Page 201 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 189

biased by an external flux Φ through an auxiliary coil. In the flux qubit
ext
the approach to compensating the detrimental effect of Q relies on shunting
r
the junction with the superconducting wire of the loop and choosing the
condition E < E . This results in making the quantum fluctuations of q
CJ J
much larger than those of Q∆ r . The Hamiltonian, with potential shown in
Fig. 4-21(e), is given by,

º
H = q 2 + φ 2 − E J cos ª e2 ( −φ φ ext ) , (4 )
4
»
«
2 C 2 L ¬ = ¼
J
where φ is the integral of the voltage across the inductor L , which gives the
flux through the superconducting loop, and q is its conjugate variable,
which represents the charge on the junction capacitance C . Both obey the
J
commutation relation [ ] iq, φ = = . The prototypical flux qubit consists of
three Josephson junctions forming a loop and being controlled by an applied
magnetic field perpendicular to the loop to control the phase, see Fig. 4-24.

E/E J E/E J 2E2 E Φ Φ

Φ Φ

0.5
0.5
Φ Φ Φ Φ
0 0
(a) (b)
Figure 4-24. Flux qubit. (a) A qubit is created by the superposition of the two classical states
embodied by the loop phase of zero and π2 . While one or two junctions would be sufficient,
three junctions allow greater control over the behavior of the system. (b) Energy levels as a
function of controlling magnetic flux. The energy gap, E = ζ (Φ 2 0 2 / L )(N − ) 2 / 1 ,
Φ
Φ
plays the same role as E . ζ is a numerically determined parameter and
J
N = Φ ext / Φ . [207], [208].
Φ
0
In this case the two qubit states 0 and 1 are embodied in transitions in
phase from loop phases of 0 to π2 , which are associated with currents
circulating around the loop in clockwise and anti-clockwise directions. In
particular, states of zero and π2 phase difference around the loop, are
“coupled” when the flux through the loop equals half the quantum magnetic
flux in the superconductor, i.e., when Φ = Φ 2 / . Under this state of
0