1. 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.

2. 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 )
(x, ) EJ L i C
(/2)0 +  = n 0 (n = integer)
 = x + iL BCS: i = Ic sin()
 -- phase difference across junction

3. The Hamiltonian of the rf-SQUID qubit
(x, ) EJ L i C

4. A double well potential

5. Dimensionless Hamiltonian

6. 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 ε3 ε2
-1/2
|E>1st ε1 ε0
|G>

7. 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
→SQUID loop size

8. “0” and “1” of an rf-SQUID qubit
：Wave function is localized at left well.
flux quanta n=0, clockwise current. i
⊙x=0.50
：Wave function is localized at right well.
Flux quanta n=1, counter-clockwise
current. i
⊙x=0.50

9. Approximated two-state system
Where Δ= E1(x = ½) - E0(x = ½)
ε～difference of two local minimum  (x - ½)

10. spin analog Rabi frequency  = g(q Bx/2 m)Bz Bz Bx
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>)

11. Time evolution and one qubit rotation
Consider an arbitrary state at time t：
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：
Rabi frequency: =
The system will oscillate between |0> and |1>
(Macroscopic Quantum Coherence Oscillation)

12. Try to measure the coherence time

13. 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

14. Superconducting Josephson qubits
Advantage: scalable, easy manipulation
Disadvantage: short coherence time
dissipative quantum system

15. Flux measurement I ? V
Qubit
DC SQUID