Page 197 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 185

the capacitor before it was connected to the inductor [206]. Q originates
r
from the inevitable work function difference and/or the presence of excess
charged impurities on the capacitor electrodes of the junction.
In the course of developing approaches to minimize the effect of Q ,
r
while retaining the nonlinearity of the resonator, three fundamental types of
Josephson-based superconducting qubits have been developed, namely, the
charge qubit, the flux qubit, and the phase qubit, see Fig. 4-21.

Φ Φ ext
ext
C C C
g g g
V V V g g g
L L I I I
b b b
(a) (b) (c)

E E
2 2
E E 1 1
E E
0 0
2 2
L~L
IÆI
<δ > large
<δ > large L~L IÆI
J0 J0 0 0
(d) (e) (f)
Figure 4- 1. Fundamental types of superconducting qubits. (a) Charge qubit. (b) Flux qubit.
2
(c) Phase qubit. (d), (e), (f) Potential (dotted line), showing qualitatively different shapes for
these three respective qubit types. In (e) the nonlinearity of the first levels comes about from
the cancellation between the superconducting loop inductance and the junction inductance
near Φ = Φ 2 / . No closed-form expressions exist for the eigenvalues and
ext 0
eigenfunctions of the potential, but its features are captured by two aspect ratios, namely,
E J / E CJ and λ = L J / L − 1. Ground-state wavefunction is also indicated (dashed-
double-dot line). The “x” represents a Josephson junction. (After [206] and [207].)
The nature of the Josephson-based qubit is a function of the relationship
between the relative magnitudes of the Josephson energy, E , which reflects
J
the strength of the coupling across the junction, and the Coulomb charging
energy, E , which reflects the energy needed to increase the charge on the
CJ
junction by a Cooper pair, e2 [208].