
0
ε1
ε0
|E>1st
|G>
parameters
βJ > ~ 1 →2-local well, small barrier height
ε2 > βJ >ε1
→only 2 states bellow the barrier
ε2-ε1 >>ε1-ε0＞kT/U0
→No thermal excitation to high levels
→definite Rabi frequency
βJ >>βc
→flux quantum number is a good
quantum number
0
1
U0 

→SQUID loop size
2
4 L L
2
0 
1
2
“0” and “1” of an rf-SQUID
qubit
( G  E )：Wave function is localized at left well.
1st
flux quanta n=0, clockwise current.
1 
1
(G  E
2
1st
⊙x=0.50
i
⊙x=0.50
i
)：Wave function is localized at right well.
Flux quanta n=1, counter-clockwise
current.
Approximated two-state system
x～0.5 0
H 2 state
 
 2



 2

 
2      
x
z
 
2
2

2 
Where Δ= E1(x = ½) - E0(x = ½)
ε～difference of two local minimum  (x - ½)
spin analog
Bz
Bx
Bz
Rabi frequency  = g(q Bx/2 m)
-pulse: a pulse of Bx apply with a duration  =  / |1> → |0>
/2-pulse:  =  /2 |1> → 1/√2(|0> + |1>)
Time evolution and one qubit rotation
Consider an arbitrary state at time t：
Ψ t  C0 (t) G  C1 (t) E
i
 C0 (0)e
E0
t

1st
i
G  C1 (0)e
E1
t

E
C 0 (0)  C1 (0)  1
2
1st
2
If, initially, the wave function of the rf-SQUID is localized in left well,
i.e. |>t=0 = |0>t=0 = 1/√2(|G>+|E>1st), so C0(0)=C1(0)=1/√2, then the
probability of finding it in right well at time t is：
P(t) |0 1 0 t |2 
1
Δ
(1  cos t)
2

Δ  E1  E 0

Rabi frequency: =

The system will oscillate between |0> and |1>
(Macroscopic Quantum Coherence Oscillation)
Try to measure the coherence time
Physical systems actively considered
for quantum computer implementation
• Liquid-state NMR
• NMR spin lattices
• Linear ion-trap
spectroscopy
• Neutral-atom optical
lattices
• Cavity QED + atoms
• Linear optics with single
photons
• Nitrogen vacancies in
diamond
• Electrons on liquid He
• Small Josephson junctions
– “charge” qubits
– “flux” qubits
• Spin spectroscopies,
impurities in semiconductors
• Coupled quantum dots
– Qubits:
spin,charge,excitons
– Exchange coupled, cavity
coupled
From IBM
Superconducting Josephson qubits
Advantage: scalable, easy manipulation
Disadvantage: short coherence time
dissipative quantum system
Flux measurement
I
V
?
Qubit
DC SQUID