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

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

A double well potential

Dimensionless Hamiltonian

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>

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

“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

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

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>)

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)

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