Page 181 - Principles and Applications of NanoMEMS Physics

4. NANOMEMS APPLICATIONS: CIRCUITS AND SYSTEMS 169

states, namely, a low energy state denoted by 0 , and a higher energy state
denoted by 1 .
Analytically, an NMR-based QC system is described in terms of two
Hamiltonians, namely, the system Hamiltonian, which captures the energy of
single and coupled spins in the presence of a magnetic field, and the control
Hamiltonian, which captures the effects of applied RF pulses controlling the
operations with qubits.
The system Hamiltonian for single spins is given by,
=
ª− Ȧ /2 0 º
H = = I = = I = 0 , (11)
− Ȧ
− ȖB
0 0 z 0 z « »
¬ 0 = Ȧ 0 /2 ¼
where I is the z-component of the angular momentum
z
I = I x ˆ + I y ˆ + I z ˆ . In general, the three components of the angular
x y z
momentum are related to the Pauli spin matrices as follows [60],
ı = I 2 , ı = I 2 ; ı = I 2 , (12)
x x y y z z
where,
ª0 º 1 ª0 − º i ª1 0 º
ı ≡ , ı ≡ ; ı ≡ . (13)
x « » y « » z « »
¬ 1 0 ¼ ¬ i 0 ¼ ¬ 0 1 - ¼
H embodies the time evolution given by the U = e − iH 0 t/ = , which represents
0
the precession of the overall state vector (the so-called Bloch vector) with
G
respect to the axis B , defined by the static magnetic field, see Fig. 4-1 6
[194].

0 0
Z Z
Y Y
0 + + i i 1 1
0
0 + + 1 1 X X
0
2 2
2 2
1 1
Figure 4-16. Precession of a spin-1/2 about the axis of a static magnetic field. (After [194].)
Vandersypen and Chuang [194] indicate that in the most general case, the
system Hamiltonian for a molecule possessing N isolated nuclei is given by,
N N
i
H = − ¦ = ( − ı1 ) BȖ I i = − ¦ = Ȧ i I , (14)
0 i i 0 z 0 z
i =1 i =1